d i s s e r t a t i o n e s m a t h e m a t i c a...

P O L S K A A K A D E MI A N A U K, I N S TY TU T MA TE MA TY CZ N Y

D I S S E R T A T I O N E SM A T H E M A T I C A E(ROZPRAWY MATEMATYCZNE)

KOMITET REDAKCYJNY

BOGDAN BOJARSKI redaktor

WIES LAW ZELAZKO zastepca redaktora

ANDRZEJ BIA LYNICKI-BIRULA, ZBIGNIEW CIESIELSKI,

JERZY LOS, ZBIGNIEW SEMADENI

CCCXX

K. M. GARG

Relativization of some aspects of the theory of

functions of bounded variation

W A R S Z A W A 1992

Krishna M. GargDepartment of MathematicsFaculty of ScienceUniversity of AlbertaEdmonton, AlbertaCanada T6G 2G1

1991Mathematics Subject Classification: Primary 26-02, 26A42, 26A45, 28-02;Secondary 26A15, 26A24, 26A30, 26A46, 28A75.

Received October 5, 1990; revised version September 20, 1991.

Published by the Institute of Mathematics, Polish Academy of Sciences

Typeset in TEX at the Institute

Printed and bound by

P R I N T E D I N P O L A N D

c© Copyright by Instytut Matematyczny PAN, Warszawa 1992

ISBN 83-85116-48-6 ISSN 0012-3862

CONTENT S

I. Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 51. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . 83. Normalization of functions of bounded variation . . . . . . . . . . . . . . . 94. Derivatives of variation functions . . . . . . . . . . . . . . . . . . . . . 11

II. Mutual singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. Mutual singularity and lower and upper singularities . . . . . . . . . . . . 146. Additivity of mutual singularities and their characterizations . . . . . . . . . 197. Reduction theorem for mutual singularities . . . . . . . . . . . . . . . . . 218. Comparison of mutual singularities with those of

normalizations and induced signed measures . . . . . . . . . . . . . . . . 259. Mutual singularities in terms of derivatives . . . . . . . . . . . . . . . . . 29

III. Relative absolute continuities . . . . . . . . . . . . . . . . . . . . . . . . . 3310. Relative absolute continuity and lower and upper ACs . . . . . . . . . . . . 3311. Bounded variation under relative ACs . . . . . . . . . . . . . . . . . . . 3612. Relative continuity and lower and upper continuities . . . . . . . . . . . . . 3813. Reduction theorem for relative ACs and their characterizations . . . . . . . . 4114. Comparison of relative continuities and ACs with those of

normalizations and induced signed measures . . . . . . . . . . . . . . . . 4515. Relative ACs in terms of derivatives . . . . . . . . . . . . . . . . . . . . 48

IV. Normalized relative derivative . . . . . . . . . . . . . . . . . . . . . . . . 5016. Existence of normalized relative derivative . . . . . . . . . . . . . . . . . 5017. Relative AC in terms of LS-integral and

a Radon–Nikodym theorem for such integrals . . . . . . . . . . . . . . . . 5518. The fundamental theorem of calculus for LS-integral . . . . . . . . . . . . . 5719. Relative LAC in terms of LS-integral . . . . . . . . . . . . . . . . . . . 6520. Mutual singularities in terms of normalized relative derivative . . . . . . . . . 6821. Reconstruction of relative primitive . . . . . . . . . . . . . . . . . . . . 71

V. Relativization of other classical theorems . . . . . . . . . . . . . . . . . . . . 7322. Lebesgue’s montonicity theorem . . . . . . . . . . . . . . . . . . . . . . 7323. Lebesgue’s decomposition theorem . . . . . . . . . . . . . . . . . . . . . 7424. Lusin’s property (N) and the Banach–Zarecki theorem . . . . . . . . . . . . 7825. Integration by parts for LS-integral . . . . . . . . . . . . . . . . . . . . 8226. Relative Lebesgue points . . . . . . . . . . . . . . . . . . . . . . . . . 8427. Arc length of rectifiable curves under relative AC . . . . . . . . . . . . . . 8628. A general formula for arc length and a problem of Denjoy . . . . . . . . . . . 91

VI. Convergence in B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9529. Stability of variations and components under norm convergence . . . . . . . . 9530. Norm closed sets and subspaces of B . . . . . . . . . . . . . . . . . . . . 9931. Strong convergence and term-by-term differentiation . . . . . . . . . . . . . 10232. Stability of arc length under strong convergence . . . . . . . . . . . . . . . 10733. Approximation in some subspaces of B by elementary functions . . . . . . . . 10934. Approximation by relative polynomials . . . . . . . . . . . . . . . . . . . 114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Index of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Index of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Abstract

We present here relativized versions of some aspects of the theory of functions of boundedvariation, viz. relative to a function of bounded variation, without going into relative boundedvariation. A few results have been known in this direction for some time when the functionsinvolved are continuous, but due to the complications that arise when the functions are discon-tinuous, no systematic attempt seems to have been made in this direction in the past.

Let B denote the linear space of all real-valued functions of bounded variation defined on agiven compact interval I = [a, b]. Given f, g ∈ B, we present here a notion of mutual singularityof f and g, and a notion of absolute continuity (or AC) of f relative to g, which are similarto these notions in the case of signed measures. Further, we present decompositions of thesetwo properties into mutual lower and upper singularities and relative lower and upper absolutecontinuities.

Several characterizations of the above six properties are obtained here in terms of variationsof f and g. Further, additivity theorems dealing with the additivity of these properties are ob-tained, and reduction theorems are obtained which reduce these properties to the discontinuous,AC and continuous singular components of f and g. Also, characterizations of these propertiesare obtained in terms of derivatives of f and g. These characterizations are based on a refinedversion of a theorem of de La Vallee Poussin which deals with derivatives of the three variationsof f ∈ B in terms of the derivative of f .

Next, with the help of the above new notions and results we present relativized versions ofsome other aspects of the theory of functions of bounded variation. A new notion of normalizationf∗ of f ∈B and a related normalized version of relative derivative also play significant roles inthis development.

Firstly, characterizations of all the above six properties are obtained here in terms of nor-malized relative derivative and the Lebesgue–Stieltjes integral (or LS-integral). Following aresome other highlights of the developments:

A Radon–Nikodym theorem is obtained for LS-integral where the normalized relative deriva-tive turns out to be the Radon–Nikodym derivative in general. Also, two versions of the fun-damental theorem of calculus are obtained for LS-integral, and a theorem dealing with thereconstruction of a function from its relative derivative is obtained.

Further, relativized versions of (i) a monotonicity theorem of Lebesgue, (ii) the Lebesguedecomposition theorem, (iii) Lusin’s property (N) and the Banach–Zarecki theorem on AC,(iv) the results on Lebesgue points, and (v) a theorem of Tonelli on arc length are obtained.Also, characterizations of mutual singularity and relative AC in terms of arc length, a generalformula for arc length based on relative Lebesgue decomposition, and a solution of an old problemof Denjoy on arc length in higher dimensions are obtained.

Next, we consider convergence in B under variation norm relative to which B is known to bea Banach space. Some theorems dealing with the stability of variations and components undernorm convergence are obtained here for sequences and series of functions in B.

Further, a relativized version of Fubini’s theorem on term-by-term differentiation is obtained,and an extension of Fubini’s (relativized) theorem is obtained which holds in general under aform of convergence which is stronger than norm convergence. Finally, some approximationtheorems are obtained which deal with approximation in some closed subspaces of B by certainelementary functions in those subspaces. E.g. the functions in B which are AC relative to someu ∈ B can be approximated in the variation norm by piecewise linear functions relative to u,and also in a sense by polynomials in u.

I. Introduction and preliminaries

1. Introduction. In this introductory section we discuss the present work insome detail along with its organization. Let B denote the space of all real-valuedfunctions of bounded variation on a given compact interval I = [a, b]. Also, foreach f ∈ B, let µf denote the LS-measure (or signed measure) induced by f .

In the next three sections of this chapter we present some notations, no-menclature and preliminary results that are used throughout the work (see §2 inparticular). In §3, we present a new form of normalization f∗ of any regulatedfunction f , which seems to be more useful in many situations. We include here, in§4, a refined version of a theorem of de La Vallee Poussin [5; 6] which deals withderivatives of all the three variation functions of f ∈ B in terms of the derivativeof f .

Chapter II deals with three notions of mutual singularity between two func-tions f, g ∈ B. We first present, in §5, a notion of mutual singularity of f and gwhich is similar to the mutual singularity of two signed measures. Then we pre-sent a decomposition of this property into mutual lower and upper singularities(or LS and US); applications of these mutual singularities appear in subsequentchapters. Several characterizations of the three mutual singularities in terms ofthe variations of f and g are obtained in §§5 and 6.

Also, in §6, we obtain an “additivity theorem” dealing with the additivity ofthe three forms of mutual singularity. In §7, a “reduction theorem” is obtainedaccording to which f, g ∈ B are mutually singular in any of the three senses if andonly if the same relation holds separately between the discontinuous, absolutelycontinuous and continuous singular components of f and g.

Next, in §8, we compare the mutual singularities of f and g with those oftheir normalizations, and of the signed measures induced by them. Finally, in §9,characterizations of the three mutual singularities of f, g ∈ B are obtained interms of derivatives of f and g.

Chapter III deals similarly with three notions of relative absolute continuity(or AC). We first present, in §10, a notion of AC of a function f : I → R

relative to another function g ∈ B which is similar to the relative AC of signedmeasures. Then we present a decomposition of this property into relative lowerand upperACs (or LAC and UAC). Also, in this section, we obtain an “additivity

6 K. M. Garg

theorem” dealing with these properties, and when f ∈ B, their characterizationsare obtained in terms of the variations of f and g.

Next, in §11, the question whether f is of bounded variation if it is AC, LACor UAC relative to g ∈ B is investigated. In §12, we present some notions ofcontinuity and lower and upper continuities (or LC and UC) of f relative to gwhich are found to hold when f is AC, LAC or UAC respectively relative to g.Further, when f ∈ B, we obtain in §13 a “reduction theorem” for relative ACssimilar to the one on mutual singularities, and a characterization of relative ACsin terms of mutual singularity with other functions.

Next, in §14, we compare the relative continuities and ACs of f relative to gwith those of f∗ relative to g∗, and of µf relative to µg. Finally, in §15, charac-terizations of various ACs of f relative to g are obtained in terms of derivativesof f and g.

It should be pointed out here that on choosing g in the above mentioneddefinitions to be the identity function, the following useful decompositions of threeordinary properties are obtained: (i) a decomposition of ordinary singularity intolower and upper singularities which have been used earlier in differentiation theory[13], (ii) a decomposition of ordinary AC into LAC and UAC which are founduseful in connection with the theory of nonabsolute integration [32], and (iii) adecomposition of ordinary continuity into LC and UC, which are different fromusual lower and upper semicontinuities, and are found useful in differentiationtheory [13].

In the next three chapters we utilize the above new notions and results toobtain relativized versions of some other aspects of the theory of functions ofbounded variation. The results of the first three chapters thus play a basic rolein the developments of Chapters IV, V and VI.

Chapter IV is devoted to a new normalized version of relative derivative andLebesgue–Stieltjes integral (or LS-integral). Characterizations of all the abovesix properties are obtained here in terms of normalized relative derivative andLS-integral.

We first present, in §16, the definition of normalized relative derivative whichis found more useful in many situations. In the case of continuous functions thisderivative becomes the same as the ordinary relative derivative, but the latter isusually not available at the points of discontinuity. Given f, g ∈ B, we establishin this section the existence of normalized derivative of f relative to g, denotedby D∗

gf , µg-almost everywhere, and its summability relative to µg.Next, in §17, we obtain a Radon–Nikodym theorem for LS-integral. This the-

orem provides a characterization of relative AC of normalized functions, and D∗gf

turns out to be in general the Radon–Nikodym derivative of µf relative to µg.Further, in §18, we obtain two versions of the fundamental theorem of calculus forLS-integral. Parts of these two theorems were obtained earlier by Lebesgue [26]in particular cases (see Remark 18.12).

In §19, a characterization of relative LAC is obtained in terms of LS-integral

Functions of bounded variation 7

of the normalized relative derivative. Further, in §20, characterizations of the threeforms of mutual singularity are obtained in terms of normalized relative derivative.Finally, in §21, we deal with the problem of reconstruction of a function f : I → R

from its derivative relative to g ∈ B when the latter exists and is finite at all buta countable set of points.

Next, in Chapter V, we deal with relativization of some other aspects of thetheory of functions of bounded variation. To be specific, a relativized version ofa monotonicity theorem due to Lebesgue is obtained in §22. In §23, a relativizedversion of the Lebesgue decomposition theorem is obtained, including some resultson the properties of f, g ∈ B that are reflected in the AC and singular componentsof f relative to g.

Further, in §24, a relativized version of Lusin’s property (N) [27] is presented, acharacterization of this property similar to Rademacher’s theorem [30] is obtained,and a relativized version of the well known Banach–Zarecki theorem [28, p. 250]is obtained which provides a similar characterization of relative AC.

In §25, two formulae for integration by parts for LS-integral are obtained oneof which is known. In §26, a relative notion of Lebesgue points is presented interms of which relativized versions of some of the known theorems on Lebesguepoints are obtained.

Next, in §27, relativized versions of two known theorems on arc length, one ofwhich is due to L. Tonelli [35; 36], are obtained. One of these relativized versionsprovides a characterization of relative AC in terms of arc length. Finally, in §28,a general formula for arc length is obtained which holds without any hypothesisand is based on the relative Lebesgue decomposition. Also, a characterization ofmutual singularity in terms of arc length is obtained here, along with a solutionof an old problem of Denjoy [7] on arc length in higher dimensions.

The final Chapter VI is devoted to convergence in B under variation normrelative to which B is known to be a Banach space. First, in §29, we obtain sometheorems dealing with the stability of variations and components under normconvergence for both sequences and series of functions in B; components relativeto other functions in B are also considered here. In §30, we obtain some normclosed subsets and subspaces of B.

Next, in §31, we first obtain a relativized version of Fubini’s theorem on term-by-term differentiation [9], which turns out to hold for any norm convergent seriesin B whose elements are pairwise mutually LS. Then another extension of thistheorem is obtained which holds in general under a form of convergence in Bwhich is stronger than norm convergence. Every norm convergent sequence in Bis found on the other hand to admit strongly convergent subsequences, which leadsto a result on term-by-term differentiation of subsequences of norm convergentsequences. Further, in §32, a theorem is obtained dealing with the stability of arclength under strong convergence.

Next, in §33, we obtain some theorems dealing with the denseness of certainclasses of elementary functions in some closed subspaces of B. Thus, given u ∈ B,

8 K. M. Garg

the functions that are AC relative to u can be approximated by piecewise linearfunctions relative to u, the functions that are singular relative to u can in turn beapproximated by generalized u-step functions, and when u is normalized, everynormalized function in B can be approximated by a generalized linear functionrelative to u (see §33 for definitions). Finally, in §34, we obtain similar theoremson the approximation of AC functions relative to u by polynomials in u, or itsappropriate components. Once again, on choosing u to be the identity function,similar results are obtained on ordinary AC and singular functions.

The origins of the present work go back to the dissertation [10] where some ofthe results (viz. Theorems 4.2, 9.3 and the last part of Theorem 31.2) appearedin somewhat different forms. As the work progressed its results were presented atvarious international mathematical meetings. Some of the results of Chapters IIand VI have been quoted and utilized earlier in [14] in the construction of certainclasses of AC and continuous singular functions.

2. Notations and preliminaries. In this section we present some notationsand preliminary results which are used throughout the work.

For any set E ⊂ R, we will follow Saks [34] to denote the Lebesgue outermeasure of E by |E|. The Lebesgue outer measure and the Lebesgue measure onR will in turn be denoted by m∗ and m respectively.

Further, we will use I to denote an arbitrary but fixed compact subinterval[a, b] of R, and I0 to denote the open interval (a, b). If E is on the other hand anysubset of I, we will use E0 and E to denote the interior and closure of E relativeto I. The reason for defining I0 differently will be clear from the context.

Also, we will use B to denote the σ-algebra of all Borel sets in I, and ifE = A ∪ B, A,B ∈ B and A ∩ B = ∅, then (A,B) will be called a Borel decom-

position of E.

Next, we will use B to denote the linear space of all real-valued functions ofbounded variation on I, and B+ to denote the set of nondecreasing functionsin B.

Given f ∈ B, if a ≤ x ≤ y ≤ b, we will use V +x,yf , V

−x,yf and Vx,yf to denote

the positive, negative and total variations respectively of f on the closed interval[x, y]. Further, we use V +f , V −f and Vf to denote these variations respectivelyon I, and f+, f− and f to denote the positive, negative and total variationfunctions respectively of f , viz.

f+(x) = V +a,xf, f−(x) = V −

a,xf, f(x) = Va,xf, x ∈ I .

Given f ∈ B, we use further

(i) fd, fc, fa, fs and fcs to denote the discontinuous, continuous, absolutelycontinuous, singular and continuous singular components (or parts) respectivelyof f ,

(ii) ωf (x) to denote the oscillation of f at any point x ∈ I,


(iii) Cf and ∆f to denote the sets of points in I where f is continuous orderivable (in the wider sense) respectively, and

(iv)∆∞f ,∆+∞

f and∆−∞f to denote the sets of points in∆f where the derivative

of f is infinite, +∞ or −∞ respectively.

Also, we will use τ to denote the identity function on I, and, for any set E ⊂ I,χE will denote the characteristic function of E on I.

Now we state a few preliminary results which will be used frequently. Thefollowing lemma is quite obvious.

2.1. Lemma. Given f ∈ B, each of the functions f and fd is continuous at a

point x ∈ I from any given side iff f is so.

2.2. Theorem. If f ∈ B, then

(a) ∆f is an Fσδ-set in I, and the derivative f ′ is of Baire class 1 relative

to ∆f ,

(b) |I ∼ ∆f | = |∆∞f | = 0, and

(c) if f is singular , then |f(I ∼ ∆∞f )| = 0.

The part (b) of this theorem is of course the classical differentiability theoremof Lebesgue (see e.g. [28], p. 219). For the part (a) see [13], pp. 315, 322, and for(c) see [11], pp. 1443, 1444.

Next, when f ∈ B+, we will use µf to denote the metric (or Caratheodory)outer measure induced by f on I (see [34], pp. 64, 99), viz. if E ⊂ I, then

µf (E) = inf∑

n

{f(bn)− f(an)} ,

where the inf is taken over all sequences of closed intervals {[an, bn] : n = 1, 2, . . .}in I for which E ⊂

⋃

n[an, bn]0. The restriction of µf to B is then a positive Borel

measure. Further, for an arbitrary f ∈ B, µf is defined to be the finite setfunction µf+ − µf− on the power set of I. The restriction of µf to the σ-algebraof µf -measurable sets is then a finite signed measure.

The above relativization of the Lebesgue measure is due to Radon [31], and µf

is called the signed measure, or Radon or LS-measure, induced by f (see e.g. [17],p. 67). For any signed measure µ we will use in turn µ+, µ− and µ to denote theupper, lower and absolute variations respectively of µ (see [17], pp. 122, 123; or[34], p. 10, for definitions).

In this connection the following known theorem will be used frequently (seeSaks [34], p. 100).

2.3. Theorem. Let f ∈ B and E ⊂ I. Then |f(E)| ≤ µf (E). Moreover , if fis nondecreasing and E ⊂ Cf , then |f(E)| = µf (E).

3. Normalization of functions of bounded variation. In this section wepresent a new form of normalization of functions of bounded variation, or, more

10 K. M. Garg

generally, of regulated functions, which is used throughout the present work. Also,we include here some elementary results on this operation.

A function f : I → R is called regulated if it has finite unilateral limits f(x−0)and f(x+ 0) at every point x ∈ I which is a left or right limit point respectivelyof I (see e.g. [3]).

Given f ∈ B, it is usually either the left limit f(x − 0), or the right limitf(x + 0), which is used for normalization of f (see e.g. [19] and [33]). We willadopt here the golden mean of these two normalizations which will be found moreuseful in the present work.

Given any regulated function f : I → R, we thus define the normalization of fto be the function f∗ defined as follows:

f∗(x) =

{

f(x) if x = a or b,12{f(x+ 0) + f(x− 0)} if a < x < b.

Further, f will be called normalized if f = f∗. (Such a function has also beencalled “regular” in the literature; see e.g. [34], p. 97.)

It is clear that f∗ is regulated, and in case f ∈ B, then f∗ ∈ B.We include here a few simple facts about normalization in the form of lemmas,

which are used frequently in the following chapters.

3.1. Lemma. Suppose f, g : I → R are regulated and α, β ∈ R. Then

(αf + βg)∗ = αf∗ + βg∗ .

Consequently , if f and g are normalized , then so is αf + βg.

P r o o f. Define h=αf +βg. Then the identity h∗(x) = αf∗(x)+βg∗(x) holdsclearly when x = a or b. When x ∈ I0, this identity follows on the other handfrom the following two relations:

h(x± 0) = αf(x± 0) + βg(x± 0) .

The last part is of course an obvious consequence of the first.

A regulated function f : I → R is said to have a removable discontinuity atx ∈ I0 if the two limits f(x+0) and f(x− 0) at x are equal but they are not thesame as f(x). We will use Rf to denote the set of all points in I0 where f has aremovable discontinuity.

3.2. Lemma. Suppose f : I → R is regulated. Then

(a) f∗(x+0) = f(x+0) for a ≤ x < b and f∗(x−0) = f(x−0) for a < x ≤ b,

(b) f∗ is normalized , and

(c) Cf∗ = Cf ∪Rf .

P r o o f. Since f∗ also is regulated, the part (a) follows from the fact that f∗

agrees with f on the set Cf which is dense in I.

The part (b) follows directly from (a) and the definition of f∗.


According to (a), f∗ is continuous at a or b iff f is so. In the case of aninterior point x of I, it follows again from (a) that f∗ is continuous at x ifff(x+ 0) = f(x− 0). The identity in (c) is now obvious.

3.3. Lemma. Let f ∈ B. Then

(a) (f∗)c = fc, and(b) µf∗ = µf .

P r o o f. To prove (a), we first observe that fc = f − fd, and similarly, (f∗)c =f∗ − (f∗)d. Hence, it is enough to prove the following identity:

(1) (f∗)d(x)− fd(x) = f∗(x)− f(x) for x ∈ I .

This identity holds trivially when x = a, for fd(a) = 0 = (f∗)d(a) (see [28],pp. 219, 220 for the definition of fd). When x > a, we have, on the other hand,

fd(x) =∑

t∈[a,x)∼Cf

{f(t+ 0)− f(t− 0)} + f(x)− f(x− 0), and

(f∗)d(x) =∑

t∈[a,x)∼Cf∗

{f∗(t+ 0)− f∗(t− 0)}+ f∗(x)− f∗(x− 0) .

Now since f(t + 0) = f(t − 0) whenever t ∈ Rf , the identity (1) follows easilyfrom these two equations with the help of the parts (a) and (c) of Lemma 3.2.

Next, to prove (b), it is enough to prove the identity µf∗(U) = µf (U) forevery subinterval U of I that is open relative to I. First, suppose U = (x, y)where a ≤ x < y ≤ b. Then it follows from part (a) of Lemma 3.2 that

µf∗(U) = f∗(y − 0)− f∗(x+ 0) = f(y − 0)− f(x+ 0) = µf (U) .

In case U = [a, y), then we have, similarly,

µf∗(U) = f∗(y − 0)− f∗(a) = f(y − 0)− f(a) = µf (U) ,

and a similar argument holds in the case when U = (x, b].

4. Derivatives of variation functions. Given a function f ∈ B, de LaVallee Poussin proved (see [5], [6] or [34], p. 127) that there exists a set E ⊂ Cf

such that |Cf ∼ E| = µf (Cf ∼ E) = 0 and, at each x ∈ E, f and f are derivable

and (f)′(x) = |f ′(x)|.With the help of another decomposition theorem of de La Vallee Poussin

([34], p. 127), we obtain here a refinement of the above result which relates thederivatives of all the three variations of f with that of f .

We need here the following lemma.

4.1. Lemma. Given g, h ∈ B, if f = g + h, then µf = µg + µh.

P r o o f. First, suppose g and h are nondecreasing. Given E ⊂ I, let ε > 0.Let δf , δg and δh denote the interval functions corresponding to f , g and hrespectively, e.g., if J = [x, y] ⊂ I, then δf (J) = f(y) − f(x). Then there exist,

12 K. M. Garg

by definition, three sequences of closed intervals {Ik,n : n = 1, 2, . . .}, k = 1, 2, 3,in I such that E ⊂

⋃

n I0k,n for each k,

(1)∑

n

δg(I1,n) < µg(E) +ε

2,

∑

n

δh(I2,n) < µh(E) +ε

2

and

(2)∑

n

δf (I3,n) < µf (E) + ε .

Now, since

E ⊂(

⋃

n

I1,n

)0

∩(

⋃

n

I2,n

)0

⊂(

⋃

n

I1,n

)

∩(

⋃

n

I2,n

)

,

there exists an open set G ⊂ I such that E ⊂ G ⊂ (⋃

n I1,n)∩(⋃

n I2,n). Let {I4,n}denote the sequence of closed intervals obtained by replacing the components ofG by their closures. Then

(3) E ⊂⋃

n

I04,n and⋃

n

I4,n ⊂⋃

n

Ik,n for k = 1, 2 .

Further, since f = g + h, δf = δg + δh. Now since δg and δh are nonnegative andadditive, it follows easily form (1) and (3) that

µf (E) ≤∑

n

δf (I4,n) =∑

n

δg(I4,n) +∑

n

δh(I4,n)

≤∑

n

δg(I1,n) +∑

n

δh(I2,n) < µg(E) + µh(E) + ε .

Moreover, since E ⊂⋃

n I03,n, it follows similarly from (2) that

µg(E) + µh(E) ≤∑

n

δg(I3,n) +∑

n

δh(I3,n)

=∑

n

δf (I3,n) < µf (E) + ε .

Hence, |µf (E)−µg(E)−µh(E)| < ε, and since this holds for an arbitrary ε, thisproves the required identity when g and h are nondecreasing.

Next, to deal with the general case, let ϕ = g+ + h+ − f+ = g− + h− − f−.Then it is easy to see that ϕ is nondecreasing. Hence it follows from the abovethat µg+ + µh+ = µf+ + µϕ and µg− + µh− = µf− + µϕ. Consequently,

µg + µh = µg+ + µh+ − µg− − µh− = µf+ − µf− = µf .

4.2. Theorem. If f ∈ B, then there is a decomposition of I into four Borel

sets A, B, C and D such that

(a) for each x ∈ A, f , f+, f− and f have finite derivatives at x, (f+)′(x) =max{f ′(x), 0}, (f−)′(x) = −min{f ′(x), 0} and (f)′(x) = |f ′(x)|;

(b) for each x ∈ B, (f)′(x) = (f+)′(x) = f ′(x) = ∞;


(c) for each x ∈ C, (f)′(x) = (f−)′(x) = −f ′(x) = ∞; and(d) |B ∪ C ∪D| = |f(D)| = |f(D)| = |f+(C ∪D)| = |f−(B ∪D)| = 0.

P r o o f. There exists, by the above-mentioned theorem of de La Vallee Poussin,a set E ⊂ Cf such that

(4) |E| = µf (E) = 0 ,

and for each x ∈ F ≡ Cf ∼ E, f ′(x) and (f)′(x) exist and

(5) (f)′(x) = |f ′(x)| .

We can choose E to be a Gδ-set in I, for since m∗ and µf are two metric outermeasures, there exist two Gδ-sets E1 and E2 in I, each including E, such that|E1| = |E| = 0 and µf(E2) = µf (E) = 0, and then E1 ∩ E2 ∩ Cf is the requiredGδ-set.

Let A, B and C denote the sets of points x of F where f ′(x) is finite, +∞ or−∞ respectively, and set

(6) D = E ∪ (I ∼ Cf ) .

Then A, B, C and D decompose I into four sets. Since Cf is a Gδ-set, D,F ∈ B.Hence it follows from Theorem 2.2(a) that A,B,C ∈ B. Further, given x ∈ I,since f(x) = f+(x) + f−(x) and f(x)− f(a) = f+(x)− f−(x), we have

(7) f+(x) = 12{f(x) + f(x)− f(a)}, f−(x) = 1

2{f(x)− f(x) + f(a)} .

The parts (a), (b) and (c) follow now from (5) and (7).Next, since the nondecreasing functions f , f+ and f− are continuous at the

points of E, it follows from (4), Theorem 2.3 and the above lemma that |f(E)| ≤µf (E) = 0, |f(E)| = µf (E) = 0 and

|f+(E)| + |f−(E)| = µf+(E) + µf−(E) = µf(E) = 0 .

Hence it follows from (4) and (6), since I ∼ Cf is countable, that

(8) |D| = |f(D)| = |f(D)| = |f+(D)| = |f−(D)| = 0 .

Further, |B ∪ C| = 0 by Theorem 2.2(b), and hence |B ∪ C ∪D| = 0.Now since C ⊂ ∆−∞

f , µf (C) ≤ −|C| = 0 (see Lemma 9.4 of [34], p. 126).Hence it follows from the decomposition theorem of de La Vallee Poussin ([34],p. 127) that µf (C) = |µf (C)| = −µf (C). Consequently, by the above lemma,

µf+(C) + µf−(C) = − µf+(C) + µf−(C) .

Thus µf+(C) = 0, for µf−(C) is finite. Hence, by Theorem 2.3, |f+(C)| = 0, andconsequently, by (8),

|f+(C ∪D)| ≤ |f+(C)|+ |f+(D)| = 0 .

The remaining relation |f−(B ∪D)| = 0 is proved similarly.

4.3. R ema r k. Regarding the sets A, B and C in Theorem 4.2, it should beobserved here that although at each point of A one of the functions f− and f+

14 K. M. Garg

has a zero derivative, this does not hold in general at the points of B or C. Infact, nothing can be said in general on the derivatives of f− and f+ at the pointsof B and C respectively, they may or may not exist, and if they exist, they mayeven be infinite.

Let I = [0, 1] and g be any continuous nondecreasing singular function onI such that g(0) = 0 < g(1), e.g. Cantor’s step function. Since |∆∞

g | = 0 (seeTheorem 2.2), there exists a Gδ-set E ⊃ ∆∞

g such that |E| = 0. Then there existsa nondecreasing AC function h on I such that h(0) = 0 and h is not derivable atany point of E (see Zahorski [38], pp. 175, 176). Now define f = g − h. Clearly,f ∈ B. Further, as we see subsequently in Lemma 7.5, g and h are mutuallysingular as defined in §5. Hence it follows from Theorem 5.5 that g = f+ andh= f−. Now let A, B, C and D denote a decomposition of I into four sets forwhich Theorem 4.2 holds for f . Then B⊂E and |g(C∪D)| = 0. Further, since g issingular, |g(A)| = 0 by Theorem 2.2(c). Consequently, |g(B)| = |g(I)| = g(1) > 0.Thus B is nonempty and f− = h is not derivable at any point of B.

Next, since E is a Gδ-set of measure zero, there also exists a nondecreasingAC function h1 on I such that h1(0) = 0 and E ⊂ ∆∞

h1(see e.g. [28], p. 214).

Define f1 = g− h1. Then g=f+1 and h1=f

−1 as before. Also, it follows as before

that the set B corresponding to f1 is nonempty and f−1 has an infinite derivative

at each point of B.

II. Mutual singularities

5.Mutual singularity and lower and upper singularities. In this sectionwe define mutual singularity and lower and upper singularities of two functionsf, g ∈ B, and then present some elementary results on them including theircharacterizations in terms of additivity of various variations of f and g.

Given any positive integer n, we will use from now on Sn to denote the follo-wing index set:

Sn = {1, . . . , n} .

Given f, g ∈ B, we will call f and g mutually singular if for every ε > 0there exists a partition a = x0 < x1 < . . . < xn = b of I for which there is adecomposition (S−, S+) of the index set Sn such that

(1)∑

i∈S+

{f(xi)− f(xi−1)}+∑

i∈S−

{g(xi)− g(xi−1)} < ε .

Further, f and g will be called mutually lower singular (or LS) if for everyε > 0 there exists a partition a = x0 < x1 < . . . < xn = b of I for which there isa decomposition (S−, S+) of the index set

(2) S = {i ∈ Sn : (f(xi)− f(xi−1))(g(xi)− g(xi−1)) < 0}


such that

(3)

∑

i∈Sn∼S−

|f(xi)− f(xi−1)| > V f − ε ,

∑

i∈Sn∼S+

|g(xi)− g(xi−1)| > V g − ε .

The mutual upper singularity (or US) of f and g is defined similarly by reversingthe inequality in the definition (2) of the index set S.

When f and g are mutually singular, LS or US, f will also be said to besingular, LS or US respectively relative to g, and we will write f ⊥ g, f ⊥− g orf ⊥− g respectively.

It is clear from the above definitions that f ⊥ g iff f ⊥ g, or, equivalently, ifff ⊥− g. Further, f ⊥− g iff f ⊥− (−g), and so it is enough to consider LS.

When f and g are simultaneously nondecreasing, or nonincreasing, they areautomatically mutually LS, and f ⊥ g iff f ⊥− g. Similarly, when f is nondecre-asing and g is nonincreasing, they are automatically mutually US, and f ⊥ g ifff ⊥− g.

The following result follows directly from the above definitions.

5.1. Theorem. Let f, g ∈ B and α, β ∈ R.

(a) If f ⊥ g, then αf ⊥ βg.

(b) If f ⊥− g and αβ ≥ 0, then αf ⊥− βg.

To obtain the desired characterizations of mutual singularities, we need thefollowing

5.2. Lemma. Let f, g ∈ B and ε > 0. If a = x0 < x1 < . . . < xn = b is a

partition of I such that

(4)

n∑

i=1

|f(xi) + g(xi)− f(xi−1)− g(xi−1)| > V f + V g − ε ,

then there exists a decomposition (S−, S+) of the index set S defined in (2) for

which (3) holds.

P r o o f. Suppose (4) holds for the given partition of I and let S be the indexset defined in (2). Set

S− = {i ∈ S : |f(xi)− f(xi−1)| < |g(xi)− g(xi−1)|}

and S+ = S ∼ S−. It is then clear that

n∑

i=1

|f(xi)− f(xi−1) + g(xi)− g(xi−1)|

≤∑

i∈Sn∼S−

|f(xi)− f(xi−1)|+∑

i∈Sn∼S+

|g(xi)− g(xi−1)| .

16 K. M. Garg

Hence, by (4),∑

i∈Sn∼S−

|f(xi)− f(xi−1)|+∑

i∈Sn∼S+

|g(xi)− g(xi−1)| > V f + V g − ε .

The two inequalities of (3) follow clearly from this inequality.

The following theorem characterizes LS in terms of additivity of various va-riations.

5.3. Theorem. Let f, g ∈ B. Then f ⊥− g iff any of the following equivalent

conditions holds:

(a) V (f + g) = V f + V g, (a′) f + g = f + g ,

(b) V +(f + g) = V +f + V +g, (b′) (f + g)+ = f+ + g+,

(c) V −(f + g) = V −f + V −g, (c′) (f + g)− = f− + g− .

P r o o f. We will first prove the equivalence of f ⊥− g with (a).Suppose f ⊥− g. Then given ε > 0, there exists a partition a = x0 < x1 <

. . . < xn = b of I for which there is a decomposition (S−, S+) of the index set Sdefined in (2) such that (3) holds. It is clear from (3) that

∑

i∈S−

|f(xi)− f(xi−1)| < ε and∑

i∈S+

|g(xi)− g(xi−1)| < ε .

Hence, by (2) and (3),n∑

i=1

|f(xi) + g(xi)− f(xi−1)− g(xi−1)|

≥∑

i∈Sn∼S

{|f(xi)− f(xi−1)|+ |g(xi)− g(xi−1)|}

+∑

i∈S−

{|g(xi)− g(xi−1)| − |f(xi)− f(xi−1)|}

+∑

i∈S+

{|f(xi)− f(xi−1)| − |g(xi)− g(xi−1)|}

=∑

i∈Sn∼S−

|f(xi)− f(xi−1)|+∑

i∈Sn∼S+

|g(xi)− g(xi−1)|

−∑

i∈S−

|f(xi)− f(xi−1)| −∑

i∈S+

|g(xi)− g(xi−1)|

> V f − ε+ V g − ε− ε− ε = V f + V g − 4ε .

Consequently, V (f + g) ≥ V f + V g. Since the reverse inequality is always valid,this proves (a). Conversely, when (a) holds, it follows clearly from Lemma 5.2that f ⊥− g. This proves the equivalence of (a).


Now since (a′)⇒(a), it is enough to prove the implications (a)⇒(b)⇒(c)⇒(c′)⇒(b′)⇒(a′).

(a)⇒(b). It follows from the definitions of positive and negative variationsthat V +(f + g) ≤ V +f + V +g and V −(f + g) ≤ V −f + V −g. Hence if (a) holds,then

V (f + g) = V +(f + g) + V −(f + g) ≤ V +f + V +g + V −f + V −g

= V f + V g = V (f + g) ,

which implies (b).(b)⇒(c). This is obvious since

V +(f + g) − V −(f + g) = f(b) + g(b) − f(a)− g(a)

= V +f − V −f + V +g − V −g .

(c)⇒(c′). Suppose (c) holds and let x ∈ I. Then since

V −(f + g) = (f + g)−(x) + V −x,b(f + g)

≤ f−(x) + g−(x) + V −x,bf + V −

x,bg

= V −f + V −g = V −(f + g) ,

it is clear that (f + g)−(x) = f−(x) + g−(x). Consequently, (c′) holds.

(c′)⇒(b′). This is obvious since, for each x ∈ I,

(f + g)+(x)− (f + g)−(x) = f(x) + g(x)− f(a)− g(a)

= f+(x)− f−(x) + g+(x)− g−(x) .

(b′)⇒(a′). Suppose (b′) holds. Then, given x ∈ I, it follows from the first of

the relations (7) of §4 that

(f + g)(x) + f(x) + g(x)− f(a)− g(a) = 2(f + g)+(x)

= 2f+(x) + 2g+(x) = f(x) + f(x)− f(a) + g(x) + g(x) − g(a) .

Consequently, (f + g)(x) = f(x) + g(x), so that (a′) holds.

On applying the above theorem to f and −g we obtain the following charac-terization of US.

5.4. Corollary. Let f, g ∈ B. Then f ⊥− g iff any of the following equivalent

conditions holds:

(a) V (f − g) = V f + V g, (a′) f − g = f + g,

(b) V +(f − g) = V +f + V −g, (b′) (f − g)+ = f+ + g−,

(c) V −(f − g) = V −f + V +g, (c′) (f − g)− = f− + g+ .

Finally, we obtain some characterizations of mutual singularity in the followingtheorem.

5.5. Theorem. Let f, g ∈ B. Then f ⊥ g iff any of the following equivalent

conditions holds:

18 K. M. Garg

(a) f ⊥− g and f ⊥− g,(b) f = (f − g)+,

(c) g = (f − g)−.

P r o o f. We will first prove that f ⊥ g iff (a) holds.Suppose f ⊥ g. Then given ε > 0, there exists a partition a = x0 < x1 <

. . . < xn = b of I for which there is a decomposition (S−, S+) of Sn such that∑

i∈S+

{f(xi)− f(xi−1)}+∑

i∈S−

{g(xi)− g(xi−1)} <ε

2.

By refining this partition of I if necessary we can assume further that∑

i∈Sn

|f(xi)− f(xi−1)| > V f −ε

2,

∑

i∈Sn

|g(xi)− g(xi−1)| > V g −ε

2.

Then∑

i∈S−

|f(xi)− f(xi−1)| > V f − ε ,∑

i∈S+

|g(xi)− g(xi−1)| > V g − ε ,

from which it is clear that f ⊥− g and f ⊥− g.Now to prove the converse, suppose (a) holds. Then by Theorem 5.3 and

Corollary 5.4, V (f + g) = V (f − g) = V f +V g. Given ε > 0, it is then clear thatthere exists a partition a = x0 < x1 < . . . < xn = b of I such that

n∑

i=1

|f(xi) + g(xi)− f(xi−1)− g(xi−1)| > V f + V g −ε

8

andn∑

i=1

|f(xi)− g(xi)− f(xi−1) + g(xi−1)| > V f + V g −ε

8.

Now, set

T1 = {i ∈ Sn : (f(xi)− f(xi−1))(g(xi)− g(xi−1)) < 0} ,

T2 = {i ∈ Sn : (f(xi)− f(xi−1))(g(xi)− g(xi−1)) > 0},

T3 = {i ∈ Sn : f(xi) = f(xi−1)} ,

T4 = {i ∈ Sn : g(xi) = g(xi−1)} .

Clearly, Sn =⋃4

k=1 Tk. If k = 1 or 2, it follows from Lemma 5.2 that there existsa decomposition (Tk−, Tk+) of Tk such that

∑

i∈Sn∼Tk−

|f(xi)− f(xi−1)|+∑

i∈Sn∼Tk+

|g(xi)− g(xi−1)| > V f + V g −ε

4.

Hence for each k = 1 and 2,∑

i∈Tk−∪T3

{f(xi)− f(xi−1)} <ε

4,

∑

i∈Tk+∪T4

{g(xi)− g(xi−1)} <ε

4.


Thus on setting S+ = T1−∪T2−∪T3 and S− = Sn ∼ S+ = T1+∪T2+∪(T4 ∼ T3),we obtain

∑

i∈S+

{f(xi)− f(xi−1)}+∑

i∈S−

{g(xi)− g(xi−1)} < ε/4 + ε/4 + ε/4 + ε/4 = ε .

Consequently, f ⊥ g.Next, it is clear from the definition of mutual singularity that f ⊥ g iff f ⊥

g. Hence by the above result f ⊥ g iff f ⊥− g, for f ⊥− g automatically.Consequently, it follows from Corollary 5.4 that f ⊥ g ⇔ (f − g)+ = f ⇔(f − g)− = g, i.e. f ⊥ g ⇔ (b) ⇔ (c).

5.6. Corollary. For each f ∈ B, f+ ⊥ f−.

For, since f+ = f+, f− = f− and (f+− f−)+ = (f − f(a))+ = f+, the resultfollows from the part (b) of the above theorem.

6. Additivity of mutual singularities and their characterizations. Inthis section we first obtain a theorem on the additivity of mutual singularities, andthen deduce from it characterizations of mutual LS and US in terms of mutualsingularity of variations of the given functions.

6.1. Theorem (Additivity). Let f, g, h ∈ B. If f and g are singular or LSrelative to h, then so is f + g. Moreover , if f ⊥− g, then f + g is singular or LSrelative to h iff both f and g are so.

P r o o f. It is enough to prove each result for LS. For, on applying it to f , gand −h, a similar result is obtained for US, and the result on singularity followson combining these two results due to Theorem 5.5.

First, suppose f ⊥− h and g ⊥− h. Given ε> 0, there clearly exists a partitiona = x0 < x1 < . . . < xn = b of I such that

n∑

i=1

|f(xi) + g(xi)− f(xi−1)− g(xi−1)| > V (f + g)− ε ,(1)

n∑

i=1

|f(xi) + h(xi)− f(xi−1)− h(xi−1)| > V (f + h)− ε ,

n∑

i=1

|g(xi) + h(xi)− g(xi−1)− h(xi−1)| > V (g + h)− ε .

Now set

T1 = {i ∈ Sn : (f(xi)− f(xi−1))(h(xi)− h(xi−1)) < 0} ,

T2 = {i ∈ Sn : (g(xi)− g(xi−1))(h(xi)− h(xi−1)) < 0} ,

S = {i ∈ Sn : (f(xi) + g(xi)− f(xi−1)− g(xi−1))(h(xi)− h(xi−1)) < 0} .

Since V (f+h) = V f+V h and V (g+h) = V g+V h by Theorem 5.3, there exist byLemma 5.2 decompositions (T1−, T1+) and (T2−, T2+) of T1 and T2 respectively

20 K. M. Garg

such that

(2)∑

i∈Sn∼T1+

|f(xi)− f(xi−1)| > V f − ε ,∑

i∈Sn∼T2+

|g(xi)− g(xi−1)| > V g− ε

and

(3)∑

i∈Sn∼Tk−

|h(xi)− h(xi−1)| > V h− ε (k = 1, 2) .

Now set S− = S ∩ (T1− ∪ T2−) and S+ = S ∼ S−. Then it is clear from (3)that

∑

i∈Sn∼S−

|h(xi)− h(xi−1)| ≥∑

i∈Sn∼T1−∪T2−

|h(xi)− h(xi−1)|

≥∑

i∈Sn∼T1−

|h(xi)− h(xi−1)| −∑

i∈T2−

|h(xi)− h(xi−1)| > V h− 2ε .

Further, since S ⊂ T1 ∪ T2,

S+ = S ∩ {(T1+ ∪ T2+) ∼ (T1− ∪ T2−)}

= S ∩ {(T1+ ∩ T2+) ∪ (T1+ ∼ T2) ∪ (T2+ ∼ T1)} .

In case i ∈ S ∩ (T1+ ∼ T2), we claim that

(4) |f(xi) + g(xi)− f(xi−1)− g(xi−1)| ≤ |f(xi)− f(xi−1)| .

For, if h(xi)− h(xi−1) > 0, then since i 6∈ T2, g(xi)− g(xi−1) ≥ 0, and hence

f(xi)− f(xi−1) ≤ f(xi) + g(xi)− f(xi−1)− g(xi−1) < 0;

otherwise h(xi)− h(xi−1) < 0, so that g(xi)− g(xi−1) ≤ 0, and hence

0 < f(xi) + g(xi)− f(xi−1)− g(xi−1) ≤ f(xi)− f(xi−1) .

Thus (4) holds in the given case, and when i ∈ S ∩ (T2+ ∼ T1), it is provedsimilarly that

|f(xi) + g(xi)− f(xi−1)− g(xi−1)| ≤ |g(xi)− g(xi−1)| .

Hence we obtain, with the help of (2),∑

i∈S+

|f(xi) + g(xi)− f(xi−1)− g(xi−1)|

≤∑

i∈S∩T1+∩T2+

{|f(xi)− f(xi−1)|+ |g(xi)− g(xi−1)|}

+∑

i∈S∩T1+∼T2

|f(xi)− f(xi−1)|+∑

i∈S∩T2+∼T1

|g(xi)− g(xi−1)|

≤∑

i∈T1+

|f(xi)− f(xi−1)|+∑

i∈T2+

|g(xi)− g(xi−1)| < 2ε .


Consequently, by (1),∑

i∈Sn∼S+

|f(xi) + g(xi)− f(xi−1)− g(xi−1)| > V (f + g)− 3ε .

Hence it follows from (3) that (f + g) ⊥− h, which proves the first part for LS.To prove the second part, we thus need to prove only the necessity of the

condition. Hence, suppose f ⊥− g and (f + g) ⊥− h. Then, by Theorem 5.3,V (f+g) = V f+V g and V (f+g+h) = V (f+g)+V h. Consequently, V (f+g+h) =V f + V g + V h. But this implies that V (f + h) = V f + V h, for otherwise

V (f + g + h) ≤ V (f + h) + V g < V f + V h+ V g .

Hence f ⊥− h by Theorem 5.3, and similarly g ⊥− h.

Next, we obtain the desired characterization of LS.

6.2. Theorem. Let f, g ∈ B. Then f ⊥− g iff f+ ⊥ g− and f− ⊥ g+, or ,equivalently , iff (f+ + g+) ⊥ (f− + g−).

P r o o f. Define h(x) = f(a), x ∈ I. Then f = f+ + (−f−) + h, where f+ ⊥(−f−) by Corollary 5.6 and Theorem 5.1, and the constant function h is obviouslysingular relative to each function in B. Hence it follows from Theorem 6.1 thatf ⊥− g iff f+ ⊥− g and (−f−) ⊥− g. Now each of the functions f+ and −f− isby a similar argument LS relative to g iff it is so relative to both g+ and −g−.But since f+ ⊥− g+ and (−f−) ⊥− (−g−) automatically, it thus follows thatf ⊥− g iff f+ ⊥− (−g−) and (−f−) ⊥− g+, or, equivalently, iff f+ ⊥ g− andf− ⊥ g+.

Next, suppose f+ ⊥ g− and f− ⊥ g+. Then since g+ ⊥ g− by Corollary 5.6,(f+ + g+) ⊥ g− by Theorem 6.1. Similarly, (f+ + g+) ⊥ f−. Consequently, byTheorem 6.1, (f++g+) ⊥ (f−+g−). The converse also follows from Theorem 6.1by a similar argument since f+ ⊥− g+ and f− ⊥− g− automatically.

On applying the above theorem to f and −g, we obtain the following charac-terization of US.

6.3. Corollary. Let f, g ∈ B. Then f ⊥− g iff f+ ⊥ g+ and f− ⊥ g−, or ,equivalently , iff (f+ + g−) ⊥ (f− + g+).

In the case when one of the functions f and g is nondecreasing, the abovecharacterizations of LS and US assume a much simpler form as follows.

6.4. Corollary. Let f, g ∈ B, and suppose g is nondecreasing. Then f ⊥− giff f− ⊥ g, and f ⊥− g iff f+ ⊥ g.

7.Reduction theorem for mutual singularities. In this section we obtaina reduction theorem which reduces the mutual singularity and LS of two functionsf, g ∈ B to those of continuous and discontinuous components of f and g, and alsoto those of discontinuous, AC and continuous singular components of f and g.

22 K. M. Garg

As it will be seen in the next section, the mutual singularity of f and g isnot comparable in general with that of the signed measures µf and µg inducedby them on B. We will investigate this question in the next section. To meetthe present needs we begin with a comparison of mutual singularity of µf and µg

with that of f∗ and g∗. For this purpose we need a lemma which requires somenomenclature.

We will call a function f ∈ B internal at a point x ∈ I0 if f(x) is in betweenthe two limits f(x− 0) and f(x+ 0), i.e. if f does not have an external saltus atx. Thus f is internal at x iff

min f(x± 0) ≤ f(x) ≤ max f(x± 0) .

Further, f will be called simply internal if it is so at every point of I0.The following lemma generalizes an identity which is known to hold when f

is continuous at the points of E (see [34], p. 99).

7.1. Lemma. Let f ∈ B and E ∈ B. Then µf (E) ≤ µf (E). Moreover , if f is

internal at each point of E, then µf (E) = µf(E).

P r o o f. Set E0 = E ∩ Cf and E1 = E ∼ E0. Then µf (E0) = µf (E0) by thetheorem cited above. Further, for each x ∈ E1,

µf ({x}) = |µf ({x})| = |f(x+ 0)− f(x− 0)|

and

µf ({x}) = f(x+ 0)− f(x− 0) = |f(x+ 0)− f(x)|+ |f(x)− f(x− 0)| .

Hence µf ({x}) ≤ µf ({x}), where equality holds clearly iff f is internal at x.Now, since E1 is countable, both the parts of the lemma follow from the above

relations on using countable additivity of µf and µf on B.

7.2. Theorem. Let f, g ∈ B. Then f∗ ⊥ g∗ iff µf ⊥ µg.

P r o o f. Since µf = µf∗ and µg = µg∗ by Lemma 3.3, we can assume heref and g to be normalized. Then it follows from the above lemma that µf ⊥ µg

iff µf ⊥ µg . But since f ⊥ g iff f ⊥ g, f and g can also be assumed to benondecreasing.

First, suppose f ⊥ g. Then, for every positive integer k, there exists a partitiona = xk,0 < xk,1 < . . . < xk,nk

= b of I for which there is a decomposition(Sk−, Sk+) of Snk

such that∑

i∈Sk+

{f(xk,i)− f(xk,i−1)}+∑

i∈Sk−

{g(xk,i)− g(xk,i−1)} < 2−k .

But since f and g are normalized, it follows clearly from this inequality that∑

i∈Sk+

{f(xk,i + 0)− f(xk,i−1)}+∑

i∈Sk−

{g(xk,i + 0)− g(xk,i−1)} < 21−k .

Now, given k, set Jk,1 = [xk,0, xk,1] and Jk,i = (xk,i−1, xk,i] for i = 2, . . . , nk.Let Ak =

⋃

i∈Sk+Jk,i and Bk =

⋃

i∈Sk−

Jk,i. Now set E =⋂

n

⋃

k>nAk and


F =⋂

n

⋃

k>nBk. Then, for each n,

µf (E) ≤∑

k>n

µf (Ak)

≤∑

k>n

∑

i∈Sk+

{f(xk,i + 0)− f(xk,i−1)} <∑

k>n

21−k = 21−n .

Hence µf (E) = 0, and similarly µg(F ) = 0. But since

I ∼ E =⋃

n

⋂

k>n

(I ∼ Ak) =⋃

n

⋂

k>n

Bk ⊂ F ,

this proves that µf ⊥ µg.

Next, to prove the converse, suppose µf ⊥ µg. Then there exists a set E ∈ Bsuch that µf (E) = µg(I ∼ E) = 0. Hence, given ε > 0, there exists by definition adisjoint sequence of open intervals {Un} in I, say Un = (an, bn), n = 1, 2, . . . , suchthat E ⊂

⋃

n Un and∑

n{f(bn)− f(an)} < ε/2. Further, since∑

n µg(Un) <∞,there exists an integer k such that

∑

n>k µg(Un) < ε/2. Set F = I ∼⋃

n≤k Un.Then

µg(F ) ≤ µg(I ∼ E) +∑

n>k

µg(Un) <ε

2.

Now let a = x0 < x1 < . . . < xj = b be the partition of I determined by thepoints {an, bn : n = 1, . . . , k}. Set S+ = {i ∈ Sj : xi−1 = an for some n ∈ Sk}and S− = Sj ∼ S+. Then

∑

i∈S+

{f(xi)− f(xi−1)}+∑

i∈S−

{g(xi)− g(xi−1)}

≤∑

n

{f(bn)− f(an)}+∑

i∈S−

µg([xi−1, xi]) <ε

2+ µg(F ) < ε .

This proves that f ⊥ g.

In the case of continuous functions the above theorem leads to the followinganalogue of mutual singularity of signed measures.

7.3. Corollary. If f, g ∈ B are continuous, then f ⊥ g iff I has a de-

composition, or , equivalently , a Borel decomposition, (A,B) such that |f(A)| =|g(B)| = 0.

For, suppose f and g are continuous. Then it follows from the above theoremand Lemma 7.1 that f ⊥ g iff I has a Borel decomposition (A,B) such thatµf (A) = µg(B) = 0. The result in terms of Borel decomposition follows nowwith the help of Theorem 2.3. The equivalence of the Borel decomposition witha general decomposition follows on the other hand from the fact that µf and µg

are metric outer measures. For, suppose I has an arbitrary decomposition (A,B)such that |f(A)| = |g(B)| = 0. Then µf (A) = |f(A)| = 0 by Theorem 2.3, andhence there exists a Gδ-set A1 ⊃ A such that µf (A1) = 0. Set B1 = I ∼ A1.

24 K. M. Garg

Then (A1, B1) is a Borel decomposition of I such that |f(A1)| = µf (A1) = 0 and|g(B1)| ≤ |g(B)| = 0.

Next, we obtain two lemmas which are further needed to prove the reductiontheorem.

When two functions f, g ∈ B are not simultaneously discontinuous at anypoint of I from any of the two sides, it will be convenient to call them simplynowhere simultaneously discontinuous from the same side.

It is interesting to note here that when f and g are simultaneously left, orright, continuous, or one of them is normalized, then they are nowhere simulta-neously discontinuous from the same side iff they have simply no common pointof discontinuity.

7.4. Lemma. Let f, g ∈ B and suppose f is a jump function. Then f ⊥ g iff

f and g are nowhere simultaneously discontinuous from the same side.

P r o o f. Since f ⊥ g iff f ⊥ g, it is clear from Lemma 2.1 that there is no lossof generality in assuming f and g to be nondecreasing.

Suppose f ⊥ g but there exists a point x in I where f and g are simultaneouslydiscontinuous from the same side, say from the right. Then f(x) < f(x+ 0) andg(x) < g(x + 0). Set

(1) ε = min{f(x+ 0)− f(x), g(x + 0)− g(x)} .

Then ε > 0. Hence there exists a partition a = x0 < x1 < . . . < xn = b of I forwhich there is a decomposition (S−, S+) of Sn such that

(2)∑

i∈S+

{f(xi)− f(xi−1)}+∑

i∈S−

{g(xi)− g(xi−1)} < ε .

Choose i such that xi−1 ≤ x < xi. For this i, by (1), f(xi)−f(xi−1) ≥ f(x+0)−f(x) ≥ ε, and similarly g(xi) − g(xi−1) ≥ ε, which contradicts (2). It is provedsimilarly that f and g are not simultaneously discontinuous from the left at anypoint of I.

Next, to prove the converse, suppose the condition holds. Let {an} and {bn}be the points where f is discontinuous from the left or right respectively. Givenε > 0, choose positive integers p and q such that

(3)∑

n>p

{f(an)− f(an − 0)} <ε

4,

∑

n>q

{f(bn + 0)− f(bn)} <ε

4.

Now set A =⋃p

n=1{an}, B =⋃q

n=1{bn} and C = A ∪ B. For each n ≤ p, sinceg is by hypothesis continuous at an from the left, there exists a′n ∈ [a, an) suchthat C ∩ (a′n, an) = ∅ and g(an)− g(a′n) < ε/(4p). Set D = C ∪

⋃pn=1{a

′n}. Now,

for each n ≤ q, there exists as before b′n ∈ (bn, b] such that D ∩ (bn, b′n) = ∅ and

g(b′n)− g(bn) < ε/(4q). Set E = D∪⋃q

n=1{b′n}∪{a}∪{b} and let a = x0 < x1 <

. . . < xk = b be the partition of I determined by the points of E. Now set

S− = {i ∈ Sk : xi ∈ A or xi−1 ∈ B} and S+ = Sk ∼ S− .


Then, for each i ∈ S−, the interval [xi−1, xi] coincides either with [a′n, an] forsome n ≤ p or with [bn, b

′n] for some n ≤ q. Hence

∑

i∈S−

{g(xi)− g(xi−1)} <

p∑

n=1

ε

4p+

q∑

n=1

ε

4q=ε

2.

Further, for each i ∈ S+, it is clear that A∩ (xi−1, xi] = ∅ and B ∩ [xi−1, xi) = ∅.Hence it follows from (3), since f is a jump function, that

∑

i∈S+

{f(xi)− f(xi−1)} <ε

4+ε

4=ε

2.

Consequently, f ⊥ g.

7.5. Lemma. If f ∈ B is AC and g ∈ B is continuous and singular , thenf ⊥ g.

P r o o f. Suppose the hypothesis holds. Set A = ∆∞g and B = I ∼ A. Then

since g also is singular, |A| = 0 and |g(B)| = 0 (see Theorem 2.2). Further, sincef is AC, |f(A)| = 0 (see [28], p. 249). Consequently, f ⊥ g by Corollary 7.3.

7.6. Theorem (Reduction). Two functions f, g ∈ B are mutually singular or

LS iff the pairs (fd, gd) and (fc, gc) are so, or , equivalently , iff the pairs (fd, gd),(fa, ga) and (fcs, gcs) are so.

P r o o f. Since fd ⊥ fc by Lemma 7.4, it follows from Theorem 6.1 that f is LSrelative to g iff fd and fc are so. Similarly, fd or fc is LS relative to g iff it is sorelative to gd and gc. However, fd ⊥ gc and fc ⊥ gd by Lemma 7.4. Consequently,f ⊥− g iff fd ⊥− gd and fc ⊥− gc.

Using a similar argument it follows from Lemma 7.5 that fc ⊥− gc iff fa ⊥− gaand fcs ⊥− gcs. Hence the other equivalence for LS.

A similar argument holds for mutual singularity.

Here is an easy consequence of the above theorem which also follows directlyfrom the above two lemmas.

7.7. Corollary. For every function f ∈ B, fd ⊥ fc, and each pair of func-

tions in fd, fa and fcs are mutually singular.

8. Comparison of mutual singularities with those of normalizations

and induced signed measures. In this section we investigate conditions underwhich the mutual singularities of two functions f, g ∈ B can be compared withthose of their normalizations f∗ and g∗. They are of course not comparable ingeneral (see Remark 8.5). The results obtained also provide a comparison ofthe mutual singularity of f and g with that of their induced signed measures µf

and µg.Also, we obtain here the variation functions of f∗ in terms of the variations of

f . We begin with this result which is found useful in dealing with normalizations.

26 K. M. Garg

Given f ∈ B, we will use f∗+ to denote (f∗)+. The brackets will be droppedsimilarly in other situations where there is no possibility of confusion. Further,since (fd)

+ = (f+)d, we will use simply f+d to denote both of these functions.

The same holds for f−d , f+

c and f−c .

8.1. Theorem. Given f ∈ B, the following are equivalent :

(a) f∗+ = f+∗, (b) f∗− = f−∗ ,

(c) f∗ = (f)∗, (d) f is internal .

Consequently , if f is normalized , then so are f+, f− and f .

P r o o f. Since f − f(a) = f+− f− and f = f++ f−, we have, by Lemma 3.1,

f∗ − f∗(a) = f+∗ − f−∗, (f)∗ = f+∗ + f−∗ .

Now, since the functions f+∗ and f−∗ are nondecreasing, it follows easily from thelast two equations that the relations (a), (b) and (c) are equivalent, and further,due to Theorem 5.5, that (a) holds iff f+∗ and f−∗ are mutually singular.

Next, since f+ = f+c +f+

d and f− = f−c +f−

d , where f+c and f−

c are continuous,we obtain from Lemma 3.1,

f+∗ = f+c + f+∗

d and f−∗ = f−c + f−∗

d .

Now since f+c ⊥ f−

c by Corollary 5.6, and f+∗d and f−∗

d are jump functions, itfollows from Theorem 7.6 that f+∗ ⊥ f−∗ iff f+∗

d ⊥ f−∗d . Thus (a) has been

proved to be equivalent to the relation (e) f+∗d ⊥ f−∗

d . Consequently, it is enoughto show that (d) and (e) are equivalent.

First, suppose f is internal. Then it is easy to see that f+ and f− do not haveany common point of discontinuity. The same holds for f+

d and f−d by Lemma 2.1,

and so f+∗d and f−∗

d also do not have any common point of discontinuity. Conse-quently, (e) holds by Lemma 7.4.

Next, to prove the converse, suppose (e) holds but f is not internal. Thenthere exists a point c ∈ I0 where either f(c) < min f(c±0) or f(c) > max f(c±0).At such a point c, it is easy to see that each of the functions f+ and f− is dis-continuous from one and only one side. The same holds therefore for f+

d and f−d ,

and consequently f+∗d and f−∗

d are simultaneously discontinuous at c from bothsides. This, however, contradicts (e) by Lemma 7.4. This completes the proofof the first part.

The last part follows directly from the first since every normalized function isinternal.

In the next theorem we compare the mutual singularity of f, g ∈ B with thatof f∗ and g∗, and that of µf and µg. Some nomenclature is needed here.

Let a function f ∈ B be called unilaterally discontinuous at a point x ∈ I0

if it is discontinuous from one and only one side at x. We call two functionsf, g ∈ B nowhere unilaterally discontinuous from opposite sides if there is no point


in I0 where f and g are simultaneously unilaterally discontinuous and furtherdiscontinuous from opposite sides.

It is interesting to note here that if f and g are simultaneously left, or ri-ght, continuous, or if one of them is normalized, then f and g are automaticallynowhere unilaterally discontinuous from opposite sides.

8.2. Theorem. Let f, g ∈ B. If f ⊥ g and f and g are nowhere unilaterally

discontinuous from opposite sides, then f∗ ⊥ g∗ and µf ⊥ µg. Conversely , iff∗ ⊥ g∗, or µf ⊥ µg, and each of f and g is continuous at the points where the

other has a removable discontinuity , then f ⊥ g.

P r o o f. On account of Theorem 7.2 it is enough to prove the results for f∗

and g∗. Let us begin by recalling a result proved earlier in Lemma 3.2, viz. Cf∗ =Cf∪Rf where Rf denotes the set of points where f has a removable discontinuity.

First, suppose f ⊥ g and that f and g are nowhere unilaterally discontinuousfrom opposite sides. Then it follows from Lemma 3.2 and Theorem 7.6 that fd ⊥gd and (f∗)c = fc ⊥ gc = (g∗)c. Hence under the present hypothesis it followseasily from Lemma 7.4 that at every point of I either f or g is continuous, andconsequently, either f∗ or g∗ is continuous. Thus (f∗)d ⊥ (g∗)d by Lemma 7.4,and hence it follows from Theorem 7.6 that f∗ ⊥ g∗.

Next, to prove the converse, suppose f∗ ⊥ g∗ and that Rf ⊂ Cg and Rg ⊂ Cf .Then, by Theorem 7.6 and Lemma 3.2, (f∗)d ⊥ (g∗)d and fc = (f∗)c ⊥ (g∗)c = gc.Thus under the present hypothesis it follows from Lemma 7.4 that

I = Cf∗ ∪ Cg∗ = (Cf ∪Rf ) ∪ (Cg ∪Rg) ⊂ Cf ∪ Cg .

Consequently, fd ⊥ gd by Lemma 7.4, and hence f ⊥ g by Theorem 7.6.

The following theorem deals with mutual LS.

8.3. Theorem. Suppose f, g ∈ B are internal. If f ⊥− g and f and g are no-

where unilaterally discontinuous from opposite sides, then f∗ ⊥− g∗. Conversely ,if f∗ ⊥− g∗, then f ⊥− g.

P r o o f. First, suppose f ⊥− g, and that f and g are nowhere unilaterallydiscontinuous from opposite sides. Then f+ ⊥ g− and f− ⊥ g+ by Theorem 6.2.We claim that the functions f+ and g−, and similarly f− and g+, are nowhereunilaterally discontinuous from opposite sides.

Suppose f+ is discontinuous at some point x ∈ I0. Then since f is internal, wehave clearly f(x− 0) ≤ f(x) ≤ f(x+0) where at least one of the two inequalitiesis strict. It is then further clear that

f+(x+ 0)− f+(x) = f(x+ 0)− f(x) ,

f+(x)− f+(x− 0) = f(x)− f(x− 0) .

Consequently, f+ is discontinuous from each of the two sides at x iff f is so.As the same holds for f−, g+ and g−, the claim follows from the correspondinghypothesis on f and g.

28 K. M. Garg

It follows now from Theorem 8.2 that f+∗ ⊥ g−∗ and f−∗ ⊥ g+∗. Consequ-ently, by Theorem 8.1, f∗+ ⊥ g∗− and g∗− ⊥ g∗+, which implies by Theorem 6.2that f∗ ⊥− g∗.

Next, to prove the converse, suppose f∗ ⊥− g∗. Then f∗+ ⊥ g∗− and f∗− ⊥g∗+ by Theorem 6.2, and so f+∗ ⊥ g−∗ and f−∗ ⊥ g+∗ by Theorem 8.1. Con-sequently, f+ ⊥ g− and f− ⊥ g+ by Theorem 8.2, and so f ⊥− g by Theo-rem 6.2.

When f and g are normalized, or simultaneously left, or right, continuous, itis easy to see that all the continuity hypotheses of Theorems 8.2 and 8.3 holdautomatically. Hence in that case we obtain

8.4. Corollary. If f, g ∈ B are normalized , or simultaneously left , or right ,continuous, then

(a) f ⊥ g iff f∗ ⊥ g∗, or , equivalently , iff µf ⊥ µg, and(b) f ⊥− g iff f∗ ⊥− g∗.

8.5. R ema r k. We include here some simple examples which show on theone hand that the mutual singularities of f, g ∈ B are not comparable in generalwith those of f∗ and g∗, and on the other that the continuity hypotheses ofTheorems 8.2 and 8.3 are not dispensable.

Let c be any interior point of I. Let us first observe here that if f, g ∈ B aretwo jump functions, then as we see subsequently in Theorem 9.6, f ⊥− g iff

{f(x+ 0)− f(x)}{g(x + 0)− g(x)} ≥ 0 for a ≤ x < b and

{f(x− 0)− f(x)}{g(x − 0)− g(x)} ≥ 0 for a < x ≤ b .

(a) Let us first consider the need of the hypothesis in the first parts of thetwo theorems for f and g to be nowhere unilaterally discontinuous from oppositesides. Define f(x) = 0 or 2 according as x < c or x ≥ c respectively, and g(x) = 2or 0 according as x ≤ c or x > c respectively. Then f and g are internal althoughthey are unilaterally discontinuous from opposite sides at c. Clearly, f ⊥ g byLemma 7.4, but f∗ and g∗ are not even mutually LS since

{f∗(c+ 0)− f∗(c)}{g∗(c+ 0)− g∗(c)} = (2− 1)(0 − 1) < 0.

(b) The need of the continuity hypothesis in the converse part of Theorem 8.2is somewhat obvious. For let f be as before and define this time g to be thecharacteristic function of the singleton set {c}. Then since g∗ ≡ 0, f∗ and g∗ aretrivially mutually singular, but f and g are not so by Lemma 7.4.

Next, we show the need of the hypothesis for f and g to be internal in thetwo parts of Theorem 8.3.

(c) Define f(x) = 1, 0 or 3 according as x < c, x = c or x > c respectively,and g(x) = 3, 0 or 1 according as x < c, x = c or x > c respectively. Then since

{f(c+ 0)− f(c)}{g(c + 0)− g(c)} = (3− 0)(1 − 0) > 0 ,

{f(c− 0)− f(c)}{g(c − 0)− g(c)} = (1− 0)(3 − 0) > 0 ,


f and g are mutually LS by the result stated at the beginning, but f∗ and g∗ arenot so by the same result since

{f∗(c+ 0)− f∗(c)}{g∗(c+ 0) − g∗(c)} = (3− 2)(1 − 2) < 0 .

(d) Define f(x) = 0, 3 or 2 according as x < c, x = c or x > c respectively,and g(x) = 0 or 2 according as x ≤ c or x > c respectively. Then since f∗ and g∗

are nondecreasing, they are automatically mutually LS, but f and g are not sosince

{f(c+ 0)− f(c)}{g(c + 0)− g(c)} = (2− 3)(2 − 0) < 0 .

9.Mutual singularities in terms of derivatives. In this section we obtaincharacterizations of mutual singularities of f, g ∈ B in terms of derivatives of fand g.

9.1. Lemma. Suppose f, g ∈ B and f is AC. Then

(a) f ⊥− g iff f ′(x)g′(x) ≥ 0 for almost every x, and(b) f ⊥ g iff f ′(x)g′(x) = 0 for almost every x.

P r o o f. (a) According to Theorem 7.6, f ⊥− g iff f ⊥− ga. Now since f , gaand f + ga are AC, we have

V f =b∫

a

|f ′(x)| dx, V ga =b∫

a

|g′a(x)| dx ,

V (f + ga) =b∫

a

|f ′(x) + g′a(x)| dx

(see [28], p. 259). Hence, by Theorem 5.3, f ⊥− ga iff

b∫

a

|f ′(x) + g′a(x)| dx =b∫

a

{|f ′(x)|+ |g′a(x)|} dx .

But since g′a(x) = g′(x) for almost every x, it follows that f ⊥− g iff |f ′(x) +g′(x)| = |f ′(x)|+ |g′(x)| for almost every x, or, equivalently, iff f ′(x)g′(x) ≥ 0 foralmost every x.

The part (b) follows on the other hand from (a) due to Theorem 5.5.

9.2. Lemma. If f ∈ B and A ⊂ I, then |f cs(A)| ≤ |f(A)|. Moreover , if

|A| = 0, then |f cs(A)| = |f(A)|.

P r o o f. Let B = A ∩ Cf and C = A ∼ B. Since C is countable, we have

|f(A)| = |f(B)| and |f cs(A)| = |f cs(B)| .

Now since f = fd + fa + f cs, where each of these functions is continuous atthe points of B, it follows clearly from Theorem 2.3 and Lemma 4.1 that

(1) |f(B)| = |fd(B)|+ |fa(B)|+ |f cs(B)| .

30 K. M. Garg

Consequently,

|f cs(A)| = |f cs(B)| ≤ |f(B)| = |f(A)| .

Next, suppose |A| = 0. Then |B| = 0, and since fa satisfies Lusin’s condition(N) (see [34], p. 227), |fa(B)| = 0. Also, since fd is a jump function, |fd(B)| = 0.Hence, by (1),

|f(A)| = |f(B)| = |f cs(B)| = |f cs(A)| .

9.3. Theorem. Two functions f, g ∈ B are mutually singular iff the following

conditions hold :

(a) f and g are nowhere simultaneously discontinuous from the same side,

(b) f ′(x)g′(x) = 0 for almost every x, and

(c) the set E of points where both f and g have infinite derivatives has a

decomposition (A,B) such that |f(A)| = |g(B)| = 0.

P r o o f. According to Theorem 7.6, f ⊥ g iff fd ⊥ gd, fa ⊥ ga and fcs ⊥ gcs.But since fd and gd are jump functions, it follows from Lemmas 2.1 and 7.4 thatfd ⊥ gd iff (a) holds. Further, since fa and ga are AC and f ′

a(x) = f ′(x) andg′a(x) = g′(x) for almost every x, it follows from Lemma 9.1 that fa ⊥ ga iff (b)holds. Hence it is enough to show that fcs ⊥ gcs iff (c) holds.

First, suppose fcs ⊥ gcs. Then, by Corollary 7.3, I has a decomposition (G,H)such that |f cs(G)| = |gcs(H)| = 0. Let E = ∆∞

f ∩∆∞g and set A = E ∩ G and

B = E ∩H. Then since |E| = 0 (see Theorem 2.2), it follows from Lemma 9.2that |f(A)| = |g(B)| = 0. Consequently, (c) holds.

Next, to prove the converse, suppose (c) holds. Then the set E = ∆∞f ∩∆∞

g

has a decomposition (A,B) such that |f(A)| = |g(B)| = 0. Set

A1 = ∆∞fcs

∼ ∆∞f , A2 = I ∼ ∆∞

fcs, G = A ∪A1 ∪A2,

B1 = ∆∞gcs

∼ ∆∞g , B2 = I ∼ ∆∞

gcs, M = B ∪B1 ∪B2

and H = I ∼ G. Then (G,H) is a decomposition of I. We claim that |f cs(G)| =|gcs(H)| = 0.

For each x ∈ A1, since f−f cs = fd+fa is nondecreasing, Df(x) ≥ (f cs)′(x) =

∞, i.e. (f)′(x) = ∞. Hence it is clear from Theorem 4.2 that |f(A1)| = 0. Further,since f cs is singular, |f cs(A2)| = 0 by Theorem 2.2. Hence, by Lemma 9.2,

|f cs(G)| ≤ |f cs(A)| + |f cs(A1)|+ |f cs(A2)|

≤ |f(A)|+ |f(A1)| = 0 .

It is proved similarly that |gcs(M)| = 0. But since

I ∼ (A ∪B) = (I ∼ ∆∞f ) ∪ (I ∼ ∆∞

g ) ⊂ A1 ∪A2 ∪B1 ∪B2 ,

it is clear that H ⊂ M . Consequently, |gcs(H)| ≤ |gcs(M)| = 0. Hence it followsfrom Corollary 7.3 that fcs ⊥ gcs.


The conditions (a), (b) and (c) of the above theorem clearly become redundantwhen one of the functions f and g is continuous, singular or AC respectively. Thusin case f or g is AC, the mutual singularity of f and g is totally determined bythe derivatives of the two functions as suggested by this section’s title (see part(b) of Lemma 9.1).

From this consequence of the above theorem we obtain the following relation-ship between the ordinary singularity of a function f ∈ B and mutual singularity.

9.4. Corollary. A function f ∈ B is singular iff it is so relative to the

identity function τ, or , equivalently , iff for every ε > 0 there exists a finite set of

nonoverlapping intervals {[ai, bi] : i = 1, . . . , n} in I such that

n∑

i=1

(bi − ai) < ε and

n∑

i=1

|f(bi)− f(ai)| > V f − ε .

To deal with LS we need another lemma.

9.5. Lemma. Given f, g ∈ B, f+cs ⊥ g−cs iff the set ∆+∞

f ∩∆−∞g has a decom-

position (A,B) such that |f(A)| = |g(B)| = 0.

P r o o f. Set E = ∆+∞f ∩∆−∞

g and F = ∆∞f+cs

∩∆∞g−

cs

.

First, suppose f+cs⊥g

−cs. Then, by Theorem 9.3, F has a decomposition (G,H)

such that |f+cs(G)| = |g−cs(H)| = 0. Set

A1 = E ∩G, A2 = E ∼ ∆∞f+cs, A = A1 ∪A2,

B1 = E ∩H, B2 = E ∩∆∞f+cs

∼ ∆∞g−

cs, B = B1 ∪B2 .

Then

E = (E ∼ ∆∞f+cs) ∪ (E ∩∆∞

f+cs∩∆∞

g−

cs) ∪ (E ∩∆∞

f+cs

∼ ∆∞g−

cs)

= (E ∼ ∆∞f+cs) ∪ (E ∩G) ∪ (E ∩H) ∪ (E ∩∆∞

f+cs

∼ ∆∞g−

cs)

= A2 ∪A1 ∪B1 ∪B2 = A ∪B .

Hence (A,B) is a decomposition of E. We claim that this is the required decom-position.

Since A1 ⊂ G, and |f+cs(A2)| = 0 by Theorem 2.2,

|f+cs(A)| ≤ |f+

cs(A1)|+ |f+cs(A2)| = 0 .

Further, since A ⊂ E ⊂ ∆+∞f , it is clear from Theorem 4.2 that |f−(A)| = 0.

Hence if C = A∩Cf , then since A ∼ C is countable and |C| = 0 (see Theorem 2.2),we obtain as before from Theorem 2.3 and Lemma 4.1,

|f(A)| = |f(C)| = µf (C) = µf+cs(C) + µf+

a(C) + µf−(C)

= |f+cs(C)|+ |f+

a (C)|+ |f−(C)| = 0 .

It is proved similarly that |g(B)| = 0.

32 K. M. Garg

Next, to prove the converse, suppose E has a decomposition (A,B) such that|f(A)| = |g(B)| = 0. Set

G1 = F ∩A, G2 = F ∼ ∆+∞f , G = G1 ∪G2 ,

H1 = F ∩B, H2 = F ∩∆+∞f ∼ ∆−∞

g , H = H1 ∪H2 .

Then (G,H) is clearly a decomposition of F . Further, since f+− f+cs is nondecre-

asing, F ⊂ ∆∞f+cs

⊂ ∆∞f+ . Hence it follows from Theorem 4.2 that |f+(G2)| = 0.

Thus if C = G ∩ Cf , then by Theorem 2.3,

|f+cs(G)| = |f+

cs(C)| = µf+cs(C)

= µf+cs(C ∩G1) + µf+

cs(C ∩G2)

≤ µf (C ∩G1) + µf+(C ∩G2)

= |f(C ∩G1)|+ |f+(C ∩G2)| = 0 .

It is proved similarly that |g−cs(H)| = 0. Consequently, it follows from Theorem 9.3that f+

cs ⊥ g−cs.

9.6. Theorem. Let f, g ∈ B. Then f ⊥− g iff the following conditions hold :

(a) {f(x − 0) − f(x)}{g(x − 0) − g(x)} ≥ 0 for a < x ≤ b and {f(x + 0) −f(x)}{g(x + 0)− g(x)} ≥ 0 for a ≤ x < b,

(b) f ′(x)g′(x) ≥ 0 for almost every x, and

(c) the set E of points where f and g have unequal infinite derivatives has a

decomposition (A,B) such that |f(A)| = |g(B)| = 0.

P r o o f. According to Theorem 7.6, f ⊥− g iff fd ⊥− gd, fa ⊥− ga andfcs ⊥− gcs. But since, by Theorem 6.2, fd ⊥− gd iff f+

d ⊥ g−d and f−d ⊥ g+d , it

follows easily from Lemma 7.4 that fd ⊥− gd iff (a) holds. Further, by Lemma9.1, fa ⊥− ga iff f ′

a(x)g′a(x) ≥ 0 for almost every x, or, equivalently, iff (b) holds.

Hence it is enough to show that fcs ⊥− gcs iff (c) holds.

Now set E1 = ∆+∞f ∩ ∆−∞

g and E2 = ∆−∞f ∩ ∆+∞

g . Clearly, E = E1 ∪ E2

where E1 ∩E2 = ∅. Hence it is clear that (c) holds iff each of the sets E1 and E2

has a decomposition (A,B) such that |f(A)| = |g(B)| = 0. Now, by Lemma 9.5,such a decomposition exists for E1 iff f+

cs ⊥ g−cs. On applying this lemma to thefunctions −f and −g it follows, on the other hand, that such a decompositionexists for E2 iff f−

cs ⊥ g+cs. Hence it follows from Theorem 6.2 that (c) holds ifffcs ⊥− gcs.

The conditions (a), (b) and (c) of the above theorem become redundant, asbefore, when f or g is continuous, singular or AC respectively. Thus in case f org is AC, the mutual LS of f and g is totally determined by the derivatives of thetwo functions (see part (a) of Lemma 9.1).

A function f ∈ B is called in [13] lower singular (LS) or upper singular

(US) if f ′(x) ≥ 0 or ≤ 0 respectively for almost every x. It is clear from the


above-mentioned consequence of Theorem 9.6 that the LS of f is related withmutual LS as follows:

9.7. Corollary. A function f ∈ B is LS iff it is so relative to the identity

function τ, or , equivalently , iff for every ε > 0 there exists a partition a = x0 <x1 < . . . < xn = b of I for which there is a decomposition (S−, S+) of the

index set S = {i ∈ Sn : f(xi) < f(xi−1)} such that∑

i∈S−

(xi − xi−1) < ε and∑

i∈Sn∼S+|f(xi)− f(xi−1)| > V f − ε.

9.8. R ema r k. In each of Theorems 9.3 and 9.6, and in Lemma 9.5, thedecomposition (A,B) of the set in question, say E, can be chosen to be a Boreldecomposition. For, let (A,B) be any decomposition of E for which |f(A)| =|g(B)| = 0. Set C = A ∩ Cf and D = A ∼ C. Then µf (C) = |f(C)| = 0 byTheorem 2.3. Hence there exists a Gδ-set F in I such that C ⊂ F and µf (F ) = 0.Now set G = D∪F andH = E ∼ G. Since E ∈ B in each case by Theorem 2.2(a),and D is countable, (G,H) is clearly a Borel decomposition of E. Further, byTheorem 2.3, |f(G)| = |f(F )| ≤ µf (F ) = 0, and since A ⊂ G, H ⊂ B, so that|g(H)| ≤ |g(B)| = 0. Hence (G,H) is the desired Borel decomposition of E.

Now let f and g be any two nondecreasing functions on [0, 1]. H. Kober [23]called f and g “contravariations” if they are the positive and negative variationfunctions respectively of f − g. Since f ⊥− g automatically, it follows from The-orem 5.5 that f and g are contravariations iff f(0) = g(0) = 0 and any of thefollowing equivalent conditions holds:

(a) f = (f − g)+, (b) g = (f − g)−, (c) f ⊥ g, (d) f ⊥− g .

Consequently, the results of this chapter on mutual singularity hold also forcontravariance on adding the hypotheses that f and g are nondecreasing andf(0) = g(0) = 0. Some of the results of Kober ([23], p. 579) on contravariancefollow immediately from the characterization of contravariance that follows fromTheorem 9.3.

III. Relative absolute continuities

10. Relative absolute continuity and lower and upper ACs. In thissection we define absolute continuity and lower and upper absolute continuitiesof a function f : I → R relative to another function g ∈ R, and then presentsome elementary results on them including their additivity and characterizationsin terms of variations of f and g.

Given f : I → R and g ∈ B, we will call f absolutely continuous relative to g iffor each ε > 0 there exists a δ > 0 such that for every finite set of nonoverlappingintervals {[ai, bi] : i = 1, . . . , n} in I,

(1)n∑

i=1

|f(bi)− f(ai)| < ε whenevern∑

i=1

{g(bi)− g(ai)} < δ .

34 K. M. Garg

Further, f will be called lower or upper absolutely continuous relative to g ifthe above condition holds with the first inequality in (1) replaced by

n∑

i=1

{f(bi)− f(ai)} > −ε or

n∑

i=1

{f(bi)− f(ai)} < ε

respectively.

We will use the abbreviations AC, LAC and UAC for absolutely continuous,lower absolutely continuous and upper absolutely continuous respectively, or forthe corresponding nouns. Further, when f is AC, LAC or UAC relative to g, wewill write f ≪ g, f ≪− g or f ≪− g respectively.

Clearly, f is AC, LAC or UAC relative to g iff it is so relative to g. Further,since f ≪− g iff −f ≪− g, it is enough to consider LAC. It is also easy to seethat f ≪ g iff f is LAC and UAC relative to g.

Given f, g, h ∈ B, it is interesting to note the following transitive propertiesof the relations ≪− and ≪ . If f ≪− g ≪ h, then f ≪− h, and, similarly, iff ≪ g ≪ h, then f ≪ h.

It is further clear that f is AC in the ordinary sense iff it is so relative to theidentity function τ . Also, f is called LAC or UAC if it is so relative to τ (see [13]and [32]).

The following result follows directly from the above definitions.

10.1. Theorem. Let f : I → R, g ∈ B, α, β ∈ R and β 6= 0.

(a) If f ≪− g and α ≥ 0, then αf ≪− βg.

(b) If f ≪ g, then αf ≪ βg.

In case f ∈ B, the AC and LAC of f relative to g can be characterized interms of variations of f as follows.

10.2. Theorem. Suppose f, g ∈ B. Then

(a) f ≪− g iff f− ≪ g, and

(b) f ≪ g iff f ≪ g.

P r o o f. To prove (a), suppose f ≪− g. Then, given ε > 0, there exists aδ > 0 such that if I ≡ {[ai, bi] : i = 1, . . . , n} is any finite set of nonoverlappingintervals in I for which

∑ni=1{g(bi)− g(ai)} < δ, then

n∑

i=1

{f(bi)− f(ai)} > −ε

2.

Suppose I is such a set of intervals. Then, by the definition of f−, there exists foreach i a finite set of nonoverlapping intervals {[ai,j , bi,j ] : j = 1, . . . , ki} in [ai, bi]such that

ki∑

j=1

{f(bi,j)− f(ai,j)} < −{f−(bi)− f−(ai)}+ε

2n.


Hence it follows from the choice of δ thatn∑

i=1

{f−(bi)− f−(ai)} <n∑

i=1

ε

2n−

n∑

i=1

ki∑

j=1

{f(bi,j)− f(ai,j)} ≤ε

2+ε

2= ε .

This proves that f− ≪ g. The sufficiency part of (a) is obvious.Next, to prove (b), suppose f≪ g. Then f≪− g and f≪− g. Hence it follows

from (a) that f− ≪ g and f+ ≪ g. Consequently, it follows from the definitionof AC, as usual, that f = f+ + f− ≪ g. The converse is again obvious.

10.3. Theorem (Additivity). Suppose f, g : I → R and h ∈ B. If f and g are

AC or LAC relative to h, then so is f + g. Moreover , if f, g ∈ B and f ⊥− g,then f + g is AC or LAC relative to h iff both f and g are so.

P r o o f. The first part of the theorem follows directly from definitions by usualarguments. To prove the second part, we thus need to prove only the necessity ofthe condition in each of the two cases.

Given f, g ∈ B, f ⊥− g, first suppose f+g ≪− h. Then f−+g− = (f+g)− ≪h by Theorems 5.3 and 10.2. Now since f− and g− are nondecreasing, it followsfrom the definition of AC that f− ≪ h and g− ≪ h. Consequently, f ≪− h andg ≪− h by Theorem 10.2. The proof in the other case is quite similar.

Now, on account of Corollary 7.7, we obtain from the above theorem

10.4. Corollary. Given f, g ∈ B, f is AC or LAC relative to g iff fd and

fc are so, or , equivalently , iff fd, fa and fcs are so.

In the case of relative AC we have also the following multiplicative property.

10.5. Theorem. Suppose f, g : I → R and h ∈ B. If f ≪ h and g ≪ h, thenfg ≪ h.

P r o o f. Suppose f ≪ h and g ≪ h. Then it follows clearly from the definitionof relative AC that f and g are bounded. Let α = sup{|f(x)| : x ∈ I} andβ = sup{|g(x)| : x ∈ I}. Then for every subinterval [x, y] of I we have

|f(y)g(y)− f(x)g(x)| ≤ |g(y)| · |f(y)− f(x)|+ |f(x)| · |g(y) − g(x)|

≤ β|f(y)− f(x)|+ α|g(y) − g(x)| .

Hence it follows easily from the definition of relative AC that fg ≪ h.

10.6. R ema r k. As the reader may have noticed by now, the above decom-position of relative AC into relative LAC and UAC is not quite similar to thatof mutual singularity into mutual LS and US presented in §5. For, as proved inTheorem 6.2, f ⊥− g iff f+ ⊥ g− and f− ⊥ g+, but according to Theorem 10.2,f ≪− g iff f− ≪ g. The reasons behind the choice of these two decompositionswill become clearer when we deal with their applications in the next three chap-ters (see in particular Theorems 6.1, 10.3, 19.3, 20.6, 29.2 and Corollary 31.3).Also, the relations f+ ≪ g− and f− ≪ g+ together turn out to be stronger thanf ≪ g.

36 K. M. Garg

11. Bounded variation under relative ACs. Since every ordinary ACfunction on I is of bounded variation, it is only natural to ask whether a functionf : I → R that is LAC or AC relative to some g ∈ B is always of boundedvariation. We investigate this question in the present section.

In the following lemma we first look into the question whether f is regulated.Suppose a function f : I → R has finite or infinite unilateral limits f(x− 0) andf(x + 0) at every point x ∈ I which is a left or right limit point respectivelyof I. We will then call f (i) lower regulated if f(x − 0) > −∞ for x > a andf(x+ 0) < ∞ for x < b, and (ii) upper regulated if f(x− 0) < ∞ for x > a andf(x+ 0) > −∞ for x < b.

It is then clear that f is regulated iff it is lower and upper regulated, and thatf is upper regulated iff −f is lower regulated.

For an arbitrary function f : I → R, x ∈ I, we will use f (x− 0) and f(x− 0)to denote the left lower and upper limits respectively of f at x, and similarlyf (x+ 0) and f(x+ 0) to denote the right lower and upper limits of f at x.

11.1. Lemma. Let f : I → R, g ∈ B and suppose f ≪− g. Then f is lower

regulated. Moreover , if g is continuous, or f ≪ g, then f is regulated.

P r o o f. Given a ≤ x < b, first suppose f does not have any finite or infinitelimit at x from the right. Then

(1) ε ≡ f(x+ 0)− f (x+ 0) > 0 .

Given any δ > 0, since g is regulated, there exists an η > 0 such that

(2) g(z) − g(y) < δ whenever x < y < z < x+ η .

But due to (1) we can clearly find y and z in (x, x + η) such that y < z andf(y) − f(z) > ε/2. Then f(z) − f(y) < −ε/2, which due to (2) contradicts thehypothesis that f ≪− g. Consequently, f(x+0) exists in the wider sense (i.e. finiteor infinite).

Now, suppose f(x + 0) = ∞. Given any δ > 0, choose η as before for which(2) holds. This time we can easily find y and z in (x, x+ η) such that y < z andf(y)− f(z) > 1. Then f(z)− f(y) < −1, which contradicts the hypothesis againdue to (2). Consequently, f(x+ 0) <∞.

The result regarding f(x − 0) is obtained by a similar argument, provingthereby that f is lower regulated.

Next, suppose g is continuous but f is not regulated. Then there is a pointx in I where either (i) f(x + 0) = −∞, or (ii) f(x − 0) = ∞. First, suppose (i)holds. Since g also is continuous, given any δ > 0, there exists an η > 0 suchthat

g(y) − g(x) < δ whenever x < y < x+ η .

But by (i) we can find y in (x, x + η) such that f(y) < f(x) − 1, which con-tradicts the hypothesis as before. A similar contradiction is obtained in thecase (ii).


In the case when f ≪ g, the result follows on applying the first part to f and−f .

11.2. Theorem. Let f : I → R and g ∈ B.

(a) If f ≪− g, and either f is regulated or g is continuous, then f is of

bounded variation.

(b) If f ≪ g, then f is of bounded variation.

P r o o f. First, suppose f ≪− g and that f is regulated. Then there existsa δ > 0 such that for every finite set of nonoverlapping intervals {[ai, bi] : i =1, . . . , n} in I,

(3)∑

i

{f(bi)− f(ai)} > −1 whenever∑

i

{g(bi)− g(ai)} < δ .

Further, since g is nondecreasing, we can find a partition a = t0 < t1 < . . . <tn = b of I by including enough points of discontinuity of g in it such that

(4) g(tk − 0)− g(tk−1 + 0) < δ for k = 1, 2, . . . , n .

Now let a = x0 < x1 < . . . < xp = b be any arbitrary partition P of I whichrefines the above partition. Then for each k = 0, 1, . . . , n, tk = xik for someik ≤ p. Let S be the index set {1, . . . , p}, and set

S+ = {i ∈ S : f(xi)− f(xi−1) ≥ 0}, S− = S ∼ S+ ,

and for each k = 1, . . . , n, set Tk = {i ∈ S− : ik−1 < i ≤ ik}. Then

p∑

i=1

|f(xi)− f(xi−1)| =∑

i∈S+

{f(xi)− f(xi−1)} −∑

i∈S−

{f(xi)− f(xi−1)}

=∑

i∈S

{f(xi)− f(xi−1)} − 2∑

i∈S−

{f(xi)− f(xi−1)}

= f(b)− f(a)− 2

n∑

k=1

∑

i∈Tk

{f(xi)− f(xi−1)} .

Now, given any k = 1, . . . , n, let αk denote the sum obtained by replacingthe terms f(tk−1) and f(tk) in

∑

i∈Tk{f(xi) − f(xi−1)}, if they occur in it, by

f(tk−1+0) and f(tk − 0) respectively. Since f is regulated, it follows easily from(3) and (4), by replacing tk−1 + 0 and tk − 0 if necessary by tk−1 + 1/q andtk − 1/q respectively with q large enough and then taking the limit as q → ∞,that αk > −1. Hence

∑

i∈Tk

{f(xi)− f(xi−1)} ≥ αk − |f(tk−1 + 0)− f(tk−1)| − |f(tk)− f(tk − 0)|

> −1− ωf (tk−1)− ωf(tk) .

38 K. M. Garg

Consequently, it follows from above,p

∑

i=1

|f(xi)− f(xi−1)| ≤ |f(b)− f(a)|+ 2n∑

k=1

{1 + ωf (tk−1) + ωf (tk)}

≤ |f(b)− f(a)|+ 2n+ 4n∑

k=1

ωf (tk) .

As the last sum is independent of the choice of P, this proves that f is of boundedvariation.

The remaining parts of the theorem follow directly from the above result onaccount of the preceding lemma.

On choosing g = τ in the above theorem, we obtain

11.3. Corollary. Every LAC function f : I → R is of bounded variation.

11.4. R ema r k. We include here a simple example to show that when g ∈ Bis not continuous, a function f ≪− g is not regulated in general, and so notof bounded variation either. Let f(a) = g(a) = 0, and f(x) = 1/(a − x) andg(x) = 1 for x > a. Then it is clear that f ≪− g, but since f(a+ 0) = −∞, f isnot regulated.

12. Relative continuity and lower and upper continuities. As ordinaryabsolute continuity implies continuity, it is only natural to expect that some formsof relative continuity are implicit in the properties of relative AC, LAC and UAC.In this section we will present these relative continuities in two forms, viz. theirglobal and local forms, and investigate their equivalence.

The global forms of these relative continuities are easier to define and aresimilar to uniform continuity.

Let f : I → R and g ∈ B. We will call f uniformly continuous relative to g iffor every ε > 0 there exists a δ > 0 such that for each pair of points x, y ∈ I, if0 < y − x < δ and g(y)− g(x) < δ, then

(1) |f(y)− f(x)| < ε .

The notions of uniform lower continuity (or LC) and uniform upper continuity

(or UC) of f relative to g are defined similary by replacing the inequality (1) inthe above definition by

f(y)− f(x) > −ε or < ε respectively.

Clearly, f is uniformly continuous iff it is so relative to the identity functionτ . The function f is defined similarly to be uniformly lower continuous (LC) oruniformly upper continuous (UC) if it is so relative to τ .

It is further clear that f is uniformly continuous relative to g iff it is uniformlyLC and UC relative to g, and that f is uniformly UC relative to g iff −f isuniformly LC relative to g. Hence it is enough to consider uniform LC relativeto g.


The following result follows directly from the definitions.

12.1. Theorem. Let f : I → R and g ∈ B. If f is AC or LAC relative to g,then it is uniformly continuous or LC respectively relative to g.

We now come to the local definitions of relative continuity, LC and UC whichare found to be more useful. For this purpose we need to consider first the proper-ties of lower and upper continuities, independent of g, which are different fromlower and upper semicontinuities and have been found useful in differentiationtheory [13].

Given f : I → R and x ∈ I, let f be called

(i) lower continuous (or LC) from the left or right at x if

f(x− 0) ≤ f(x) or f(x) ≤ f (x+ 0)

respectively, provided x > a or x < b respectively, and

(ii) upper continuous (or UC) from the left or right at x if

f (x− 0) ≥ f(x) or f(x) ≥ f(x+ 0)

respectively, provided x > a or x < b respectively.

Further, f will be called LC or UC at x if it so from both the sides at x, andf will be called simply LC or UC if it is so at every point of I.

It should be noted here that f is UC from any side at x iff −f is LC fromthat side at x, and that f is continuous from any side at x iff it is LC and UCfrom that side at x. Further, f is LC at every point where Df > −∞, and everynondecreasing function is automatically LC.

In terms of the above definitions, we now define f to be continuous, LC orUC relative to g from the left or right at a point x ∈ I if it is continuous, LCor UC respectively from that side at x whenever g is continuous from the side inquestion at x.

Further, f will be called continuous, LC or UC relative to g at x if it is so fromboth the sides at x, and f will be called simply continuous, LC or UC relative to

g if it is so at every point of I.

Thus f is continuous, LC or UC relative to g iff it is continuous, LC or UCrespectively at every point x ∈ I from the side from which g is continuous. Henceif f is continuous, LC or UC, then it is so relative to every function g ∈ B, andwhen g is continuous, the converse also holds.

Also, it is clear as before that f is continuous relative to g iff it is LC and UCrelative to g, and that f is UC relative to g iff −f is LC relative to g. Hence itis again enough to consider relative LC.

12.2. Theorem. Let f : I → R and g ∈ B. If f is uniformly continuous or LCrelative to g, then it is continuous or LC respectively relative to g. Consequently ,if f ≪ g or f ≪− g, then f is continuous or LC respectively relative to g.

40 K. M. Garg

P r o o f. We need to prove here only the first part of the theorem since thesecond part follows directly from the first due to Theorem 12.1.

First, suppose f is uniformly LC relative to g but it is not LC relative to gat some point x ∈ I, say from the right. Then g is right continuous at x andf(x) > f (x + 0). Choose ε = 1

2{f(x) − f (x + 0)} which is > 0. Given anyδ > 0, since g also is right continuous at x, there clearly exists a point y ∈ I suchthat 0 < y − x < δ, g(y) − g(x) < δ and f(y) < f (x + 0) + ε = f(x) − ε, orf(y) − f(x) < −ε, which contradicts the hypothesis. A similar contradiction isobtained when f is not LC relative to g from the left at x. This proves the firstpart for uniform LC; on applying this result to −f a similar result is obtained onuniform UC, and on combining the two results the result on uniform continuityis obtained.

Next, we obtain the equivalence of global and local definitions of relativecontinuity and LC in the case when f ∈ B. For this purpose we need thefollowing lemma.

12.3. Lemma. Let f, g ∈ B. Then f is continuous or LC relative to g iff f or

f− respectively is continuous relative to g. Moreover , the same holds on replacing

f by fd, or g by g or gd.

P r o o f. It is easy to see that f is LC from any given side at a point x ∈ I ifff− is continuous from that side at x. The result on LC follows directly from thisfact. The result on continuity follows similarly from the fact that f is continuousfrom any given side at a point x ∈ I iff f is continuous from that side at x.

The concluding remark also holds since f and fd have the same parity ofcontinuity from each side, and the same holds for g, g and gd.

12.4. Theorem. Let f, g ∈ B. Then f is continuous or LC relative to g iff it

is uniformly so.

P r o o f. It is enough to prove the result for LC, for on applying this result to−f a similar result is obtained on UC, and the result on continuity follows oncombining these two results. Further, due to Theorem 12.2, we need to provehere only the necessity part of the result.

Hence suppose f is LC relative to g. Then, by the above lemma, f− is conti-nuous relative to g. To prove that f is uniformly LC relative to g, let ε > 0. Weneed to find a δ > 0 such that f(y)− f(x) > −ε whenever x, y ∈ I, 0 < y−x < δand g(y)− g(x) < δ.

Let {xn} and {yn} be the sequences of points where f− is discontinuous fromthe left or right respectively. Then, for each n,

αn ≡ f−(xn)− f−(xn − 0) > 0, βn ≡ f−(yn + 0)− f−(yn) > 0 ,

and, since f− is continuous relative to g,

γn ≡ g(xn)− g(xn − 0) > 0, δn ≡ g(yn + 0)− g(yn) > 0 .


Now since∑

n αn +∑

n βn ≤ f−(b)− f−(a) <∞, there exists an integer k suchthat

∑

n>k αn+∑

n>k βn < ε/2. Further, since f−c is uniformly continuous, there

exists an η > 0 such that

|f−c (y)− f−

c (x)| < ε/2 whenever x, y ∈ I and |y − x| < η .

Now choose δ = min{γn, δn, η : n ≤ k}. Clearly, δ > 0.

Next, let x, y be any pair of points in I for which 0 < y − x < δ and g(y) −g(x) < δ. Then for each n ≤ k, xn 6∈ (x, y] since γn ≥ δ, and yn 6∈ [x, y) sinceδn ≥ δ. Consequently, we obtain

f−(y)− f−(x) = f−c (y)− f−

c (x) + f−d (y)− f−

d (x)

<ε

2+

∑

n>k

αn +∑

n>k

βn < ε ,

so that

f(y)− f(x) = {f+(y)− f+(x)} − {f−(y)− f−(x)} > −ε .

We conclude this section with a new result on ordinary lower continuity offunctions that follows from the above theorem and Lemma 12.3 on choosing g = τ .

12.5. Corollary. A function f ∈ B is lower continuous iff it is uniformly

so, or , equivalently , iff f− is continuous.

13. Reduction theorem for relative ACs and their characterizations.

In this section we obtain a reduction theorem similar to the one in §7 whichreduces the AC and LAC of f ∈ B relative to g ∈ B to the continuous anddiscontinuous components of f and g, and also to the discontinuous, AC andcontinuous singular components of f and g. Further, characterizations of relativeAC and LAC are obtained in terms of mutual singularity with other functions.

As was the case with mutual singularity, the AC of f relative to g is notcomparable in general with that of µf relative to µg (see Remark 14.7). We willinvestigate this question in the next section. To meet the present needs we beginhere with a comparison of AC of µf relative to µg with that of f∗ relative to g∗.

13.1. Theorem. Let f, g ∈ B. Then f∗ ≪ g∗ iff µf ≪ µg.

P r o o f. Since µf = µf∗ and µg = µg∗ , we can assume f and g to be normal-ized. Then it follows from Lemma 7.1 that µf ≪ µg iff µf ≪ µg. Hence, due toTheorem 10.2, f and g can also be assumed to be nondecreasing.

First, suppose f ≪ g. Then given ε > 0, there exists a δ > 0 such thatif {[ai, bi] : i = 1, . . . , n} is any finite set of nonoverlapping intervals in I with∑n

i=1{g(bi)− g(ai)} < δ, then

(1)n∑

i=1

{f(bi)− f(ai)} <ε

2.

42 K. M. Garg

Now let E ∈ B and suppose µg(E) < δ. Then there exists, by definition, a disjointsequence of open intervals {(an, bn) : n = 1, 2, . . .} in I such that E ⊂

⋃

n(an, bn)and

∑

n{g(bn)−g(an)} < δ. Hence it follows from (1) that∑

n{f(bn)−f(an)} ≤ε/2. Consequently, by definition,

µf (E) ≤∑

n

{f(bn)− f(an)} < ε ,

which proves that µf ≪ µg .

Next, to prove the converse, suppose µf ≪ µg. Given ε > 0, then there existsa δ > 0 (for µg is finite) such that µf(E) < ε whenever E ∈ B and µg(E) < 2δ.Now let I ≡ {[ai, bi] : i = 1, . . . , n} be any finite set of nonoverlapping intervalsin I such that

∑ni=1{g(bi)− g(ai)} < δ. Let {[ci, di] : i = 1, . . . , k} be the disjoint

family of closed intervals obtained by uniting the abutting intervals in I. SetE =

⋃ki=1[ci, di]. Then since g is normalized,

µg(E) =

k∑

i=1

µg([ci, di]) =

k∑

i=1

{g(di + 0)− g(ci − 0)}

≤ 2k

∑

i=1

{g(di)− g(ci)} = 2n∑

i=1

{g(bi)− g(ai)} < 2δ .

Consequently,

n∑

i=1

{f(bi)− f(ai)} =k∑

i=1

{f(di)− f(ci)} ≤k

∑

i=1

µf ([ci, di]) = µf (E) < ε ,

which proves that f ≪ g.

Next, to prove the reduction theorem we further need the following.

13.2. Lemma. Let f, g ∈ B. If f is a jump function, then it is AC or LACrelative to g iff f is continuous or LC respectively relative to g.

P r o o f. Suppose f is a jump function. It is enough to prove the result forAC, for, by Theorem 10.2, f ≪− g iff f− ≪ g, and by Lemma 12.3, f− iscontinuous relative to g iff f is LC relative to g. Further, since f ≪ g iff f ≪ g(see Theorem 10.2), it follows from Lemma 12.3 that there is no loss of generalityin assuming f and g to be nondecreasing.

When f ≪ g, f is continuous relative to g by Theorem 12.2. Hence, to provethe converse, suppose f is continuous relative to g. Given ε > 0, let {xn} and{yn} be the points where f is discontinuous from the left or right respectively.Then, for each n,

αn ≡ f(xn)− f(xn − 0) > 0, βn ≡ f(yn + 0)− f(yn) > 0 .

Hence, by hypothesis,

γn ≡ g(xn)− g(xn − 0) > 0, δn ≡ g(yn + 0)− g(yn) > 0 .


Now since∑

n αn+∑

n βn = f(b)−f(a)<∞, there exists an integer n0 such that∑

n>n0αn+

∑

n>n0βn < ε. Choose δ to be the minimum of γn and δn for n ≤ n0.

Clearly, δ > 0. Now let {[ai, bi] : i = 1, . . . , k} be any finite set of nonoverlapping

intervals in I such that∑k

i=1{g(bi)− g(ai)} < δ. Then for each i ≤ k and n ≤ n0

it is clear that xn 6∈ (ai, bi] and yn 6∈ [ai, bi). Since f is a jump function, thus itfollows that

k∑

i=1

{f(bi)− f(ai)} ≤∑

n>n0

αn +∑

n>n0

βn < ε .

Consequently, f ≪ g.

13.3. Theorem (Reduction). Let f, g ∈ B. Then

(a) f ≪ g iff fd ≪ gd and fc ≪ gc, or , equivalently , iff fd ≪ gd, fa ≪ ga and

fcs ≪ gcs; and

(b) f ≪− g iff fd ≪− gd and fc ≪− gc, or , equivalently , iff fd ≪− gd,fa ≪− ga and fcs ≪− gcs.

P r o o f. It is enough to prove (a), for (b) follows easily from (a) with thehelp of Theorem 10.2 on using the fact that (f−)d = (fd)

−, (f−)c = (fc)−,

(f−)a = (fa)− and (f−)cs = (fcs)

−.

We can assume, as before, without loss of generality that f and g are nonde-creasing. We will first prove the equivalence of f ≪ g with fd ≪ gd and fc ≪ gc.

First, suppose f ≪ g. Then by Corollary 10.4, fd ≪ g and fc ≪ g. Now sincefd is a jump function, it follows clearly from Lemmas 2.1 and 13.2 that fd ≪ gd.To prove that fc ≪ gc, let ε > 0. Since fc ≪ g, there exists a δ > 0 such that foreach finite set of nonoverlapping intervals I ≡ {[ai, bi] : i = 1, . . . , k} in I,

(2)k∑

i=1

{fc(bi)− fc(ai)} <ε

2whenever

k∑

i=1

{g(bi)− g(ai)} < δ .

What is needed to show fc ≪ gc is however that

(3)k

∑

i=1

{fc(bi)− fc(ai)} < ε wheneverk

∑

i=1

{gc(bi)− gc(ai)} < δ .

Suppose the second inequality of (3) holds for I. Let {xn} be the points of

discontinuity of g in⋃k

i=1[ai, bi]. Then since∑

n ωg(xn) < ∞, there exists aninteger n0 such that

∑

n>n0

ωg(xn) < δ −k

∑

i=1

{gc(bi)− gc(ai)}.

Further, there exists an η > 0 such that

|fc(x)− fc(y)| <ε

2n0whenever x, y ∈ I and |x− y| ≤ 2η .

44 K. M. Garg

Now for each n ≤ n0 there is a unique integer in ≤ k such that xn ∈ [ain , bin ].Set

[cn, dn] = [ain , bin ] ∩ [xn − η, xn + η], n = 1, . . . , n0 ,

and let {[a′j , b′j ] : j = 1, . . . , k′} be the set of closures of the intervals obtained by

deleting⋃

n≤n0[cn, dn] from

⋃

i≤k[ai, bi]. Then

n0∑

n=1

{fc(dn)− fc(cn)} <n0∑

n=1

ε

2n0=ε

2,

and since

k′

∑

j=1

{g(b′j)− g(a′j)} <k

∑

i=1

{gc(bi)− gc(ai)}+∑

n>n0

ωg(xn) < δ ,

it follows from (2) that

k∑

i=1

{fc(bi)− fc(ai)}

=

n0∑

n=1

{fc(dn)− fc(cn)}+k′

∑

j=1

{fc(b′j)− fc(a

′j)} <

ε

2+ε

2= ε .

Consequently, fc ≪ gc.

Next, to prove the converse, suppose fd ≪ gd and fc ≪ gc. Then since gdand gc are clearly AC relative to g, it follows from the transitivity of the relation≪ that fd and fc are AC relative to g. Consequently, f = fd + fc ≪ g byTheorem 10.3. This proves the first equivalence.

To prove the second equivalence, it is now enough to show that fc ≪ gc ifffa ≪ ga and fcs ≪ gcs. The sufficiency part follows again from Theorem 10.3 bya similar argument. To prove the necessity suppose fc ≪ gc. Then fa and fcsare AC relative to gc by Corollary 10.4. Hence, by Theorem 13.1, µfa ≪ µgc andµfcs ≪ µgc , and by the same theorem it is enough to show that µfa ≪ µga andµfcs ≪ µgcs .

Given E ∈ B, first suppose µga(E) = 0. Then since gcs ⊥ τ by Lemma 9.1and µτ = m∗, it follows from Theorem 7.2 that µgcs ⊥ m∗. Hence there is a Boreldecomposition A ∪ B of E such that µgcs(A) = m∗(B) = 0. Now since A ⊂ E,we have by Lemma 4.1, µgc(A) = µga(A) + µgcs(A) = 0. Hence µfa(A) = 0.Further, since fa ≪ τ , µfa ≪ m∗ by Theorem 13.1. Hence µfa(B) = 0, so thatµfa(E) = µfa(A) + µfa(B) = 0. Consequently, µfa ≪ µga .

Next, suppose µgcs(E) = 0. Then since µfcs ⊥ m∗ as before, there is a Boreldecomposition A ∪ B of E such that µfcs(A) = m∗(B) = 0. Now since B ⊂ Eand µga ≪ m∗ as before, µgc(B) = µga(B) + µgcs(B) = 0 by Lemma 4.1. Henceµfcs(B) = 0, so that µfcs(E) = µfcs(A) + µfcs(B) = 0. Therefore, µfcs ≪ µgcs .


In the following theorem we obtain characterizations of relative AC and LACin terms of mutual singularity which will be needed subsequently. It may berecalled here that we use B+ to denote the set of nondecreasing functions in B.

13.4. Theorem. Let f, g ∈ B. Then

(a) f ≪ g iff for each h ∈ B, if g ⊥ h, then f ⊥ h; and(b) f ≪− g iff for each h ∈ B+, if g ⊥ h, then f ⊥− h.

P r o o f. First, to prove the necessity part of (a), suppose f ≪ g, h ∈ B andg ⊥ h. Then fd ≪ gd and fc ≪ gc by Theorem 13.3, and gd ⊥ hd and gc ⊥ hcby Theorem 7.6. It is now easy to see from Lemmas 7.4 and 13.2 that fd ⊥ hd.Further, since µfc ≪ µgc and µgc ⊥ µhc

by Theorems 13.1 and 7.2, it is clear thatµfc ⊥ µhc

. Hence fc ⊥ hc by Theorem 7.2. Consequently, f ⊥ h by Theorem 7.6.Next, to prove the sufficiency part of (a), suppose the condition holds but f

is not AC relative to g. Then, by Theorem 13.3, either (i) fd is not AC relativeto gd, or (ii) fc is not AC relative to gc.

First, suppose (i) holds. Then it follows from Lemmas 2.1 and 13.2 that there isa point x0 in I where f is discontinuous from some side from which g is continuous,say from the right. Define h(x) = 0 or 1 according as a ≤ x ≤ x0 or x0 < x ≤ brespectively. Then h ∈ B and g ⊥ h by Lemma 7.4, but f and h are not mutuallysingular by the same lemma, a contradiction. A similar argument holds in theother case.

Next, suppose (ii) holds. Then by Theorem 13.1 there exists a set A ∈ B suchthat µgc

(A) = 0 < µfc(A). Define

ν(E) = µfc(E ∩A), E ∈ B .

Then there exists a unique normalized function h ∈ B such that ν = µh. Nowsince ν(I ∼ A)=0, ν ⊥ µg, so that g ⊥ h by Theorem 7.2. But since ν agrees withµfc on A, and µfc(A) > 0, it is clear that µfc and ν are not mutually singular,and so fc and h are not mutually singular by Theorem 7.2. But since fc ⊥ fd,thus it follows from Theorem 6.1 that f and h are not mutually singular, whichagain contradicts the hypothesis.

This completes the proof of (a), and it is further clear that (a) holds also onreplacing B by B+ in (a). Hence it follows from Theorem 10.2 that f ≪− g iff foreach h ∈ B+, if g ⊥ h, then f− ⊥ h. But since h ∈ B+, by Corollary 6.4, f− ⊥ hiff f ⊥− h. Consequently, (b) holds.

14. Comparison of relative continuities and ACs with those of nor-

malizations and induced signed measures. Given f, g ∈ B, we investigatehere conditions under which the various continuities and ACs of f relative tog can be compared with the ones of f∗ relative to g∗. These properties are ofcourse not comparable in general (see Remark 14.7). The results obtained leadto a comparison of AC of f relative to g with that of µf relative to µg, and areused frequently in the subsequent chapters.

46 K. M. Garg

We need here some nomenclature. Given two regulated functions f, g : I → R,we will call f partially continuous, LC or UC relative to g if it is continuous,LC or UC respectively relative to g at the points of I0 where g is unilaterallydiscontinuous.

It is interesting to note here that in the special cases when either (i) g is nor-malized, or (ii) f and g are simultaneously left, or right, continuous, the functionf is automatically partially continuous relative to g.

We begin with relative LC.

14.1. Theorem. Let f and g be two regulated functions on I.

(a) Suppose f(x − 0) ≤ f(x + 0) for x ∈ Rg. Then if f is LC relative to g,then so is f∗ relative to g∗.

(b) Suppose f is internal and it is partially LC relative to g. Then if f∗ is

LC relative to g∗, then so is f relative to g.

P r o o f. To prove (a), suppose its hypothesis holds and f isLC relative to g. Toprove that f∗ is LC relative to g∗, it is enough to verify this at the interior pointsof I, for this holds clearly for x = a and b by Lemma 3.2(a). Hence supposex ∈ I0. Now since g∗ cannot be continuous at x from one side alone, supposex ∈ Cg∗ . Then by Lemma 3.2(c), either x ∈ Cg or x ∈ Rg . In case x ∈ Cg, thensince f is LC relative to g, it follows from Lemma 3.2 that

f∗(x− 0) = f(x− 0) ≤ f(x) ≤ f(x+ 0) = f∗(x+ 0) .

Hence it follows from the definition of f∗ that it is LC at x. If on the other handx ∈ Rg, the same follows from the hypothesis. Consequently, f∗ is LC relativeto g∗.

Next, to prove (b), suppose its hypothesis holds and f∗ is LC relative to g∗.To prove that f is LC relative to g, we need to verify this only at the pointsx ∈ I0 where g is continuous from both sides, for this holds clearly as beforewhen x = a or b. Hence suppose x is such a point. Then since g∗ is continuousat x,

f(x− 0) = f∗(x− 0) ≤ f∗(x) ≤ f∗(x+ 0) = f(x+ 0) .

But since f is internal, this clearly implies that f is LC at x.

The following thoerem deals with LAC.

14.2. Theorem. Let f, g ∈ B.

(a) Suppose f(x− 0) ≤ f(x+0) for x ∈ Rg. Then if f ≪− g, then f∗ ≪− g∗.

(b) Suppose f is internal and it is partially LC relative to g. Then if f∗ ≪−

g∗, then f ≪− g.

P r o o f. To prove (a), suppose its hypothesis holds and f ≪− g. Then f is LCrelative to g by Theorem 12.2, and so f∗ is LC relative to g∗ by Theorem 14.1.It is now clear that (f∗)d is LC relative to (g∗)d, and hence (f∗)d ≪− (g∗)d by


Lemma 13.2. Further, due to Theorem 13.3 (and Lemma 3.3), (f∗)c = fc ≪−

gc = (g∗)c, and so it follows from the same theorem that f∗ ≪− g∗.

Next, to prove (b), suppose its hypothesis holds and f∗ ≪− g∗. Then it followsfrom Theorems 12.2 and 14.1 that f is LC relative to g. Consequently, fd is LCrelative to gd, and so fd ≪− gd by Lemma 13.2. Further, by Theorem 13.3, fc =(f∗)c ≪− (g∗)c = gc, and so it follows from the same theorem that f ≪− g.

The next two theorems dealing with relative continuity and AC are obtainedfrom Lemmas 3.2 and 13.2 and Theorems 12.2 and 13.3 by similar arguments.

14.3. Theorem. Let f and g be two regulated functions on I.

(a) Suppose f(x− 0) = f(x+ 0) for x ∈ Rg. Then if f is continuous relative

to g, then so is f∗ relative to g∗.

(b) Suppose Rf ∩Cg = ∅ and that f is partially continuous relative to g. Then

if f∗ is continuous relative to g∗, then so is f relative to g.


(a) Suppose f(x− 0) = f(x+ 0) for x ∈ Rg. Then if f ≪ g, then f∗ ≪ g∗.

(b) Suppose Rf ∩Cg = ∅ and that f is partially continuous relative to g. Then

if f∗ ≪ g∗, then f ≪ g.

When f and g are simultaneously left, or right, continuous, it is easy to seethat all the continuity hypotheses of the above four theorems hold automatically.Hence in that case we obtain

14.5. Corollary. Suppose f and g are two regulated functions on I which are

simultaneously left , or right , continuous. Then f is continuous or LC relative to

g iff f∗ is so relative to g∗. Moreover , if f, g ∈ B, then f is AC or LAC relative

to g iff f∗ is so relative to g∗.

On account of Theorem 13.1, we further obtain from Theorem 14.4 the follo-wing result comparing AC of f relative to g with that of µf relative to µg.

14.6. Corollary. Let f, g ∈ B. If f ≪ g and f(x−0) = f(x+0) for x ∈ Rg ,then µf ≪ µg. Conversely , if µf ≪ µg, Rf ∩Cg = ∅ and f is partially continuous

relative to g, then f ≪ g.

Consequently , if f and g are simultaneously left , or right , continuous, thenf ≪ g iff µf ≪ µg.

14.7. R em a r k. It is easy to see that the continuities and ACs of f ∈ Brelative to g ∈ B are not comparable in general with the ones of f∗ relative tog∗, or of µf relative to µg. Let a < c < b, and let us recall here that for any setE ⊂ I, χE denotes the characteristic function of E on I.

First, define f = χ[a,c] and g = χ{c}. Then f is clearly continuous and ACrelative to g, but since g∗ ≡ 0, f∗ is not LC or LAC relative to g∗. This alsoshows the necessity of the hypotheses in the first parts of the above four theorems.

48 K. M. Garg

Next, define f = χ{c} and g ≡ 0. Then f∗ ≡ 0 is trivially continuous and ACrelative to g∗, but since f(c) = 1 > 0 = f(c+ 0), f is not LC or LAC relative tog. This in turn shows the necessity of the hypotheses in the second parts of theabove theorems.

Further, due to Theorem 13.1, the above two examples show also that therelation f ≪ g is not comparable in general with µf ≪ µg.

15. Relative ACs in terms of derivatives. Given f, g ∈ B, in this sec-tion we obtain characterizations of AC and LAC of f relative to g in terms ofderivatives of f and g.

If P and Q are any two pointwise propositions, we shall say that P holds for

almost all x for which Q holds provided the set of points in I where Q holdsbut P does not hold is of measure zero.

We begin with the characterization of AC.

15.1. Theorem. Let f, g ∈ B. Then f ≪ g iff the following conditions hold :

(a) f is continuous relative to g,(b) f ′(x) = 0 for almost all x for which g′(x) = 0,(c) |f(∆∞

f ∼ ∆∞g )| = 0, and

(d) |f(E)| = 0 whenever E ⊂ ∆∞f ∩∆∞

g and |g(E)| = 0.

P r o o f. On account of Theorem 13.3, f ≪ g iff (i)fd ≪ gd, (ii) fa ≪ ga and(iii) fcs ≪ gcs. However, (i)⇔(a) by Lemmas 2.1 and 13.2, and since f ′

a = f ′ andg′a = g′ a.e., it follows easily from Lemma 9.1 and Theorem 13.4 that (ii)⇔(b).Hence it is enough to show that (iii) is equivalent to (c) and (d) together.

Set C = Cf ∩ Cg and ∆ = C ∩ ∆∞f . We will first show that (iii) holds iff

(iv) µf ≪ µg on ∆. It is clear from Theorems 10.2 and 13.1 that (iii) holdsiff µfcs ≪ µgcs

. However, since I ∼ C is countable, µfcs(I ∼ C) = 0, andµfcs(C ∼ ∆∞

fcs) = 0 by Theorems 2.2 and 2.3. Further, since ∆∞

fcs⊂ ∆∞

f, it

follows from Theorem 4.2 that

µfcs(C ∩∆∞fcs

∼ ∆∞f ) ≤ µf (C ∩∆∞

f ∼ ∆∞f ) = |f(∆∞

f ∼ ∆∞f )| = 0 .

Thus µfcs(I ∼ ∆) = 0. Hence µfcs ≪ µgcsiff this holds on ∆. But since |∆| = 0,

it is clear from Lemma 9.2 and Theorem 2.3 that µfcs and µgcsagree with µf and

µg respectively on ∆. Consequently, it is enough to show that (iv) is equivalentto (c) and (d) together.

First, suppose (iv) holds, i.e. µf ≪ µg on ∆. Let A = ∆∞f ∼ ∆∞

g , and setB = A ∩ C, B1 = B ∩∆∞

g and B2 = B ∼ B1. Since B1 ⊂ Cg ∩ (∆∞g ∼ ∆∞

g ), itfollows from Theorem 4.2 that µg(B1) = |g(B1)| = 0. Further, since µga

≪ m∗

and |B2| = 0, it follows from Theorems 2.2 and 2.3 that

µg(B2) = µga(B2) + µgcs

(B2) = 0 .

Hence µg(B) = 0, so that |f(B)| = µf (B) = 0. Now since A ∼ B is countable,

we thus have |f(A)| = 0, i.e. (c) holds.


Further, let E ⊂ ∆∞f ∩∆∞

g and suppose |g(E)| = 0. Set F = E ∩ C. Then

µg(F ) = 0 by Theorem 2.3, so that |f(F )| = µf (F ) = 0. Now since E ∼ F is

countable, we thus have |f(E)| = 0, so that (d) also holds.Next, to prove the converse, suppose (c) and (d) hold. Let G be a subset of

∆ such that µg(G) = 0. Set G1 = G ∼ ∆∞g and G2 = G∩∆∞

g . Then |f(G1)| = 0

by (c), and since |g(G2)| = 0, |f(G2)| = 0 by (d). Thus by Theorem 2.3, µf (G) =

|f(G)| = 0, which proves that µf ≪ µg on ∆.

The following theorem deals with the characterization of relative LAC.

15.2. Theorem. Let f, g ∈ B. Then f ≪− g iff the following conditions hold :

(a) f is LC relative to g,(b) f ′(x) ≥ 0 for almost all x for which g′(x) = 0,(c) |f−(∆−∞

f ∼ ∆∞g )| = 0, and

(d) |f−(E)| = 0 whenever E ⊂ ∆−∞f ∩∆∞

g and |g(E)| = 0.

P r o o f. Since f ≪− g iff f− ≪ g (see Theorem 10.2), the result is obtained,with the help of Lemma 12.3 and Theorem 4.2, on applying Theorem 15.1 to f−

and g. For, by Theorem 4.2,

|f−(∆∞f− ∼ ∆−∞

f )| = |f−(∆−∞f ∼ ∆∞

f−)| = 0 .

In each of the above two theorems, the conditions (a), (c) and (d) clearlybecome redundant when f is AC. Hence in that case the AC and LAC of frelative to g are totally determined by the derivatives of f and g, as suggested bythe title of this section, as follows:

15.3. Corollary. Given f, g ∈ B, suppose f is AC. Then

(a) f ≪ g iff f ′(x) = 0 for almost all x for which g′(x) = 0, and(b) f ≪− g iff f ′(x) ≥ 0 for almost all x for which g′(x) = 0.

Since f ∈ B is AC or LAC iff it is so relative to τ , we obtain further from theabove two theorems

15.4. Corollary. A function f ∈ B is

(a) AC iff it is continuous and |f(∆∞f )| = 0,

(b) LAC iff it is LC and |f−(∆−∞f )| = 0.

Now, with the help of this corollary, we obtain from the two theorems

15.5. Corollary. Let f, g ∈ B.

(a) Given f ≪ g, if g is continuous, AC or singular , then so is f .(b) If f ≪− g and g is continuous, AC or singular , then f is LC, LAC or

LS respectively.

15.6. R ema r k. Given f, g ∈ B, let f be called weakly absolutely continuous

(or WAC) relative to g if for each ε > 0 there exists a δ > 0 such that if {[ai, bi] :i = 1, . . . , n} is any finite set of mutually disjoint closed intervals in I such that g

50 K. M. Garg

is continuous at each ai and bi and∑n

i=1{g(bi)− g(ai)} < δ, then∑n

i=1{f(bi)−f(ai)} < ε. Clearly, WAC is weaker than AC. Let us write f ≪w g when f isWAC relative to g.

For I=[0, 1], Kober [23] called g a “covariation” of f if it is nondecreasing andf≪w g. He obtained a result on covariance similar to the part (a) of Theorem13.4(see [23], pp. 568, 575).

When any of the functions f and g is continuous, it is easy to see that f ≪w giff f ≪ g. Further, when f is a jump function, it follows by an argument similarto the one given for Lemma 13.2 that f ≪w g iff (a′) f is continuous at eachpoint where g is (bilaterally) continuous. Hence all the results on relative AC in§§10, 13 and 15 can be generalized by similar arguments to WAC, and so alsoto covariance. In particular, Theorem 15.1 leads to a characterization of WAC,and hence of covariance when g is nondecreasing, by replacing its condition (a)by (a′). Some of the results on covariance in [23] follow immediately from thischaracterization.

IV. Normalized relative derivative

16. Existence of normalized relative derivative. Given f, g ∈ B, inthis section we define a normalized version of derivative of f relative to g whichexists µg-a.e., and establish its summability relative to µg. A characterizationof relative AC will be obtained in the next section in terms of this normalizedrelative derivative.

Let f, g : I → R and x be any point of I such that g is not constant on anyneighbourhood of x. Then the lower and upper derivates of f relative to g at xare defined (see e.g. [34], p. 108) to be the lower and upper limits of the ratio

f(x+ h)− f(x)

g(x+ h)− g(x)

as h → 0 through those values for which x + h ∈ I and g(x + h) 6= g(x). Wewill use Dgf(x) and Dgf(x) to denote these two derivates respectively, and whenthey are equal, their common value is called the derivative of f relative to g at x,and is denoted by f ′

g(x).

If g ∈ B and x is any point of I where g has a non-removable discontinuity,it is clear that µg({x}) > 0, but the derivative f ′

g does not always exist at x.Hence to obtain the results of this chapter in full generality it is found necessaryto modify this relative derivative.

Now suppose f and g are regulated. Then on replacing f and g by theirnormalizations, when f∗ has a derivative relative to g∗ at some point x ∈ I, wewill call this derivative the normalized derivative of f relative to g at x, and denoteit by D∗

gf(x).


We begin with the existence of normalized relative derivative at all the pointsof non-removable discontinuity of g, or, equivalently, the points of discontinuityof g∗ (see Lemma 3.2).

16.1. Lemma. Suppose f, g : I → R are regulated , and let A = I ∼ Cg∗ . Then

D∗gf exists at each point x ∈ A, where

D∗gf(x) =

f(a+ 0)− f(a)

g(a+ 0)− g(a)if x = a,

f(x+ 0)− f(x− 0)

g(x+ 0)− g(x− 0)if a < x < b,

f(b)− f(b− 0)

g(b) − g(b− 0)if x = b.

Moreover , if f, g ∈ B, then D∗gf is µg-summable on A and

µf (E) =∫

E

D∗gf dµg for E ⊂ A .

P r o o f. The first part follows easily from the definitions of f∗ and g∗ given in§3 and Lemma 3.2(a).

Further, when f, g ∈ B, it is clear from the first part that

D∗gf(x) = µf ({x})/µg({x}) for each x ∈ A .

Now since A is countable, the second part follows from this relation on using thecountable additivity of µf and µg.

16.2. Lemma. Let f, g ∈ B, where f is increasing and g is nondecreasing.

Suppose E ⊂ Cf ∩ Cg and α ≥ 0.

(a) If Dgf(x) ≥ α for each x ∈ E, then αµg(E) ≤ µf(E).(b) If Dgf(x) ≤ α for each x ∈ E, then µf (E) ≤ αµg(E).

P r o o f. First, to prove (a), suppose Dgf(x) ≥ α for each x ∈ E. We canassume here α > 0, for otherwise the result holds trivially. Let C be the countableset of values that g assumes more than once, and set A = g−1(C). Then µg(E ∩A) = |g(E ∩ A)| ≤ |C| = 0 (see Theorem 2.3). Hence E can be assumed not tocontain any point of A.

Let ε > 0 and 0 < β < α. Choose an open set U ⊃ f(E) such that |U | <|f(E)|+ε. Then for each x ∈ E, since f is continuous at x, there exists a sequenceof points {yx,n} in I with limit x such that for each n, Ix,n ≡ co{f(x), f(yx,n)} ⊂U and

f(yx,n)− f(x)

g(yx,n)− g(x)> β .

Now set Jx,n = co{g(x), g(yx,n)} for each n. Then |Ix,n| > β|Jx,n| for each n, andsince x ∈ Cg, |Jx,n| → 0 as n → ∞. Thus {Jx,n : x ∈ E, n = 1, 2 . . .} is a Vitali

52 K. M. Garg

covering of g(E), and so it contains a disjoint sequence {Jxi,ni: i = 1, 2, . . .} such

that |g(E) ∼⋃

i Jxi,ni| = 0. Consequently,

|g(E)| ≤∑

i

|Jxi,ni| <

1

β

∑

i

|Ixi,ni| .

But since f is increasing, and the intervals {co{xi, yxi,ni} : i = 1, 2, . . .} are

obviously disjoint, the intervals {Ixi,ni: i = 1, 2, . . .} also are disjoint. Hence

β|g(E)| <∣

∣

∣

⋃

Ixi,ni

∣

∣

∣≤ |U | < |f(E)|+ ε .

Now on making ε → 0, and then β → α, it follows that α|g(E)| ≤ |f(E)|.Consequently, by Theorem 2.3, αµg(E) ≤ µf (E), which proves (a).

Next, to prove (b), suppose Dgf(x) ≤ α for each x ∈ E. Let ε > 0 and β > α.Choose this time an open set U ⊃ g(E) such that |U | < |g(E)| + ε. Then foreach x ∈ E there is a sequence {yx,n} which converges to x such that for eachn, Jx,n ≡ co{g(x), g(yx,n)} ⊂ U and

f(yx,n)− f(x)

g(yx,n)− g(x)< β .

Now set Ix,n = co{f(x), f(yx,n)} for each n. Then |Ix,n| < β|Jx,n| for each n, andsince x ∈ Cf , |Ix,n| → 0 as n → ∞. Thus {Ix,n : x ∈ E, n = 1, 2, . . .} is a Vitalicovering of f(E), and so it contains a disjoint sequence {Ixi,ni

: i = 1, 2, . . .} suchthat |f(E) ∼

⋃

i Ixi,ni| = 0. Thus

|f(E)| ≤∑

i

|Ixi,ni| < β

∑

i

|Jxi,ni| .

But since f is increasing, the intervals co{xi, yxi,ni}, i = 1, 2, . . . , are disjoint,

and hence the intervals {Jxi,ni} are at least nonoverlapping. Consequently,

|f(E)| < β∣

∣

∣

⋃

Jxi,ni

∣

∣

∣≤ β|U | < β{|g(E)| + ε} .

Now on making ε → 0, and then β → α, it follows that |f(E)| ≤ α|g(E)|.Consequently, by Theorem 2.3, µf (E) ≤ αµg(E).

16.3. Lemma. Let f ∈ B and g ∈ B+. Then there exists a Borel set P ⊂ Cg

such that µg(Cg ∼ P ) = 0, f has a finite derivative relative to g at the points of

P , and

(1) µf (E) =∫

E

f ′g dµg for E ∈ B, E ⊂ P .

P r o o f. It is enough to prove the result when f is increasing. For, the resultfor a general f follows on applying this particular result to the increasing functions

f1(x) = f+(x) + f(a) + x, f2(x) = f−(x) + x, x ∈ I ,

since f = f1 − f2 and µf = µf1 − µf2 .


Hence suppose f is increasing. Let C = Cf ∩Cg. Since Cg ∼ C = Cg ∼ Cf iscountable, µg(Cg ∼ C) = 0. Let A denote the set of points in C where f ′

g doesnot exist. Given any pair of rational numbers α, β, α < β, let

Aα,β = {x ∈ C : Dgf(x) ≤ α < β ≤ Dgf(x)} .

Since A is the union of all such sets, it is enough to show that µg(Aα,β) = 0. Butby the above lemma,

µf (Aα,β) ≤ αµg(Aα,β) ≤ βµg(Aα,β) ≤ µf (Aα,β) ,

which holds only if µg(Aα,β) = 0. Consequently, µg(A) = 0.Now, let B denote the set of points in C ∼ A where f ′

g = ∞. Given anypositive integer n, it follows from Lemma 16.2 that nµg(B) ≤ µf (B), i.e. µg(B) ≤(1/n)µf (B). Consequently, µg(B) = 0. Thus if N = A ∪B ∪ (Cg ∼ C), we haveproved that µg(N) = 0. Hence there exists a set N1 ∈ B such that N ⊂ N1

and µg(N1) = 0. Then P ≡ Cg ∼ N1 ∈ B, µg(Cg ∼ P ) = 0, and f has a finitederivative relative to g at each point of P .

Next, we claim that f ′g is B-measurable on P , or, equivalently, on P0 = P ∼

{b}. Define f(x) = f(b) and g(x) = g(b) for x > b, and for each positive integern, define

ϕn(x) =f(x+ 1/n)− f(x)

g(x+ 1/n)− g(x), x ∈ P0 .

Then {ϕn} is a sequence of B-measurable functions which converges pointwise tof ′g on P0. Hence the claim.

Now, to prove (1), let E ∈ B, E ⊂ P , and let n be any positive integer. Set

En = {x ∈ E : f ′g(x) < n} .

Given ε > 0, let 0 = α0 < α1 < . . . < αk = n be a partition of the interval [0, n]such that αi − αi−1 < ε for each i. Set

En,i = {x ∈ En : αi−1 ≤ f ′g(x) < αi}, i = 1, . . . , k .

Then for each i, En,i is B-measurable, and we obtain from Lemma 16.2

αi−1µg(En,i) ≤ µf (En,i) ≤ αiµg(En,i) .

Hence,

|µf (En,i)− αiµg(En,i)| ≤ εµg(En,i) ,

and so on using finite additivity of µf and µg on B, we obtain

∣

∣

∣µf (En)−

k∑

i=1

αiµg(En,i)∣

∣

∣≤

k∑

i=1

|µf (En,i)− αiµg(En,i)| ≤ εµg(En) .

Now on making ε→ 0, we obtain

µf (En) =∫

En

f ′g dµg ,

and (1) follows from this equation on making n→ ∞.

54 K. M. Garg

16.4. Theorem. Given f, g ∈ B, there exists a Borel set P ⊂ I such that

µg(I ∼ P ) = 0, f has a finite normalized derivative relative to g at the points of

P , and

(2) µf (E) =∫

E

D∗gf dµg for E ∈ B, E ⊂ P .

Consequently , D∗gf is µg-summable on I. Furthermore, D∗

gf = f ′g µg-a.e. in Cg.

P r o o f. On applying Lemma 16.3 to the pairs of functions f , g and g, g itfollows that there exists a Borel set A ⊂ Cg = Cg such that µg(Cg ∼ A) = 0, f ′

g

and g′g exist and are finite at the points of A, and if E ∈ B and E ⊂ A, then

(3) µf (E) =∫

E

f ′g dµg and µg(E) =

∫

E

g′g dµg .

Let (E+, E−) be a Hahn decomposition for µg, and set A+ = A ∩ E+ andA− = A ∩ E−. Since A+ ⊂ Cg, it follows from Lemma 7.1 that µg = µg = µg onBorel subsets of A+. Hence, by (3),

µg(E) =∫

E

g′g dµg for E ∈ B, E ⊂ A+ .

Consequently, g′g = 1 µg-a.e. in A+. Thus there exists a Borel set B+ ⊂ A+ suchthat µg(A+ ∼ B+) = 0 and g′g = 1 on B+. Now, given x ∈ B+, since

limy→x

f(y)− f(x)

g(y)− g(x)= lim

y→x

f(y)− f(x)

g(y)− g(x)

{

limy→x

g(y)− g(x)

g(y)− g(x)

}−1

= f ′g(x) ,

f ′g(x) exists and is equal to f ′

g(x). Hence, by (3),

(4) µf (E) =∫

E

f ′g dµg for E ∈ B, E ⊂ B+ .

Using a similar argument we obtain a Borel subset B− of A− such thatµg(A− ∼ B−) = 0, f ′

g = −f ′g on B−, and such that (4) holds for every Borel sub-

set E of B−. Consequently, (4) holds for every Borel subset E of B ≡ B+ ∪B−.Further, B ∈ B, B ⊂ Cg, µg(Cg ∼ B) = µg(Cg ∼ B) = 0 (see Lemma 7.1), andf ′g exists and is finite on B.

Next, on replacing f and g in the result just established for B by f∗ and g∗

respectively, we obtain a Borel set B∗ ⊂ Cg∗ such that µg(Cg∗ ∼ B∗) = 0, D∗gf

exists and is finite on B∗, and such that

µf (E) =∫

E

D∗gf dµg for E ∈ B, E ⊂ B∗ .

Now set P = B∗ ∪ (I ∼ Cg∗). Then it follows from Lemma 16.1 that P has allthe required properties. Also, since µg(I ∼ P ) = 0 and µf (P ) < ∞, it is clearfrom (2) that D∗

gf is µg-summable on I.


Now set Q = B ∩B∗. It is then clear that µg(Cg ∼ Q) = 0, and that

∫

E

f ′g dµg =

∫

E

D∗gf dµg for E ∈ B, E ⊂ Q .

Consequently, f ′g = D∗

gf µg-a.e. in Q, and hence also in Cg.

16.5. R ema r k. In connection with Theorem 16.4 it is natural to ask whetherµg(I ∼ P ) also is zero. This does hold in the case when Rg = ∅ but not in general.For if Rg = ∅, then g is continuous on Cg∗ (see Lemma 3.2), so that µg = µg onCg∗ by Lemma 7.1, and as I ∼ P ⊂ Cg∗ (see the above proof), it follows fromTheorem 16.4 that µg(I ∼ P ) = 0. But if Rg 6= ∅, say x ∈ Rg , then clearlyµg({x}) > 0, and in case f∗ is discontinuous at x, it is clear that f∗ cannot havea finite derivative relative to g∗ at x.

17. Relative AC in terms of LS-integral and a Radon–Nikodym the-

orem for such integrals. In this section we obtain from the results of §16two versions of a Radon–Nikodym theorem for Lebesgue–Stieltjes integral (orLS-integral) which provide characterizations of relative AC in the case of norma-lized functions of bounded variation.

17.1. Theorem. Let f, g ∈ B. Then f∗ ≪ g∗ iff

(1) µf(E) =∫

E

D∗gf dµg , E ∈ B .

P r o o f. Since, by Theorem 13.1, f∗ ≪ g∗ iff µf ≪ µg, the result followsclearly from Theorem 16.4.

Thus if f∗ ≪ g∗, then the normalized relative derivative D∗gf is the Radon–

Nikodym derivative of µf relative to µg.

In the case when g is continuous the above theorem holds also for the ordinaryrelative derivative, for in that case the following result is obtained from the abovetheorem with the help of Theorem 16.4.

17.2. Corollary. Let f, g ∈ B, and suppose g is continuous. Then f∗ ≪ giff

µf (E) =∫

E

f ′g dµg, E ∈ B .

Following is another consequence of the above theorem which will be usedsubsequently.

17.3. Corollary. Given f, g ∈ B, suppose f∗ ≪ g∗. Then

µf (E) =∫

E

|D∗gf | dµg, E ∈ B .

56 K. M. Garg

Consequently , if f is internal , then

V f =∫

I

|D∗gf | dµg.

For, the first part follows clearly from (1) on using the Hahn decompositionfor µg, and the second part follows from the first with the help of Lemma 7.1.

The last part of the above corollary is well known in the case when g is theidentity function τ (see e.g. [28], p. 259).

There is another form of relative derivative for which also the above theoremholds.

Given f, g ∈ B, define f(x) = f(a) for x < a and f(x) = f(b) for x > b, andextend g similary to R. Then for each x ∈ I, if the ratio

f(x+ h)− f(x− h)

g(x+ h)− g(x− h)

has a limit as h→ 0 from the right, we will call it the symmetric derivative of frelative to g at x, and denote it by Ds

gf(x).

The following theorem has been obtained by P. J. Daniell [4] in the case whenf and g are continuous (see Theorem 13.1 and Remark 17.5).

17.4. Theorem. Let f, g ∈ B. Then f∗ ≪ g∗ iff

µf(E) =∫

E

Dsgf dµg, E ∈ B .

P r o o f. It is enough to prove here that D∗gf = Ds

gf µg-a.e. At each pointx ∈ I ∼ Cg∗ , it follows easily from Lemma 16.1 that D∗

gf(x) = Dsgf(x). Also,

since Cg∗ ∼ Cg = Rg (see Lemma 3.2) is countable, and g∗ is continuous at thepoints of Rg, it is clear that µg(Cg∗ ∼ Cg) = 0. Now since D∗

gf = f ′g µg-a.e. in

Cg (see Theorem 16.4), it is thus enough to show that Dsgf = f ′

g µg-a.e. in Cg.

Let A be the set of points in Cg where f ′g exists, and B be the set of points

in Cg where g has a nonzero derivative. Set E = A ∩ B. Then µg(Cg ∼ A) =0 by Theorem 16.4, and it follows clearly from the decomposition theorem ofde La Vallee Poussin (see [34], p. 127) that µg(Cg ∼ B) = 0. Consequently,µg(Cg ∼ E) = 0. Now let x ∈ E. If x = a or b, it is obvious that Ds

gf(x) = f ′g(x).

Hence suppose a < x < b. Set α = f ′g(x). Then given ε > 0, since g′(x) is

either > 0 or < 0, we can find a δ > 0 such that if 0 < h < δ, then x ± h ∈ I,|{f(x± h)− f(x)}/{g(x ± h)− g(x)} − α| < ε, and either (i) g(x− h) < g(x) <g(x+ h), or (ii) g(x− h) > g(x) > g(x+ h). Then

f(x+ h)− f(x− h)

g(x+ h)− g(x− h)=f(x+ h)− f(x)

g(x+ h)− g(x)·

g(x + h)− g(x)

g(x+ h)− g(x− h)

+f(x)− f(x− h)

g(x)− g(x− h)·

g(x) − g(x− h)

g(x+ h)− g(x− h)


is in each of the two cases a convex sum of {f(x+ h)− f(x)}/{g(x + h)− g(x)}and {f(x− h)− f(x)}/{g(x− h)− g(x)}, and so is in the interval (α− ε, α+ ε).Consequently, Ds

gf(x) exists and is equal to f ′g(x).

17.5. R ema r k. Theorem 17.4 has been obtained by Daniell [4] for the follo-wing weaker form of symmetric derivative which we denote by WDs

gf :

WDsgf(x) = lim

h→0+

f(x+ h+ 0)− f(x− h− 0)

g(x+ h+ 0)− g(x− h− 0).

In fact, Daniell stated his theorem for Dsgf , but from his proof the result is

obtained only forWDsgf (see [4], p. 359, line 8). However, WDs

gf clearly coincideswith Ds

gf when f and g are continuous, and so in that case Theorem 17.4 followsfrom Daniell’s proof. Also, some of the ideas in the proof of Theorem 16.4 havebeen adapted from Daniell’s proof.

18. The fundamental theorem of calculus for LS-integral. In thissection we obtain a fundamental theorem of calculus for LS-integral (see The-orem 18.2) similar to the well-known theorem of Lebesgue on the L-integral.The proof of this theorem is somewhat involved since it requires a splitting ofthe atoms of µf and µg. We include here another version of the theorem whichholds in terms of the relative AC introduced earlier (see Theorem 18.7). Parts ofthese two theorems have been obtained by Lebesgue [26] in particular cases (seeRemark 18.12).

Given g∈B and a µg-summable function ϕ : I→R, the indefinite LS-integralof ϕ relative to g, which is defined for x > a in terms of the restriction of g tothe interval [a, x], is given by

(1)x∫

a

ϕdg =

{

0 if x = a,∫

[a,x)ϕdµg + ϕ(x){g(x) − g(x− 0)} if x > a.

To obtain a characterization of this indefinite integral in full generality we needhere some new teminology.

Given a regulated function f : I → R, let If denote the set of points where fis internal, and Ef = I ∼ If . Then Ef is the set of points x in I0 where f hasan external saltus, i.e. either

(i) f(x) > max{f(x+ 0), f(x− 0)}, or(ii) f(x) < min{f(x+ 0), f(x− 0)}.

We define fi(x) = f(x) when x ∈ If , and fi(x) = max{f(x + 0), f(x − 0)}or min{f(x+ 0), f(x − 0)} when (i) or (ii) respectively hold. Further, we definefe = f−fi. It is clear that fi is an internal function, and that fe is a jump functionwhich has only removable discontinuities. We will call fi and fe the internal andexternal parts respectively of f , and any function with the properties of fe willbe called an external function. Clearly, the decomposition f = fi + fe of f intointernal and external functions is unique.

58 K. M. Garg

Let us add here a few observations about this decomposition which will beused in the sequel. Given x∈I, it is clear that fi(x+0) = f(x+0) for x< b, andfi(x− 0) = f(x− 0) for x > a. Consequently, f∗

i ≡ (fi)∗ = f∗, and so if f ∈ B,

then µfi = µf . Further, the function fe has everywhere a zero limit, and henceµfe ≡ 0.

Next, given g ∈ B, if a regulated function f : I → R is continuous relative tog and fi ≪ gi, we will call f internally absolutely continuous relative to g, andwrite f ≪i g. It is easy to see that the relation “f ≪i g” is not comparable with“f ≪ g” in general.

Further, any regulated function f : I → R will be said to satisfy Lebesgue’scondition relative to g, or simply the condition (Lg), if at every point x ∈ I0

where g is discontinuous from both sides,

(2)f(x+ 0)− f(x)

g(x+ 0)− g(x)=f(x− 0)− f(x)

g(x− 0)− g(x).

Lebesgue recognized the necessity of this condition for f to be an indefiniteLS-integral relative to g (see [26], pp. 285, 288).

Finally, it is necessary to extend here the definition of D∗gf to the points of

removable discontinuity of g. Hence we define

Dgf(x) =

{

f ′g(x) if it exists and x ∈ Rg,D∗

gf(x) if it exists and x 6∈ Rg.

The derivative Dgf(x) may be called the extended normalized derivative of frelative to g at x.

18.1. Lemma. Given f : I → R and g ∈ B, suppose f satisfies the condition

(Lg). Then at every point x ∈ I where g is discontinuous from both sides,

(3) Dgf(x) = f ′g(x) =

f(x+ 0)− f(x)

g(x+ 0)− g(x)=f(x− 0)− f(x)

g(x− 0)− g(x).

P r o o f. Suppose g is discontinuous from both sides at x. Then it followsdirectly from (2) that f ′

g(x) exists and the last two equalities of (3) hold. The firstequality needs to be proved only when x 6∈ Rg, for then Dgf(x) is by definitionD∗

gf(x). However, from the last two equalities of (3) we obtain

f(x+ 0)− f(x− 0) = {f(x+ 0)− f(x)} − {f(x− 0)− f(x)}

= f ′g(x){g(x + 0)− g(x)} − f ′

g(x){g(x − 0)− g(x)}

= f ′g(x){g(x + 0)− g(x − 0)},

and so when x 6∈ Rg, it follows from Lemma 16.1 that D∗gf(x) = f ′

g(x).

18.2. Theorem (Fundamental). Let f : I → R and g ∈ B. Then the following

are equivalent :

(a) f is an indefinite LS-integral relative to g,(b) f ≪i g and f satisfies the condition (Lg),(c) f(x) = f(a) +

∫ x

aDgf dg, x ∈ I.


P r o o f. It is enough to prove here the implications (a)⇒(b) and (b)⇒(c).

To prove the first implication, suppose (a) holds. Then there exists a µg-summable function ϕ such that

f(x) = f(a) +x∫

a

ϕdg, x ∈ I .

Define

(4) f1(x) = f(a) +x∫

a

ϕdgi and f2(x) =x∫

a

ϕdge, x ∈ I .

It is then clear that f = f1 + f2, where f2 is an external function.

We will first prove that f1 = fi ≪ gi. Define

ν(E) =∫

E

ϕdµg , E ∈ B ,

and let k be the unique normalized function in B for which µk = ν and k(a) =f(a). We claim that k = f∗

1 .

Given x ∈ I, first suppose x < b. Then since µgi = µg, we obtain from (1)and (4),

f1(x+ 0) = f(a) + limh→0+

x+h∫

a

ϕdgi = f(a) + limh→0+

∫

[a,x+h)

ϕdµg

+ limh→0+

ϕ(x+ h){gi(x+ h)− gi(x+ h− 0)} .

But since ϕ is µg-summable, and µgi= µgi = µg since gi is internal (see

Lemma 7.1),

limh→0+

|ϕ(x+ h){gi(x+ h)− gi(x+ h− 0)}| ≤ limh→0+

∫

(x,x+h]

|ϕ| dµg = 0 .

Hence we obtain

(5) f1(x+ 0) = f(a) +∫

[a,x]

ϕdµg = f(a) + ν([a, x]) = k(x+ 0) .

Also, when x > a, we obtain by a similar argument,

(6) f1(x− 0) = f(a) +∫

[a,x)

ϕdµg = k(x− 0) .

Thus when x ∈ I0, it follows from (5) and (6) that f∗1 (x) = k∗(x) = k(x). As this

equation clearly holds for x = a and b, this proves that k = f∗1 .

60 K. M. Garg

Now, given x ∈ I, when x < b, we obtain from (1), (4) and (5),

(7) f1(x+ 0)− f1(x) =∫

[a,x]

ϕdµg −x∫

a

ϕdgi = ϕ(x){gi(x+ 0)− gi(x)} ,

and when x > a, we obtain similarly from (1), (4) and (6),

(8) f1(x)− f1(x− 0) =x∫

a

ϕdgi −∫

[a,x)

ϕdµg = ϕ(x){gi(x)− gi(x− 0)} .

It follows clearly from (7) and (8) that f1 is internal, and that f1 is continuousrelative to gi. Now, since f1 = k on Cg, and k ∈ B, f1 is of bounded variationon Cg. Further, if {xn} is the countable set of points in I ∼ Cg, then since ϕis µg-summable, and µgi

= µg as seen before, it follows from (7) and (8) that∑

n ωf1(xn) ≤∫

I∼Cg|ϕ| dµg < ∞. Consequently, f1 ∈ B. Now since µk =

ν ≪ µg, it follows from Theorem 13.1 that f∗1 = k ≪ g∗ = g∗i , and hence from

Theorem 14.4 that f1 ≪ gi. Also, since f1 is internal and f2 is obviously external,it is clear that f = f1 + f2 is regulated, f1 = fi and f2 = fe.

Next, given x ∈ I, since f2 and ge are external, it is easy to see from (4) thatf2(x) = ϕ(x)ge(x). Hence when x < b, we obtain from (7),

f(x+ 0)− f(x) = fi(x+ 0)− fi(x)− fe(x)(9)

= ϕ(x){gi(x+ 0)− gi(x)} − ϕ(x)ge(x)

= ϕ(x){g(x + 0)− g(x)},

and when x > a, we obtain similarly from (8),

(10) f(x)− f(x− 0) = ϕ(x){g(x) − g(x− 0)} .

It is now clear from (9) and (10) that f is continuous relative to g and that fsatisfies the condition (Lg). Consequently, (b) holds.

Next, to prove the implication (b)⇒(c), suppose (b) holds. Then, since fi ∈ Bby Theorem 11.2, and gi is internal, it follows from Theorem 14.4 that f∗ = f∗

i ≪g∗i = g∗. Hence, by Theorem 17.1,

(11) µf(E) =∫

E

D∗gf dµg , E ∈ B .

Let x ∈ I. Since the equation in (c) holds clearly for x = a, suppose x > a.Since µg(Rg) = 0, Dgf = D∗

gf µg-a.e. in I, and so from (11) we obtain,

∫

[a,x)

Dgf dµg = µf ([a, x)) = f(x− 0)− f(a) .

Consequently, by (1),

f(a) +x∫

a

Dgf dg = f(x− 0) +Dgf(x){g(x) − g(x− 0)} .


Hence to obtain (c) it is enough to show that

(12) f(x)− f(x− 0) = Dgf(x){g(x) − g(x− 0)} .

In case g is left continuous at x, so is f by hypothesis, and so then (12) holdstrivially. Hence, suppose g(x − 0) 6= g(x). Now if g is right continuous at x, sois f by hypothesis, and since in this case Dgf(x) = D∗

gf(x), (12) follows fromLemma 16.1. When g is, on the other hand, discontinuous from both sides at x,(12) follows directly from Lemma 18.1.

The above theorem assumes the following simpler form in the case when fand g are normalized. For then f and g are internal, Dgf = D∗

gf = f ′g and the

condition (Lg) holds automatically.

18.3. Corollary. If f : I → R and g ∈ B are normalized , then f ≪ g iff fis an indefinite LS-integral relative to g, or , equivalently , iff

f(x) = f(a) +x∫

a

f ′g dg, x ∈ I .

Following is another immediate consequence of Theorem 18.2.

18.4. Corollary. Let g ∈ B. If f : I → R is the indefinite LS-integral ofsome function ϕ relative to g, then ϕ = Dgf µg-a.e.

As we see later in Examples 18.11, this corollary does not hold for the ordinaryrelative derivative f ′

g in general. However, this result does hold for f ′g in case g is

not unilaterally discontinuous at any point of I0. For then Dgf = f ′g on I0 ∼ Cg

by Theorem 18.2 and Lemma 18.1, and if g is discontinuous at a or b, it followsfrom Lemma 16.1 that Dgf = f ′

g at that point, and according to Theorem 16.4,Dgf = f ′

g µg-a.e. in Cg . Hence in that case we obtain the following result ofLebesgue (see Remark 18.12) from the above corollary.

18.5. Corollary (Lebesgue). If g ∈ B is nowhere unilaterally discontinuous

in I0, and f : I → R is the indefinite LS-integral of some function ϕ relative to

g, then ϕ = f ′g µg-a.e.

It is natural to ask here what form Theorem 18.2 assumes if in that theoremthe relation “f ≪i g” is replaced by “f ≪ g”. The next theorem deals with thisvery question.

We begin with a comparison of these two forms of relative AC. For the sake ofsimplicity, when f ≪ g and f satisfies the condition (Lg), we will call f strongly

absolutely continuous (or SAC) relative to g, and write f ≪s g.

18.6. Lemma. Let f : I → R and g ∈ B. If f ≪s g, then f ≪i g. Conversely ,if f ≪i g and f ∈ B, then f ≪ g.

P r o o f. First, suppose f ≪s g. Then f ∈ B by Theorem 11.2, and f iscontinuous relative to g by Theorem 12.2. We will first prove that fi is continuousrelative to gi.

62 K. M. Garg

Given x ∈ I, first suppose x ∈ Ig . Then since f is continuous relative to g,it follows easily from the condition (Lg) that x ∈ If . Hence fi(x) = f(x) andgi(x) = g(x), and since fi and gi have the same unilateral limits at x as f andg respectively, it follows that fi is continuous relative to gi at x. Next, supposex ∈ Eg, and that gi is continuous from the right at x. Then gi(x) = gi(x+ 0) =g(x+ 0), and so either g(x− 0) ≤ g(x+ 0) < g(x) or g(x− 0) ≥ g(x+ 0) > g(x).In each of these two cases it follows easily from the condition (Lg) that eitherf(x− 0) ≤ f(x+ 0) < f(x) or f(x− 0) ≥ f(x+ 0) > f(x). Hence, in each case,fi(x) = f(x + 0) = fi(x + 0), i.e. fi is continuous from the right at x. Whengi is left continuous at x, it is proved similarly that fi is left continuous at x.Consequently, fi is continuous relative to gi.

Now, if x ∈ Rg, it follows from the condition (Lg) that f(x− 0) = f(x+ 0).Hence, by Theorem 14.4, f∗

i = f∗ ≪ g∗ = g∗i , and since fi is continuous relativeto gi, it follows from the other part of Theorem 14.4 that fi ≪ gi. Consequently,f ≪i g.

Next, to prove the converse, suppose f ≪i g and f ∈ B. Then since gi isinternal, it follows from Theorem 14.4 that f∗ = f∗

i ≪ g∗i = g∗, and since f is byhypothesis continuous relative to g, it follows from the other part of Theorem 14.4that f ≪ g.

18.7. Theorem. Let f : I → R and g ∈ B. Then f ≪s g iff f is the indefinite

LS-integral of some µg-summable function relative to g.

P r o o f. First, suppose f ≪s g. Then, by the above lemma, f ≪i g. Hence, byTheorem 18.2, there exists a µg-summable function ϕ such that

(13) f(x) = f(a) +x∫

a

ϕdg, x ∈ I .

It will thus suffice to show that ϕ is µg-summable.Let {xn} be an enumeration of the countable set of points in Eg. We claim

that

(14)∫

Eg

|ϕ| dµg =∑

n

ωf (xn) .

For, with the help of the relations (9) and (10) deduced earlier from (13) in theproof of Theorem 18.2, we obtain

∫

Eg

|ϕ| dµg =∑

n

|ϕ(xn)|{g(xn + 0)− g(xn − 0)}

=∑

n

|ϕ(xn)|{|g(xn + 0)− g(xn)|+ |g(xn)− g(xn − 0)|}

=∑

n

[|ϕ(xn){g(xn + 0)− g(xn)}|

+ |ϕ(xn){g(xn)− g(xn − 0)}|]


=∑

n

{|f(xn + 0)− f(xn)|+ |f(xn)− f(xn − 0)|}

=∑

n

ωf (xn) .

Now since f ∈ B by Theorem 11.2, it follows from (14) that ϕ is µg-summableon Eg. Also, since µg = µg on Ig (see Lemma 7.1), ϕ is clearly µg-summable onIg. Consequently, ϕ is µg-summable on I, which proves the necessity part.

Next, to prove the sufficiency, suppose (13) holds for some µg-summable func-tion ϕ. Then, by Theorem 18.2, f ≪i g and f satisfies the condition (Lg). Hencefi ∈ B by Theorem 11.2, and since f = fi on Ig, f is of bounded variation on Ig.Further, it follows from (14) that f is of bounded variation on Eg, and so f ∈ B.It follows now from Lemma 18.6 that f ≪ g, and hence f ≪s g.

Next we state a few consequences of Theorems 18.2 and 18.7 which are obta-ined in some particular cases.

In the case when g is internal, we have Dgf = D∗gf , and µg = µg by

Lemma 7.1. Hence in that case we obtain from these two theorems

18.8. Corollary. Let f : I → R, g ∈ B, and suppose g is internal. Then

f ≪s g iff f is an indefinite LS-integral relative to g, or , equivalently , iff

f(x) = f(a) +x∫

a

D∗gf dg, x ∈ I .

Now, when g is left or right continuous, the condition (Lg) holds vacuously.Hence in that case we obtain from the above corollary

18.9. Corollary. Let f : I → R and g ∈ B. If g is left or right continuous,then f ≪ g iff f is an indefinite LS-integral relative to g, or , equivalently , iff

f(x) = f(a) +x∫

a

D∗gf dg, x ∈ I .

Finally, on combining Theorem 18.7 with Corollary 18.5 we obtain

18.10. Corollary. Given f : I → R and g ∈ B, suppose g is nowhere

unilaterally discontinuous in I0. Then f ≪s g iff f ′g is µg-summable and f(x) =

f(a) +∫ x

af ′g dg, x ∈ I.

18.11. Examples. We include here two simple examples to show that (a)Corollary 18.4 does not hold in general for f ′

g in place of Dgf unless g is nowhereunilaterally discontinuous in I0 (Corollary 18.5), and that (b) the µg-summabilityin Theorem 18.7 cannot be weakened in general to µg-summability unless g isinternal (Corollary 18.8).

(a) Let a < c < b. Define f(x) = 0 and g(x) = x for a ≤ x < c, and f(x) = 1and g(x) = x+ 1 for c ≤ x ≤ b. Then g is an increasing function, and it is easyto see that f ≪s g, and that f is the indefinite LS-integral of the characteristic

64 K. M. Garg

function ϕ of {c} relative to g. Also, f ′g(x)=0 for x 6=c, but f ′

g(c) does not exist.Now, since µg({c}) > 0, this shows that Corollary 18.4 does not hold for f ′

g ingeneral.

(b) Let {xn} be any increasing sequence of points in I. Define g(xn) = 1/n2

and ϕ(xn) = n for each n, and g(x) = ϕ(x) = 0 otherwise. Then Vg =∑

n 2/n2 <∞, so that g ∈ B. Further, since g∗ ≡ 0, µg ≡ 0, and so ϕ is µg-summable. Define

f(x) =x∫

a

ϕdg, x ∈ I .

It is then clear that f(xn) = 1/n for each n, and f(x) = 0 otherwise. Also,f satisfies the condition (Lg), and f ′

g(xn) = n = ϕ(xn) for each n, so thatϕ = f ′

g µg-a.e. However, since Vf =∑

n 2/n = ∞, f is not AC relative to g byTheorem 11.2. Hence µg-summability in Theorem 18.7 cannot be weakened toµg-summability in general.

18.12. R ema r k. The results of this section are not entirely new, they seemto be known at least when g is continuous.

In connection with Theorems 18.2 and 18.7, Lebesgue deduced two theoremsfrom his well known fundamental theorem on indefinite L-integral using an in-genious transformation of LS-integral into an L-integral (see [26], pp. 285–288).His first theorem [26, p. 288] is similar to Theorem 18.7 but is somewhat different.For the condition of “µg-summability” on the integrand ϕ is not included there,but while proving the sufficiency part of the theorem ϕ is assumed to be bounded(see the proof of bounded variation of f on p. 225 of [26]).

In his second theorem [26, p. 301], Lebesgue obtained the result presentedin Corollary 18.5. It may be pointed out here that this result was stated in [26]without the continuity hypothesis of Corollary 18.5, but this hypothesis is implicitthere in the proof of the theorem. A similar oversight seems to have occurred in[37] where W. H. Young obtained Corollary 18.5 for an increasing function g usinghis theory of integration. However, as we saw in Example 18.11(a), Corollary 18.5does not hold in general without its continuity hypothesis on g. Corollary 18.4is indeed the generalized version of this result in terms of Dgf which holds ingeneral.

It may be noted further that the present proof of Theorem 18.2 is based only onVitali’s covering theorem, and so Lebesgue’s fundamental theorem on L-integralcan be deduced from it by choosing g = τ .

The problem that the fundamental theorem for LS-integral does not hold forf ′g in general seems to have been recognized by R. L. Jeffery in [21] and [22]although he did not state it explicitly. He developed a partial solution of thisproblem by ignoring the values of f and g outside Cg. But then his conclusionholds only for x ∈ Cg. Also, his definitions of relative AC and relative derivativeare considerably involved. A more complete proof of Jeffery’s theorem has beengiven subsequently by M. C. Chakrabarty [2].


19. Relative LAC in terms of LS-integral. In this section we obtain acharacterization of relative LAC in terms of LS-integral of the normalized relativederivative (see Corollary 19.4). This in turn yields a characterization of ordinaryLAC (Corollary 19.5) which is related to the theory of non-absolute integration(see [1] and [32]).

Given f : I → R and g ∈ B, let f be said to satisfy the condition (L∗g) if

either (i) g is continuous, or (ii) f is regulated and partially continuous relativeto g (see §14 for definition) and it satisfies the condition (Lg). Since the condition(Lg) is satisfied vacuously by f when g is continuous, the condition (L∗

g) is indeedstronger than (Lg).

Let us note here that in each of the following three special cases f satisfies thecondition (L∗

g) automatically: (i) g is continuous, (ii) f is regulated and f and gare simultaneously left (or right) continuous, (iii) f and g are normalized.

We need two lemmas here.

19.1. Lemma. Let f, g ∈ B. Then for each E ∈ B,

µf (E) ≥∫

E

|D∗gf | dµg .

P r o o f. There exists, by Theorem 16.4, a set P ∈ B such that D∗gf exists and

is finite at each point of P , µg(I ∼ P ) = 0 and for each B ∈ B, B ⊂ P ,

(1) µf (B) =∫

B

D∗gf dµg .

Given E ∈ B, set E0 = E ∩ P . Then since µg(E ∼ E0) = 0, we obtain from (1),∫

E

|D∗gf | dµg =

∫

E0

|D∗gf | dµg = µf (E0) ≤ µf (E) .

19.2. Lemma. Suppose g ∈ B is internal. If f : I → R is nondecreasing and it

satisfies the condition (L∗g), then

f(y)− f(x) ≥y∫

x

D∗gf dg for a ≤ x < y ≤ b .

P r o o f. Suppose f satisfies the given conditions, and let a ≤ x < y≤b. Then,by Lemma 19.1,

∫

(x,y)

D∗gf dµg ≤

∫

(x,y)

|D∗gf | dµg(2)

≤ µf ((x, y)) = f(y − 0)− f(x+ 0) .

First, suppose g is not continuous from the right at x. Then it is clear thatD∗

gf(x) exists. We claim that

(3) D∗gf(x){g(x + 0)− g(x)} = f(x+ 0)− f(x) .

66 K. M. Garg

If g is left continuous at x, then so is f by hypothesis, and so (3) follows fromLemma 16.1. Otherwise g is discontinuous from both sides at x, and so (3) fol-lows from Lemma 18.1. Also, when g is right continuous at x, the left side of(3) is zero. Hence, f being nondecreasing, the following inequality holds in anycase:

(4) D∗gf(x){g(x + 0)− g(x)} ≤ f(x+ 0)− f(x) .

Using a similar argument we obtain the inequality

(5) D∗gf(y){g(y) − g(y − 0)} ≤ f(y)− f(y − 0) .

Now, with the help of (2), (4) and (5) we obtainy∫

x

D∗gf dg = D∗

gf(x){g(x+ 0)− g(x)} +∫

(x,y)

D∗gf dµg

+D∗gf(y){g(y) − g(y − 0)}

≤ f(x+ 0)− f(x) + f(y − 0)− f(x+ 0) + f(y)− f(y − 0)

= f(y)− f(x) .

19.3. Theorem. Let f : I → R, g ∈ B and suppose g is internal. If f ≪− gand f is internal and it satisfies the condition (L∗

g), then D∗gf is µg-summable on

I and

(6) f(y)− f(x) ≥y∫

x


Conversely , if there is a µg-summable function ϕ on I such that

(7) f(y)− f(x) ≥y∫

x

ϕdg for a ≤ x < y ≤ b ,

then f ≪− g.

P r o o f. First, suppose f ≪− g and f is internal and it satisfies the condition(L∗

g). Then since either g is continuous or f is regulated, it follows from Theo-rem 11.2 that f ∈ B. Hence by Theorem 16.4, D∗

gf exists and is finite µg-a.e.,and it is µg-summable on I.

Further, according to Theorem 10.2, f−≪g. And, since g is internal, it is easyto see that f+ and f− also satisfy the condition (L∗

g). Hence, given a ≤ x < y ≤ b,it follows from Corollary 18.8 that

(8) f−(y)− f−(x) =

y∫

x

D∗gf

−dg ,

and from Lemma 19.2 that

(9) f+(y)− f+(x) ≥y∫

x

D∗gf

+dg .


Now, on combining (8) and (9) we obtain

f(y)− f(x) = {f+(y)− f+(x)} − {f−(y)− f−(x)}

≥y∫

x

{D∗gf

+ −D∗gf

−} dg .

Also, since f is internal, we have by Theorem 8.1,

f+∗ − f−∗ = f∗+ − f∗− = f∗ − f(a) ,

and since g is internal, µg = µg by Lemma 7.1. Hence D∗gf

+ − D∗gf

− = D∗gf

µg-a.e., and consequently (6) follows from the last inequality.Next, to prove the converse, suppose (7) holds for some µg-summable function

ϕ on I. Define

h(x) =x∫

a

ϕdg, x ∈ I .

Then since g is internal, h≪ g by Corollary 18.8. Hence, given ε > 0, there existsa δ > 0 such that if {[ai, bi] : i = 1, . . . , n} is any finite set of nonoverlapping closedintervals in I for which

∑

i≤n{g(bi)− g(ai)} < δ, then∑

i≤n |h(bi) − h(ai)| < ε.

Consequently, for such a set of intervals it follows from (7) that

∑

i

{f(bi)− f(ai)} ≥∑

i

bi∫

ai

ϕdg =∑

i

{h(bi)− h(ai)} > −ε ,

which proves that f ≪− g.

In the three special cases mentioned earlier in which f satisfies the condition(L∗

g) automatically, we obtain from the above theorem

19.4. Corollary. Given f : I → R and g ∈ B, suppose either (i) f is internal

and g is continuous, or (ii) f is regulated and f and g are simultaneously left

(or right) continuous, or (iii) f and g are normalized. Then the following are

equivalent :

(a) f ≪− g,(b) there is a µg-summable function ϕ on I such that

f(y)− f(x) ≥y∫

x

ϕdg for a ≤ x < y ≤ b ,

(c) D∗gf is µg-summable on I and

f(y)− f(x) ≥y∫

x


Now when g = τ , according to Theorem 16.4 we have D∗gf = f ′

g = f ′ a.e.Hence on choosing g = τ in the above corollary we obtain the following charac-terization of ordinary LAC.

68 K. M. Garg

19.5. Corollary. If f : I → R is internal , then the following are equivalent :

(a) f is LAC,

(b) there is an L-summable function ϕ on I such that

f(y)− f(x) ≥y∫

x

ϕdx for a ≤ x < y ≤ b ,

(c) f ′ is L-summable on I and

f(y)− f(x) ≥y∫

x

f ′dx for a ≤ x < y ≤ b .

20. Mutual singularities in terms of normalized relative derivative.

In this section we obtain characterizations of mutual singularity and LS in termsof normalized relative derivative.

20.1. Theorem. Let f, g ∈ B. Then f∗ ⊥ g∗ iff D∗gf = 0 µg-a.e.

P r o o f. We can assume here without loss of generality that f and g arenormalized. Then the derivative D∗

gf becomes the same as f ′g.

First, suppose f ⊥ g. Then µf ⊥ µg by Theorem 7.2. Hence there exists aBorel decomposition (A,B) of I such that µf (A) = µg(B) = 0 (see the proof ofCorollary 7.3). But then, by Lemma 19.1,

∫

A

|f ′g| dµg ≤ µf (A) = 0 .

Consequently, f ′g = 0 µg-a.e.

Next, to prove the converse, suppose f ′g = 0 µg-a.e. Set A = {x : f ′

g(x) = 0}and B = I ∼ A. Then µg(B) = 0. Further, by Theorem 16.4, there exists a Borelsubset C of A such that µg(A ∼ C) = 0 and for each E ∈ B, E ⊂ C, µf (E) =∫

Ef ′g dµg. Hence µf (E) = 0 for every Borel subset E of C, and consequently

µf (C) = 0. Further, µg(I ∼ C) ≤ µg(B) + µg(A ∼ C) = 0, which proves thatµf ⊥ µg. Consequently, f ⊥ g by Theorem 7.2.

We include here a few consequences of the above theorem. With the help ofTheorem 8.2 we obtain

20.2. Corollary. Let f, g ∈ B. If f ⊥ g and f and g are nowhere unilaterally

discontinuous from opposite sides, then D∗gf = 0 µg-a.e. Conversely , if D∗

gf = 0µg-a.e., and each of f and g is continuous at the points where the other has a

removable discontinuity , then f ⊥ g.

The necessity of the continuity hypotheses in the above corollary can be ver-ified easily by simple examples. From this corollary we obtain on the other handthe following result in some special cases.


20.3. Corollary. Let f, g ∈ B, and suppose either (i) g is continuous, or(ii) f and g are simultaneously left or right continuous, or (iii) f and g are

normalized. Then f ⊥ g iff D∗gf = 0 µg-a.e.

The proof of the result for mutual LS is somewhat more involved. We needtwo lemmas for this purpose.

Let us recall here that for any set A ⊂ I, χA denotes the characteristicfunction of A on I.

20.4. Lemma. Suppose f ∈ B is internal , and let (A+, A−) be a Hahn decom-

position for µf . Then

D∗ff

+ = D∗ff

+ = χA+and D∗

ff− = −D∗

ff− = χA−

µf -a.e.

Consequently , |D∗ff | = 1 µf -a.e.

P r o o f. Since f+ ≪ f and f is internal, by Theorem 14.4 we have f+∗ ≪ f∗.

Hence by Theorem 17.1,

(1) µf+(E) =∫

E

D∗ff

+dµf , E ∈ B .

Further, since µf = µf (see Lemma 7.1), we obtain with the help of Lemma 4.1,

(µf )+ = 1

2(µf + µf) =

12(µf + µf ) =

12µ2f+ = µf+ .

Hence for every Borel subset E of A+,

µf+(E) = (µf )+(E) = µf(E) = µf (E) .

Consequently, it follows from (1) that D∗ff+ = 1 µf -a.e. in A+. Similarly, for

every Borel subset E of A−, µf+(E) = 0, and hence it follows from (1) thatD∗

ff+ = 0 µf -a.e. in A−. Thus we have proved that D∗

ff+ = χA+

µf -a.e. Also,

on applying this result to −f we obtain D∗ff− = χA−

µf -a.e.

Next, from the two established relations we obtain with the help of Lemma 3.1,

D∗ff = D∗

ff+ −D∗

ff− = χA+

− χA−µf -a.e.

Consequently, D∗ff

+ = D∗ff+(D∗

ff)−1 = χA+

and D∗ff

− = D∗ff−(D∗

ff)−1 =

−χA−µf -a.e.

The last part is of course an obvious consequence of the first.

20.5. Lemma. Suppose f, g ∈ B are internal , and let (A+, A−) be a Hahn

decomposition for µf . Then

D∗g+f+ = (D∗

gf)χA+and D∗

g+f− = −(D∗gf)χA−

µg+-a.e. ,(2)

D∗g−f+ = −(D∗

gf)χA+and D∗

g−f− = (D∗gf)χA−

µg− -a.e.(3)

P r o o f. It is enough to prove here (2), for (3) follows from (2) on applying itto f and −g.

70 K. M. Garg

According to Lemma 20.4 there exists a set C ∈ B such that µf (I ∼ C) = 0and

(4) D∗ff

+ = χA+and D∗

ff− = −χA−

on C .

Let (B+, B−) be a Hahn decomposition for µg. Then by Lemma 20.4, D∗gg

+ =χB+

µg-a.e., and since µg+(B−) = 0, it is clear that D∗g+g = 1 µg+ -a.e. Hence it

follows from Theorem 16.4 that there is a set D ∈ B such that µg+(I ∼ D) = 0,each of the derivatives D∗

gf , D∗g+f+ and D∗

g+f− exists and is finite at the pointsof D, and

(5) D∗g+g = 1 on D .

Now set E = D ∼ C. Then by Lemma 19.1,∫

E

|D∗gf | dµg ≤ µf (E) ≤ µf (I ∼ C) = 0 .

Hence D∗gf = 0 µg-a.e. in E. By a similar argument we obtain from Lemma 19.1

that D∗g+f+ = D∗

g+f− = 0 µg+ -a.e. in E. The two relations in (2) thus hold

clearly µg+ -a.e. in E. At the points of D ∩C we have, on the other hand, by (4)and (5),

D∗g+f+ = D∗

ff+ ·D∗

gf ·D∗g+g = (D∗

gf)χA+,

D∗g+f− = D∗

ff− ·D∗

gf ·D∗g+g = −(D∗

gf)χA−.

20.6. Theorem. Let f, g ∈ B. Then f∗ ⊥− g∗ iff D∗gf ≥ 0 µg-a.e.

P r o o f. There is as before no loss of generality in assuming f and g to benormalized. Then by Theorem 8.1 each of the variation functions of f and g isalso normalized.

First, suppose f ⊥− g. Then by Theorem 6.2, f+ ⊥ g− and f− ⊥ g+. Hence,by Theorem 20.1, (f+)′g− = 0 µg− -a.e. and (f−)′g+ = 0 µg+ -a.e. Now, since eachof the variation functions of f and g is nondecreasing, it follows from the tworelations of (2) that f ′

g ≥ 0 µg+ -a.e., and from the two relations of (3) that f ′g ≥ 0

µg− -a.e. Consequently, f ′g ≥ 0 µg-a.e.

Next, to prove the converse, suppose f ′g ≥ 0 µg-a.e. Then it follows as before

from the second relation of (2) that (f−)′g+ = 0 µg+ -a.e., and from the first

relation of (3) that (f+)′g− = 0 µg− -a.e. Hence, by Theorem 20.1, f− ⊥ g+ and

f+ ⊥ g−. Consequently, f ⊥− g by Theorem 6.2.

We conclude this section with two consequences of the above theorem similarto the ones obtained earlier. With the help of Theorem 8.3 we obtain

20.7. Corollary. Suppose f, g ∈ B are internal. If f ⊥− g and f and gare nowhere unilaterally discontinuous from opposite sides, then D∗

gf ≥ 0 µg-a.e.

Conversely , if D∗gf ≥ 0 µg-a.e., then f ⊥− g.

This in turn yields the following result in some special cases.


20.8. Corollary. Let f, g ∈ B, and suppose either (i) f is internal and g is

continuous, or (ii) f and g are simultaneously left or right continuous, or (iii) fand g are normalized. Then f ⊥− g iff D∗

gf ≥ 0 µg-a.e.

21. Reconstruction of relative primitive. In this section we deal with theproblem of reconstructing a function f from its relative derivative f ′

g if the latterexists and is finite everywhere (see Theorem 21.4). The solution of this problemin the case of ordinary derivative f ′ is well known (see e.g. [28], p. 266). The proofpresented here is significantly different from the one given in [28].

We shall need here three lemmas which relativize some of the known resultson ordinary Dini derivates.

Given f : I → R and g ∈ B, we will use D+fg and D+fg to denote the lowerand upper right derivates of f relative to g, which are defined at x ∈ I, x < b,similar to Dgf(x) and Dgf(x) by taking the lower and upper limits of the ratioconsidered at the beginning of §16 as h → 0 from the right. Also, we will useD−fg and D−fg to denote the lower and upper left derivates of f relative to gwhich are defined similarly.

21.1. Lemma. Suppose f : I → R, g ∈ B is nondecreasing and α > 0. If E is

a set of points in I where D+fg ≤ α and D−fg ≥ −α, then

|f(E)| ≤ αµg(E) .

P r o o f. Given ε > 0, let En denote, for each positive integer n, the set ofpoints x in E for which

f(t)− f(x) < (α+ ε)|g(t) − g(x)| for t ∈ I, |t− x| < 1/n .

Then {En} is a nondecreasing sequence of sets which converges to E.

Next, given n, using the denseness of Cg in I it is easy to find a sequence ofopen intervals {Ik} in I such that En ⊂

⋃

k Ik, |Ik| < 1/n for each k, and∑

k

µg(Ik) < µg(En) + ε .

Now, for each k, if x, y ∈ En ∩ Ik, then

|f(y)− f(x)| < (α+ ε)|g(y) − g(x)| ≤ (α+ ε)µg(Ik) .

Hence |f(En ∩ Ik)| ≤ (α+ ε)µg(Ik) for each k. Consequently,

|f(En)| ≤∑

k

|f(En ∩ Ik)| ≤ (α + ε)∑

k

µg(Ik) ≤ (α + ε){µg(En) + ε} .

Now, making ε→ 0, we obtain |f(En)| ≤ αµg(En), and hence

|f(E)| = limn

|f(En)| ≤ α limnµg(En) = αµg(E) .

The following is a relativized version of a well-known theorem on ordinaryderivative (see Saks [34], p. 227).

72 K. M. Garg

21.2. Lemma. Let f : I → R and g ∈ B. If f ′g exists and is finite at the points

of a set E ∈ B, then

|f(E)| ≤∫

E

|f ′g| dµg .

P r o o f. Let ε > 0, and set for each positive integer n,

En = {x ∈ E : ε(n− 1) ≤ |f ′g(x)| < εn} .

Now given n, let x ∈ En. Then if y ∈ I and g(y) 6= g(x), it is clear that∣

∣

∣

∣

f(y)− f(x)

g(y)− g(x)

∣

∣

∣

∣

≤

∣

∣

∣

∣

f(y)− f(x)

g(y)− g(x)

∣

∣

∣

∣

.

Hence each of the derivates D+fg(x) and D−fg(x) is bounded by |f ′g(x)|, and so

by εn. Consequently, it follows from Lemma 21.1 that

|f(En)| ≤ εnµg(En) ≤∫

En

|f ′g| dµg + εµg(En) .

Hence,

|f(E)| ≤∑

n

|f(En)| ≤∫

E

|f ′g| dµg + εµg(E) .

Now on making ε→ 0 we obtain the required inequality.

21.3. Lemma. Given f : I → R and g ∈ B, suppose f(x−0) ≤ f(x) ≤ f(x+0)for each x, and g is increasing. Let a ≤ x < y ≤ b. If E is the set of points in

(x, y) where D+fg ≤ 0, then

(1) f(x)− f(y) ≤ |f(E)| .

P r o o f. Suppose f(x) > f(y), for (1) is obvious otherwise. Let A and Bdenote the sets of points in (x, y) where D+f < 0 or D+f = 0 respectively, andset C = A ∪B.

Now, given f(y) < α < f(x), let z = sup{t : x ≤ t ≤ y, f(t) ≥ α}. Then itis clear from the continuity hypothesis of f that x < z < y and f(z) = α. Nowsince f(t) < α for z < t < y, we have D+f(z) ≤ 0, i.e. z ∈ C. This proves thatf(C) includes the interval (f(y), f(x)), and so f(x) − f(y) ≤ |f(C)|. But since|f(B)| = 0 (see [34], p. 272), it follows that f(x)− f(y) ≤ |f(A)|. Further, sinceg is increasing, it is clear that A ⊂ E, and so (1) holds.

21.4. Theorem. Let f : I → R and g ∈ B. Suppose, at each x ∈ I, either(i) f is left continuous and f (x+0) ≤ f(x) ≤ f(x+0), or (ii) f(x− 0) = f(x) =

f(x+ 0). Then if f ′g exists and is finite at all but a countable set of points in I,

and it is µg-summable, then

f(x) = f(a) +x∫

a

f ′g dg, x ∈ I .


P r o o f. Given a ≤ x < y ≤ b, let E be the set of points in (x, y) whereD+fg ≤ 0. Then by Lemmas 21.2 and 21.3 we have

f(x)− f(y) ≤ |f(E)| ≤∫

E

|f ′g| dµg ≤

y∫

x

|f ′g| dg .

Define ϕ = −|f ′g|. Since f

′g is µg-summable, so is ϕ, and clearly

f(y)− f(x) ≥y∫

x

ϕdg .

Hence, since g is internal, it follows from Theorem 19.3 that f ≪− g, or, equ-ivalently, f ≪− g. Also, on applying this result to −f in case (i), and to f(−x),x ∈ −I, in case (ii), we obtain f ≪− g. Consequently, f ≪ g. It follows nowfrom Lemma 11.1 that f is regulated, and so f is continuous in each of the twocases. Hence f clearly satisfies the condition (Lg). Consequently, it follows fromLemma 18.6 and Theorem 18.2 that

(2) f(x) = f(a) +x∫

a

Dgf dg, x ∈ I .

Now, since f is continuous, Dgf = 0 on I ∼ Cg, and by Theorem 16.4,D∗

gf = f ′g µg-a.e. on Cg. Hence it is clear that (2) remains valid on replacing

Dgf by f ′g.

On choosing g = τ in the above theorem, we obtain the following verison ofthe classical theorem which is known for continuous f (see [28], p. 266).

21.5. Corollary. Suppose f satisfies one of the continuity hypotheses (i) and(ii) of the above theorem. Then if f has a finite derivative at all but a countable

set of points in I, and f ′ is summable, then

f(x) = f(a) +x∫

a

f ′(t) dt, x ∈ I .

V. Relativization of other classical theorems

22. Lebesgue’s monotonicity theorem. We begin this chapter with arelativization, in the present section, of the following well-known monotonicitytheorem which follows directly from Lebesgue’s fundamental theorem on indefiniteL-integral (see [26]; or [28], p. 255): If a function f : I → R is AC and LS, then itis nondecreasing. It is of course the following consequence of this theorem whichis more commonly known: If f is AC and singular, then it is constant.

Following is the relativized version of these two results with an extension ofthe former theorem (see in particular Corollary 22.2).

74 K. M. Garg


(a) If g is nondecreasing , f ≪− g and f ⊥− g, then f is nondecreasing.

(b) If f ≪ g and f ⊥ g, then f is constant.

P r o o f. We will prove here the part (b) first. Suppose f ≪ g and f ⊥ g. Thenf ⊥ f by Theorem 13.4. Hence it follows from the definition of mutual singularitythat f is constant.

Next, to prove (a), suppose g is nondecreasing, f ≪− g and f ⊥− g. Thenf− ≪ g by Theorem 10.2, and f− ⊥ g by Corollary 6.4. Hence f− is constant by(b), and so f is nondecreasing.

It may be noted here that the converse of the two parts of the above theoremare also valid. On choosing g = τ we obtain on the other hand the following resultfrom part (a) of the above theorem with the help of Corollaries 11.3 and 9.7.

22.2. Corollary. A function f : I → R is nondecreasing iff it is LAC and

LS. Consequently , f is constant iff it is AC and singular.

See [13, p. 310] for the first part of this corollary.

23. Lebesgue’s decomposition theorem. In this section we obtain a rela-tivized version of Lebesgue’s decomposition theorem. Also, from this new versionwe deduce here a simple characterization of relative AC in the presence of relativeLAC.

The original Lebesgue decomposition (see e.g. [28], p. 246) follows from thefollowing theorem on choosing g = τ (see Corollary 9.4).

23.1. Theorem. Given f, g ∈ B, there are two unique functions ϕ,ψ ∈ Bsuch that f = ϕ+ ψ, ϕ≪ g, ϕ(a) = f(a), and ψ ⊥ g.

P r o o f. We need to deal here with the continuous and discontinuous parts off separately. Define

ϕ1(x) = fc(a) +x∫

a

(fc)′gc dgc, x ∈ I .

The function ϕ1 is well defined due to Theorem 16.4, and ϕ1(a) = fc(a) = f(a).Further, ϕ1 ≪ gc by Theorem 18.7 and Lemma 7.1, and so ϕ1 is continuousand of bounded variation by Theorems 11.2 and 12.2. Also, by Corollary 18.5,(ϕ1)

′gc = (fc)

′gc µgc -a.e.

Now, define ψ1 = fc − ϕ1. Clearly, ψ1 also is continuous and of boundedvariation, and (ψ1)

′gc = 0 µgc -a.e. Consequently, ψ1 ⊥ gc by Corollary 20.3.

Next, let D− and D+ denote the sets of points in I where g is discontinuousfrom the left or right respectively. Define ϕ2(a) = 0, and for a < x ≤ b,

ϕ2(x) =∑

t∈D−, t≤x

{f(t)− f(t− 0)} +∑

t∈D+, t<x

{f(t+ 0)− f(t)} .


Also, define ψ2 = fd − ϕ2. Then ϕ2 and ψ2 are two jump functions in B suchthat fd = ϕ2 +ψ2. Further, since ϕ2 is clearly continuous relative to gd, ϕ2 ≪ gdby Lemma 13.2. Also, since ψ2 is left continuous at the points of D− and rightcontinuous at the points of D+, ψ2 ⊥ gd by Lemma 7.4.

Now define ϕ = ϕ1 + ϕ2 and ψ = ψ1 + ψ2. Then f = fc + fd = ϕ + ψ andϕ(a) = ϕ1(a) + ϕ2(a) = f(a). Further, ϕ ≪ g by Theorem 13.3, and ψ ⊥ g byTheorem 7.6. Hence ϕ+ ψ is the desired decomposition of f .

Finally, to prove the uniqueness of this decomposition, suppose f = ϕ0 + ψ0

is some other decomposition of f such that ϕ0 ≪ g, ϕ0(a) = f(a) and ψ0 ⊥ g.Then h ≡ ϕ − ϕ0 = ψ0 − ψ, and so h ≪ g by Theorems 10.1 and 10.3, andh ⊥ g by Theorems 5.1 and 6.1. Hence h is constant by Theorem 22.1, and sinceh(a) = ϕ(a) − ϕ0(a) = 0, this proves that h ≡ 0. Consequently ϕ0 = ϕ andψ0 = ψ, which proves the uniqueness of ϕ and ψ.

The unique decomposition ϕ + ψ of f determined by the above theorem willbe called the Lebesgue decomposition of f relative to g, and the functions ϕ andψ of this decomposition will in turn be called the absolutely continuous (or AC)and singular components (or parts) respectively of f relative to g.

It is natural to ask here which of the properties of f and g are reflected inϕ and ψ. The next three theorems deal with some such properties, and so theyare in a sense mere extensions of Theorem 23.1.

23.2. Theorem. Given f, g ∈ B, let ϕ + ψ be the Lebesgue decomposition of

f relative to g. Then

(a) f− = ϕ− + ψ−, f+ = ϕ+ + ψ+;

(b) fc = ϕc + ψc, fd = ϕd + ψd; and

(c) fa = ϕa + ψa, fs = ϕs + ψs.

P r o o f. Since f = ϕ + ψ, according to Theorem 5.3, (a) holds iff ϕ ⊥− ψ.However, since ϕ≪ g and ψ ⊥ g, it follows from Theorem 13.4 that ϕ ⊥ ψ, whichof course implies that ϕ ⊥− ψ (see Theorem 5.5).

The part (b) is obvious, for

f = ϕ+ ψ = (ϕc + ψc) + (ϕd + ψd) ,

where ϕc + ψc is continuous and ϕd + ψd is a jump function.

Further, since

f = ϕ+ ψ = (ϕa + ψa) + (ϕs + ψs) ,

where ϕa + ψa is AC and ϕs + ψs is singular, (c) follows from the uniqueness ofthe ordinary Lebesgue decomposition.

In the following theorem the result on monotonicity in part (d) is well knownfor the ordinary Lebesgue decomposition (see e.g. [28], p. 246).

76 K. M. Garg

23.3. Theorem. Given f, g ∈ B, let ϕ and ψ be the AC and singular compo-

nents respectively of f relative to g. Then

(a) if f and g are normalized , then so are ϕ and ψ;(b) ϕ satisfies the condition (Lg) iff f does so;(c) f is continuous, or left , right , lower or upper continuous at any point

x ∈ I iff ϕ and ψ are so;(d) f is nondecreasing or a jump function iff ϕ and ψ are so;(e) ϕ is nondecreasing iff f− ⊥ g; and(f) ψ is nondecreasing iff f ≪− g.

P r o o f. Let D− and D+ denote as before the sets of points in I where g isdiscontinuous from the left or right respectively, and set D = D− ∩ D+. Sinceψ ⊥ g, at every point x of D− we have ψ(x − 0) = ψ(x) by Theorem 9.3, andsince f = ϕ+ ψ, we obtain

(1) ϕ(x− 0)− ϕ(x) = f(x− 0)− f(x) for x ∈ D− .

Similarly,

(2) ϕ(x+ 0)− ϕ(x) = f(x+ 0)− f(x) for x ∈ D+ .

Further, since ϕ≪ g, ϕ is continuous relative to g (see Theorem 12.2).Now, if f and g are normalized, then I0 ∼ D ⊂ Cg , and so ϕ is continuous at

the points of I0 ∼ D. Further, for each x ∈ D, it follows from (1) and (2) that

ϕ(x+ 0) + ϕ(x− 0)− 2ϕ(x) = f(x+ 0) + f(x− 0)− 2f(x) = 0 ,

so that ϕ∗(x) = ϕ(x). Consequently ϕ is normalized, and hence, by Lemma 3.1,ψ = f − ϕ also is normalized. The part (a) is thus established.

The part (b) follows on the other hand directly from (1) and (2).Next, in (c), as the sufficiency parts are obvious, we need to prove only the

necessity part of each result.Given x ∈ I, first suppose f is left continuous at x. Then if x ∈ D−, it follows

from (1) that ϕ is left continuous at x; otherwise it follows from the continuity ofϕ relative to g that ϕ is left continuous at x. Also, since ψ = f −ϕ, ψ also is leftcontinuous at x. A similar argument holds for right continuity, using (2) in placeof (1).

Next, suppose f is left LC at x. Then if x ∈ D−, it follows form (1) that ϕis left LC at x, and ψ is left continuous at x since ψ ⊥ g (see Theorem 9.3).Otherwise, as seen above, ϕ is left continuous at x, and hence ψ = f − ϕ also isleft LC at x. A similar argument holds for right LC, using (2) in place of (1).Hence if f is LC at x, then so are ϕ and ψ. Now on applying this result to −f ,the result on UC is obtained, and then on combining the two results the resulton continuity is obtained.

Next, in (d) also we need to prove only the necessity parts. Since f is non-decreasing iff f− ≡ 0, the monotonicity result follows clearly from the relationf− = ϕ− + ψ− obtained in Theorem 23.2.


To prove the other part of (d), suppose f is a jump function. Then fc ≡ 0, andso, by Theorem 23.2, ϕc = −ψc. But since ϕ ⊥ ψ, ϕc ⊥ ψc by Theorem 7.6, andhence ϕc ⊥ ϕc by Theorem 5.1. Hence it follows from the definition of mutualsingularity that ϕc is constant. Consequently, ϕ is a jump function, and so istherefore ψ = f − ϕ.

To prove the remaining two parts we need the following relations:

(i) f− = ϕ− + ψ−, (ii) ϕ− ⊥ ψ−, (iii) ϕ− ≪ g, (iv) ψ− ⊥ g.

The relation (i) has already been established in Theorem 23.2. Further, sinceϕ ⊥− ψ by Theorem 5.5, (ii) follows from Corollary 6.3. And, since ϕ≪− g, (iii)follows from Theorem 10.2. Also, since ψ ⊥ g, and ψ+ ⊥ ψ− by Corollary 5.6,(iv) follows from Theorem 6.1.

Now, to prove (e), first suppose ϕ is nondecreasing. Then ϕ− ≡ 0, and sof− = ψ− by (i). Hence f− ⊥ g by (iv). Next, to prove the converse, supposef− ⊥ g. Then it follows from (i), (ii) and Theorem 6.1 that ϕ− ⊥ g. Hence itfollows from (iii) and Theorem 22.1 that ϕ− is constant, i.e. ϕ is nondecreasing.

Next, to prove (f), first suppose ψ is nondecreasing. Then ψ− ≡ 0, and sof− = ϕ− by (i). Hence, f− ≪ g by (iii), and so f ≪− g by Theorem 10.2. Now,to prove the converse, suppose f ≪− g. Then f− ≪ g by Theorem 10.2. Henceit follows from (i), (ii) and Theorem 10.3 that ψ− ≪ g. Consequently, it followsfrom (iv) and Theorem 22.1 that ψ− is constant, i.e. ψ is nondecreasing.

23.4. Theorem. Given f, g, h ∈ B, let ϕ + ψ be the Lebesgue decomposition

of f relative to g. Then f is AC, LAC, singular or LS relative to h iff ϕ and ψare so. Consequently , f is AC, LAC, singular or LS iff ϕ and ψ are so.

P r o o f. Since ϕ ⊥ ψ as observed earlier, all the results in the first part followdirectly from Theorems 6.1 and 10.3. The second part follows on the other handfrom the first on choosing g = τ .

Finally, we deduce from Theorem 23.3 the following characterization of relativeAC in the presence of relative LAC. It is interesting to compare this characteri-zation with the ones obtained earlier in Theorem 17.1 and Corollary 18.3.

23.5. Theorem. Suppose f, g ∈ B are normalized and f ≪− g. Then f ≪ giff

(3) f(b)− f(a) =∫

I

f ′g dµg .

P r o o f. We need to prove here the sufficiency part only, for the necessityfollows directly from Theorem 17.1.

Hence suppose (3) holds. Let ϕ and ψ be the AC and singular componentsrespectively of f relative to g. Then, by Theorem 23.3, ϕ and ψ are normalized

78 K. M. Garg

and ψ is nondecreasing. Further, since ϕ≪ g, we have by Theorem 17.1,

(4) ϕ(b)− ϕ(a) =∫

I

ϕ′g dµg ,

and since ψ ⊥ g, ψ′g = 0 µg-a.e. by Theorem 20.1. Hence f ′

g = ϕ′g µg-a.e., and

so it follows from (3) and (4) that f(b) − f(a) = ϕ(b) − ϕ(a). Consequently,ψ(b) = ψ(a). Now since ψ is nondecreasing, this implies that ψ is constant, andhence f = ϕ+ ψ ≪ g.

23.6. Corollary. Given f, g ∈ B, suppose f ≪− g, and that either (i) g is

continuous, or (ii) f and g are simultaneously left , or right , continuous. Then

f ≪ g iff

f(b)− f(a) =∫

I

D∗gf dµg .

For, in each of the two cases it follows from Theorem 14.2 that f∗ ≪− g∗.Further, since f is LC relative to g (see Theorem 12.2), Rf = ∅ if (i) holds, andso in both the cases it follows easily from Theorem 14.4 that f ≪ g iff f∗ ≪ g∗.Hence the result is obtained on applying Theorem 23.5 to f∗ and g∗.

Now on choosing g = τ in the above corollary we obtain the following re-sult with the help of Corollary 11.3 and Theorem 16.4. This result is known tohold for a continuous nondecreasing function f (see e.g. [28], p. 264), but everynondecreasing function f is on the other hand LAC.

23.7. Corollary. If f : I → R is LAC, then it is AC iff

f(b)− f(a) =b∫

a

f ′dx .

23.8. R ema r k. It should be pointed out here that different versions of The-orem 22.1(b) and Theorem 23.1 have been obtained by Kober in terms of hisnotions of covariance and contravariance which are somewhat weaker (see Re-marks 9.8 and 15.6, and [23], pp. 574, 576).

24. Lusin’s property (N) and the Banach–Zarecki theorem. In thissection we present a relativized version of Lusin’s property (N) [27, p. 109] andobtain its characterization similar to Rademacher’s theorem [30]. In terms of thisproperty we obtain a relativized version of the Banach–Zarecki theorem dealingwith the characterization of AC. Also, we include here another characterizationof relative AC in terms of this property which is more general.

Given g ∈ B, let a function f : I → R be said to have the property (N) relativeto g, or, more briefly, the property (Ng), if |f(E)| = 0 for every set E ⊂ I forwhich |g(E)| = 0.

We begin with a relativization of Rademacher’s theorem (see [30] or [28,p. 248]). The following theorem, on choosing g = τ , also generalizes Radema-


cher’s theorem from continuous functions to measurable functions. A Lebesguemeasurable set in R is called here simply measurable.

24.1. Theorem. Let g ∈ B, and suppose f : I → R is µg-measurable. Then fhas the property (Ng) iff f(E) is measurable for every µg-measurable set E ⊂ I.

P r o o f. There is clearly no loss of generality in assuming g to be nondecreas-ing.

To prove the necessity, suppose f has the property (Ng), and let E be anyµg-measurable subset of I. Then there exists a nondecreasing sequence of closedsets {En} in E such that µg(E ∼

⋃

nEn) = 0. Further, since f is µg-measurable,using Lusin’s theorem ([34], p. 72) we can construct another nondecreasing se-quence of closed sets {Fn} in I such that f is continuous on each Fn andµg(I ∼

⋃

n Fn) = 0. Now set Kn = En ∩ Fn for each n, and let A =⋃

nKn

and B = E ∼ A. Then for each n, since Kn is compact, so is f(Kn). Hencef(A) is an Fσ-set. Further, by Theorem 2.3, |g(B)| ≤ µg(B) = 0, and hence byhypothesis |f(B)| = 0. Consequently, f(E) = f(A) ∪ f(B) is measurable.

Next, we will prove the sufficiency by contradiction. Suppose f does not havethe property (Ng). Then there is a set E ⊂ I such that |g(E)| = 0 but |f(E)| >0. Hence f(E) includes a set N which is not measurable. Now set A = E ∩Cg ∩ f

−1(N). Then, by Theorem 2.3, µg(A) = |g(A)| ≤ |g(E)| = 0. Hence A isµg-measurable. However, since f(A) ⊂ N and N ∼ f(A) is countable, f(A) isnot measurable.

The Banach–Zarecki theorem ([28], p. 250) follows from its following relativi-zed version on choosing g = τ .

24.2. Theorem. Let f : I → R and g ∈ B. Then f ≪ g iff the following

conditions hold : (i) f ∈ B, (ii) f is continuous relative to g, and (iii) f possesses

the property (Ng).

P r o o f. Since f is continuous relative to g iff it is so relative to g (seeLemma 2.1), it is clear from the definitions of relative AC and property (Ng)that there is no loss of generality in assuming g to be nondecreasing.

We will first prove the result for normalized f and g. Hence suppose theyare so. Now, to prove the necessity, suppose f ≪ g. Then (i) and (ii) followfrom Theorems 11.2 and 12.2. To prove (iii), let E be any subset of I for which|g(E)| = 0. Set E1 = E∩Cg and E2 = E ∼ Cg. Then, by Theorem 2.3, µg(E1) =|g(E1)| = 0, and since µf ≪ µg by Theorem 13.1, it follows that µf (E1) = 0.Hence it follows from Theorem 2.3 and Lemma 7.1 that |f(E1)| ≤ µf (E1) =µf (E1) = 0. Now since E2 is countable, we thus obtain |f(E)| ≤ |f(E1)| +|f(E2)| = 0. Consequently, (iii) holds.

Next, to prove the sufficiency, suppose (i), (ii) and (iii) hold. Then by The-orem 16.4 there is a set D ∈ B such that f ′

g exists and is finite at each point

80 K. M. Garg

of D, µg(I ∼ D) = 0, f ′g is µg-summable on I, and

(1) µf (E) =∫

E

f ′g dµg, E ∈ B, E ⊂ D .

We will use here Theorem 19.3 to prove that f ≪ g. Hence let a ≤ x < y ≤ b.Now set J = (x, y), A = J ∩ D ∩ Cg, B = J ∩D ∼ Cg and C = J ∼ (A ∪ B).Then since I ∼ Cg ⊂ D (see Lemma 16.1), we have J ∼ Cg ⊂ B. Further, dueto (ii), Cg ⊂ Cf . Hence it is easy to see that

(2) |f(y)− f(x)| ≤ |f(A ∪ C)|+∑

t∈B

|f(t+ 0)− f(t− 0)|

+|f(x+ 0)− f(x)|+ |f(y)− f(y − 0)| .

Now, by Theorem 2.3, |g(C)| ≤ µg(C) ≤ µg(I ∼ D) = 0, and so due to (iii),|f(C)| = 0. Hence, we obtain from (1),

|f(A ∪ C)| = |f(A)| ≤ µf (A) ≤∫

A

|f ′g| dµg .

Further, according to Lemma 16.1,∑

t∈B

|f(t+ 0)− f(t− 0)| =∑

t∈B

|f ′g(t){g(t + 0)− g(t− 0)}| =

∫

B

|f ′g| dµg .

Next we claim that

|f(x+ 0)− f(x)| = |f ′g(x)|{g(x + 0)− g(x)} .

When g is discontinuous at x, this relation follows directly from Lemma 18.1, orfrom Lemma 16.1 in case x = a; otherwise it follows from (ii) whether f ′

g(x) existsor not. Similarly, we have

|f(y)− f(y − 0)| = |f ′g(y)|{g(y) − g(y − 0)} .

It is now clear from the last four relations and (2) that

|f(y)− f(x)| ≤y∫

x

|f ′g| dg .

Consequently, it follows from Theorem 19.3 that f ≪ g. This proves the result inthe case when f and g are normalized.

Next, to obtain the result in general, let us first observe that f has the property(Ng) iff f

∗ has the property (Ng∗). This follows clearly from the fact that the setsof points where f 6= f∗ or g 6= g∗ are countable.

First, suppose f ≪ g. Then (i) and (ii) follow as before from Theorems 11.2and 12.2, and since g is nondecreasing, it follows from Theorem 14.4 that f∗ ≪ g∗.Hence, by the above result, f∗ has the property (Ng∗), and so (iii) holds as justobserved.

Next, to prove the sufficiency, suppose (i), (ii) and (iii) hold. Then since f ∈B,it is clear that f∗ ∈ B, and it follows from Theorem 14.3 that f∗ is continuous


relative to g∗. Further, f∗ has the property (Ng∗) as just observed. Consequently,it follows from the above result that f∗ ≪ g∗, and hence from Theorem 14.4 thatf ≪ g.

The following characterization of relative AC in terms of relative property (N)is in a sense more general. Such a characterization is known for ordinary AC (seee.g. [34], p. 228), which in turn follows from the following theorem on choosingg = τ .

24.3. Theorem. Suppose f : I → R is regulated and internal , and g ∈ B is

internal. Then f ≪ g iff the following conditions hold :

(a) D∗gf exists and is finite µg-a.e., and it is µg-summable on I,

(b) f is continuous relative to g, and

(c) f possesses the property (Ng).

P r o o f. On account of Theorem 16.4, (a) is weaker than the condition (i) ofTheorem 24.2. Hence the necessity part follows directly from that theorem.

To prove the sufficiency, let us first observe that by Lemma 20.4, |D∗gg| = 1

µg-a.e., and so |D∗gf | = |D∗

gf ·D∗gg| = |D∗

gf | µg-a.e. Consequently, there is againno loss of generality in assuming g to be nondecreasing.

Now, when f and g are normalized, then using this time (a) and Lemma 21.2in place of Theorem 16.4 and (1), the earlier proof of the sufficiency part (in theproof of Theorem 24.2) remains valid.

To obtain the sufficiency in general, suppose (a), (b) and (c) hold.Then (a) and(c) clearly also hold for f∗ and g∗, and the same holds for (b) due to Theorem 14.3.Hence by the above result f∗ ≪ g∗, and so by Theorem 11.2, f∗ ∈ B. Now sincef is internal, it is easy to see that f ∈ B, and hence it follows from Theorem 14.4that f ≪ g.

24.4. R ema r k. We include here an example to show that the hypothesis of fbeing internal, although not needed in Theorem 24.2, is essential for the validityof Theorem 24.3.

Let {xn} be any increasing sequence of points in I, and set A = {xn : n =1, 2, . . .} and B = I ∼ A. Define f(xn) = 1/n for each n, and f(x) = 0 for x ∈ B.Also define, for each n,

g(xn) = xn + n−2 + 2∑

i<n

i−2 ,

and for x ∈ B,

g(x) = x+ 2∑

i:xi<x

i−2 .

It is then easy to see that g is increasing, and that it is continuous at the pointsof B. Also, for each n,

g(xn)− g(xn − 0) = n−2 = g(xn + 0)− g(xn) .

82 K. M. Garg

Hence g is normalized, and it is discontinuous from both sides at the points ofA. Also, Cf = B, so that f is continuous relative to g, and since f∗ ≡ 0, D∗

gfexists and is zero everywhere, and so it is trivially µg-summable. Further, f hasthe property (Ng) since f(I) is countable. Thus all the three conditions (a), (b)and (c) of Theorem 24.3 hold for f and g. However, since Vf = 2

∑

n 1/n = ∞,f is not AC relative to g by Theorem 11.2.

As regards Theorem 24.2, it should be pointed out here that a somewhat simi-lar result has been obtained earlier by Chakrabarty [2]. However, the definitionsused there are considerably different and somewhat involved.

25. Integration by parts for LS-integral. This section deals with an appli-cation of one of the Radon–Nikodym theorems established earlier in §17. We ob-tain here from this theorem two formulae for integration by parts for LS-integral,one of which is known.

Given f, g ∈ B, it is necessary to modify here the definitions of f∗ and g∗ atthe end-points of I. As it was done in §17 while defining Ds

gf , define f(x) = f(a)for x < a and f(x) = f(b) for x > b. It is then more natural to define f∗ at a andb as at the interior points of I. Thus, for the purposes of this section, we define

f∗(a) = 12{f(a+ 0) + f(a)} and f∗(b) = 1

2{f(b) + f(b− 0)} .

Hence f will be normalized here at a or b iff it is continuous at that point. Thesame will apply to g.

We need here two lemmas.

25.1. Lemma. If f ∈ B, then

µf2(E) = 2∫

E

f∗dµf , E ∈ B .

P r o o f. Using the above extension of f to R we first observe that, for eachx ∈ I,

limh→0+

f2(x+ h)− f2(x− h)

f(x+ h)− f(x− h)= lim

h→0+{f(x+ h) + f(x− h)} = 2f∗(x) .

Hence, Dsf (f

2)(x) = 2f∗(x).

Further, since f ≪ f , it follows from Theorem 10.5 that f2 ≪ f . Hence it fol-lows clearly from Theorem 14.4 that (f2)∗ ≪ f∗. Consequently, by Theorem 13.1,µf2 ≪ µf . The result follows now from Theorem 17.4.

25.2. Lemma. Let f, g ∈ B. Then for each E ∈ B,

(1)∫

E

f∗dµg +∫

E

g∗dµf = µfg(E) .

P r o o f. Since 2fg = (f + g)2 − f2 − g2, with the help of Lemmas 3.1 and 4.1we obtain from Lemma 25.1,

µfg(E) = 12{µ(f+g)2 (E) − µf2(E)− µg2(E)}


=∫

E

(f + g)∗dµf+g −∫

E

f∗dµf −∫

E

g∗dµg

=∫

E

f∗dµg +∫

E

g∗dµf .

We now obtain the following known formula for integration by parts (see [34],p. 102).

25.3. Theorem. Let f, g ∈ B, and suppose either (i) f and g are normalized ,or (ii) at each point of I at least one of f and g is continuous. Then for every

closed subinterval J ≡ [x, y] of I,∫

J

f dµg +∫

J

g dµf = f(y + 0)g(y + 0)− f(x− 0)g(x − 0) .

P r o o f. In case (i) the result follows directly from the last lemma. To obtainthe result in the other case, suppose (ii) holds. Then since A ≡ {x : f∗(x) 6= f(x)}is a countable subset of Cg, µg(A) = 0, and so the first integral in (1) remainsunaltered on replacing f∗ by f . Similarly, g∗ in the second integral can be replacedby g, and thus the result follows again from the last lemma.

For the indefinite LS-integral (see §18) we obtain, on the other hand, thefollowing formula for integration by parts.

25.4. Theorem. Let f, g ∈ B, and suppose at each point of I at least one of

f and g is continuous. Then if a ≤ x < y ≤ b, theny∫

x

f dg +y∫

x

g df = f(y)g(y)− f(x)g(x) .

P r o o f. Given a ≤ x < y ≤ b, let J = (x, y). Then by Lemma 25.2, as justseen in the above proof,

∫

J

f dµg +∫

J

g dµf =∫

J

dµfg = f(y − 0)g(y − 0)− f(x+ 0)g(x + 0) .

Hence,y∫

x

f dg +

y∫

x

g df = f(y − 0)g(y − 0)− f(x+ 0)g(x + 0)

+ f(y){g(y)− g(y − 0)} + f(x){g(x+ 0)− g(x)}

+ g(y){f(y)− f(y − 0)}+ g(x){f(x + 0)− f(x)}

= f(y)g(y)− f(x)g(x) ,

where the last equality follows clearly from the continuity hypothesis.

25.5. R ema r k. The above proof of Lemma 25.2, based on Theorem 17.4, isessentially due to Daniell [4]. However, his proof of Theorem 17.4 applied only tocontinuous f and g (see Remark 17.5).

84 K. M. Garg

26. Relative Lebesgue points. In this section we present a relative notionof Lebesgue points and obtain relativized versions of some of the known resultson Lebesgue points.

Let g ∈ B and ϕ be a µg-summable function on I. Let x ∈ I, and suppose g∗

is not constant in any neighbourhood of x. We will then call x a Lebesgue point

of ϕ relative to g provided

limh→0

1

g∗(x+ h)− g∗(x)

x+h∫

x

|ϕ(t)− ϕ(x)| dg∗ = 0 ,

where h approaches 0 through those values for which g∗(x+ h) 6= g∗(x).

We need here the following lemma to obtain the normalization of the indefiniteintegral of ϕ.

26.1. Lemma. Suppose g ∈ B and ϕ is a µg-summable function on I. If f is

the indefinite LS-integral of ϕ relative to g, then f∗ is the indefinite LS-integralof ϕ relative to g∗.

P r o o f. Define k(x)=f(a) +∫ x

aϕ dg∗, x∈I. We need to show here that k=

f∗. According to Theorem 18.2, k is continuous relative to g∗ and it satisfies thecondition (Lg∗). Hence k is continuous at the points of Cg∗ , so that k∗(x) = k(x)for x ∈ Cg∗ . Also, this relation holds clearly for x = a or b. In case x ∈ I0 ∼ Cg∗ ,then since g∗(x + 0) − g∗(x) = g∗(x) − g∗(x − 0), it follows from the condition(Lg∗) satisfied by k that k(x + 0) − k(x) = k(x) − k(x − 0), i.e. k∗(x) = k(x).Consequently, k∗ = k everywhere.

Next, since µg = µg∗ , it is clear that f(x) = k(x) when x ∈ Cg or x = a orb. Hence, given x ∈ I, since f and k are regulated by Theorem 18.2, it followsfrom the denseness of Cg in I that f(x+ 0) = k(x+ 0) and f(x− 0) = k(x− 0).Consequently, f∗(x) = k∗(x) = k(x), i.e. k = f∗ everywhere.

The following three theorems relativize some of the known results on ordinaryLebesgue points (see e.g. [28], pp. 255, 256), which in turn follow from thesetheorems on choosing g = τ .

26.2. Theorem. Let g ∈ B, and suppose f is the indefinite LS-integral ofsome µg-summable function ϕ relative to g. Then if x is a Lebesgue point of ϕrelative to g, then ϕ(x) = D∗

gf(x).

P r o o f. Suppose x is a Lebesgue point of ϕ relative to g. Since f∗ is by theabove lemma the indefinite LS-integral of ϕ relative to g∗, it is clear that

∣

∣

∣

∣

f∗(x+ h)− f∗(x)

g∗(x+ h)− g∗(x)− ϕ(x)

∣

∣

∣

∣

=

∣

∣

∣

∣

1

g∗(x+ h)− g∗(x)

x+h∫

x

{ϕ(t)− ϕ(x)} dg∗∣

∣

∣

∣

≤

∣

∣

∣

∣

1

g∗(x+ h)− g∗(x)

x+h∫

x

|ϕ(t)− ϕ(x)| dg∗∣

∣

∣

∣

.


But according to the hypothesis the last expression tends to zero as h → 0.Consequently, ϕ(x) = D∗

gf(x).

26.3. Theorem. Suppose g ∈ B is internal. If a function ϕ is µg-summable

on I, then µg-almost every point of I is a Lebesgue point of ϕ relative to g.

P r o o f. Since g is internal, g∗ = g∗ by Theorem 8.1. Also, since µg∗ = µg,it is clear from the definition of relative Lebesgue points that there is no loss ofgenerality in assuming g to be normalized. Then of course g also is normalized.

Let {rn} be an enumeration of the set of all rational numbers. Define, for eachn, ϕn = |ϕ− rn|. Given n, since ϕn is µg-summable, it is also summable relativeto µg = µg (see Lemma 7.1). Hence it follows clearly from Corollary 18.4 andLemma 26.1 that there is a set An in I such that µg(I ∼ An) = 0, and for everyx ∈ An,

(1) ϕn(x) = limh→0

1

g(x+ h)− g(x)

x+h∫

x

ϕn dg .

Set A =⋂

nAn. Then µg(I ∼ A) = 0.Now, given x ∈ A and ε > 0, choose an n such that ϕn(x) < ε. Then, by (1),

there exists a δ > 0 such that if 0 < |h| < δ, then∣

∣

∣

∣

1

g(x+ h)− g(x)

x+h∫

x

ϕn dg

∣

∣

∣

∣

< ε .

Now since |ϕ(t)−ϕ(x)| = |ϕ(t)− rn + rn −ϕ(x)| ≤ ϕn(t)+ϕn(x), it follows thatif 0 < |h| < δ, then

∣

∣

∣

∣

1

g(x+ h)− g(x)

x+h∫

x

|ϕ(t) − ϕ(x)| dg(t)

∣

∣

∣

∣

≤

∣

∣

∣

∣

1

g(x+ h)− g(x)

x+h∫

x

{ϕn(t) + ϕn(x)} dg(t)

∣

∣

∣

∣

< 2ε .

Consequently,

limh→0

1

g(x+ h)− g(x)

x+h∫

x

|ϕ(t) − ϕ(x)| dg = 0 .

Next, by Lemma 20.4, there is a set B ⊂ I such that µg(I ∼ B) = 0, and forevery x ∈ B, |g′g(x)| = 1. Now if x ∈ A ∩B, then

limh→0

1

g(x+ h)− g(x)

x+h∫

x

|ϕ(t) − ϕ(x)| dg

= limh→0

1

g(x+ h)− g(x)

x+h∫

x

|ϕ(t)− ϕ(x)| dg · limh→0

g(x+ h)− g(x)

g(x+ h)− g(x)= 0 .

86 K. M. Garg

This proves the result since µg(I ∼ A ∩B) = 0.

26.4. Theorem. Suppose g ∈ B and ϕ is a µg-summable function on I. Then

every point x ∈ I where ϕ is continuous and |D∗gg(x)| > 0 is a Lebesgue point of

ϕ relative to g. Consequently , if g is nondecreasing , then every point x ∈ I where

ϕ is continuous is a Lebesgue point of ϕ relative to g.

P r o o f. Suppose ϕ is continuous at x ∈ I and α ≡ |D∗gg(x)| > 0. Then,

given ε > 0, there exists a δ > 0 such that if 0 < |h| < δ and x + h ∈ I, then|ϕ(t)− ϕ(x)| < ε and

∣

∣

∣

∣

g∗(x+ h)− g∗(x)

g∗(x+ h)− g∗(x)

∣

∣

∣

∣

>α

2.

Hence for such a value of h,∣

∣

∣

∣

1

g∗(x+ h)− g∗(x)

x+h∫

x

|ϕ(t) − ϕ(x)| dg∗∣

∣

∣

∣

≤ ε

∣

∣

∣

∣

g∗(x+ h)− g∗(x)

g∗(x+ h)− g∗(x)

∣

∣

∣

∣

<2ε

α.

Now on making ε→ 0 it follows that x is a Lebesgue point of ϕ relative to g.

27. Arc length of rectifiable curves under relative AC. In this sectionwe obtain, in Theorems 27.4 and 27.8, relativized versions of two known theoremson arc length (see Corollaries 27.5 and 27.6). The latter theorem also provides acharacterization of relative AC in terms of arc length. A similar characterizationof mutual singularity is obtained in the next section.

Given f, g ∈ B, let C be the curve in R2 given by x = f(t), y = g(t), t ∈ I.

This curve is known to be rectifiable (see e.g. [34], p. 123). For each pair of pointst1, t2 ∈ I, t1 < t2, we will use σ(f, g; t1, t2) to denote the length of the linearsegment from (f(t1), g(t1)) to (f(t2), g(t2)), viz.

(1) σ(f, g; t1, t2) = [{f(t2)− f(t1)}2 + {g(t2)− g(t1)}

2]1/2 .

Now, if P is any partition a = t0 < t1 < . . . < tn = b of I, the arc length ofthe polygon whose vertices {(f(ti), g(ti)): i = 0, 1, . . . , n} are on C is given by thesum

∑ni=1 σ(f, g; ti−1, ti). The limit of this sum as the norm of P → 0 is called

the arc length of C.We will use L(f, g) to denote the arc length of C, which is of course finite.

Also, for each t ∈ I, we will use La,t(f, g) to denote the arc length of the subarcof C obtained by restricting f and g to [a, t]. Further, we will use sf,g, or s, todenote the arc length function

(2) s(t) ≡ sf,g(t) = La,t(f, g), t ∈ I .

Clearly, s is a nondecreasing function for which s(a) = 0 and s(b) = L(f, g).We will use also L(f) to denote the arc length of the graph of f , i.e. L(f) =

L(f, τ) where τ is the identity function.We begin with some preliminary results. The following result of Jordan in

terms of the above notations is well known (see e.g. [34], p. 123).


27.1. Lemma. If a ≤ t1 < t2 ≤ b, then

max{f(t2)− f(t1), g(t2)− g(t1)}

≤ s(t2)− s(t1) ≤ {f(t2)− f(t1)}+ {g(t2)− g(t1)} .

27.2. Theorem. Given f, g, u ∈ B, suppose u is nondecreasing , and let

s = sf,g. Then

(a) s is left or right continuous at t ∈ I iff f and g are so;(b) s is continuous relative to u at t ∈ I iff f and g are so;

(c) if f and g are normalized , then so is s;

(d) if f and g satisfy the condition (Lu), then so does s;

(e) s≪ u iff f ≪ u and g ≪ u; and(f) s ⊥ u iff f ⊥ u and g ⊥ u.

P r o o f. Given t ∈ I, since f and g are left or right continuous at t iff f and gare so, the part (a) follows directly from the above lemma. The part (b) followsin turn from (a).

Now, if t ∈ I0, we have clearly

s(t+ 0)− s(t) = [{f(t+ 0)− f(t)}2 + {g(t + 0)− g(t)}2]1/2 ,

s(t)− s(t− 0) = [{f(t)− f(t− 0)}2 + {g(t) − g(t− 0)}2]1/2 .

The parts (c) and (d) follow easily from these two relations.

Next, since f ≪ u and g ≪ u implies by Theorems 10.2 and 10.3 that f+g ≪u, the part (e) follows easily from the above lemma on using the definition ofrelative AC. The part (f) follows similarly from Theorem 6.1 and the above lemmaon using the definition of mutual singularity.

27.3. Theorem. Given f, g, u ∈ B, suppose u is nondecreasing , and let s =sf,g. Then

(a) D∗us = [(D∗

uf)2 + (D∗

ug)2]1/2 µu-a.e., and

(b) L(f, g) ≥∫ b

a[(D∗

uf)2 + (D∗

ug)2]1/2du.

P r o o f. It is easy to see from the two equations considered in the previousproof that L(f∗, g∗) ≤ L(f, g). Hence there is no loss of generality here in as-suming f, g and u to be normalized. Then s also is normalized by the previoustheorem. Hence it follows from Theorem 16.4 that there is a set P ∈ B such thatµu(I ∼ P ) = 0 and each of f, g and s has a finite derivative relative to u at thepoints of P .

To prove (a), first consider any point t ∈ I ∼ Cu. If t ∈ I0, then it is easy tosee that

s(t+ 0)− s(t− 0) = [{f(t+ 0)− f(t− 0)}2 + {g(t+ 0)− g(t− 0)}2]1/2 .

Hence it follows from Lemma 16.1 that s′u(t) = [{f ′u(t)}

2+{g′u(t)}2]1/2. A similar

argument holds when t = a or b.

88 K. M. Garg

Next, suppose t ∈ P ∩ Cu, h 6= 0, t+ h ∈ I and u(t+ h) 6= u(t). Then since

|s(t+ h)− s(t)| ≥ [{f(t+ h)− f(t)}2 + {g(t + h)− g(t)}2]1/2 ,

on dividing the two sides of this inequality by |u(t + h) − u(t)| and then takinglimit as h→ 0 we obtain

s′u(t) ≥ [(f ′u(t))

2 + (g′u(t))2]1/2 .

To obtain equality, let A be the set of points in P ∩ Cu where this inequality isstrict. It is then enough to show that µu(A) = 0.

Now let An denote, for each positive integer n, the set of points t in A forwhich the following inequality holds whenever 0 < |h| < 1/n, t + h ∈ I andu(t+ h) 6= u(t):

(3)s(t+ h)− s(t)

u(t+ h)− u(t)>

[{

f(t+ h)− f(t)

u(t+ h)− u(t)

}2

+

{

g(t+ h)− g(t)

u(t+ h)− u(t)

}2]1/2

+1

n.

Clearly, A =⋃

nAn.

Now, given n and ε > 0, choose a partition a = t0 < t1 < . . . < tk = b of Isuch that ti − ti−1 < 1/n for each i ∈ Sk and

(4) L(f, g) <

k∑

i=1

σ(f, g; ti−1, ti) + ε .

Let S denote the set of indices i ∈ Sk for which [ti−1, ti]∩An 6= ∅. Then for eachi ∈ S there exists some point t ∈ An ∩ [ti−1, ti], and so we obtain from (3),

s(ti)− s(ti−1) = {s(ti)− s(t)}+ {s(t)− s(ti−1)}

> σ(f, g; t, ti) + σ(f, g; ti−1, t) + {u(ti)− u(ti−1)}/n

≥ σ(f, g; ti−1, ti) + {u(ti)− u(ti−1)}/n .

Now since An ⊂⋃

i∈S[ti−1, ti] and u is continuous at the points of An, it is thusclear that

µu(An) ≤∑

i∈S

{u(ti)− u(ti−1)}

< n∑

i∈S

{s(ti)− s(ti−1)− σ(f, g; ti−1, ti)}

≤ n

k∑

i=1

{s(ti)− s(ti−1)− σ(f, g; ti−1, ti)} .

Hence it follows from (4) that µu(An) < nε. Now on making ε → 0 we obtainµu(An) = 0. Consequently, µu(A) = 0.

This proves the part (a). The part (b) follows on the other hand from (a) andTheorem 16.4 since µs is positive.


Let us recall here that a function f ∈ B was defined in §14 to be “partiallycontinuous” relative to g ∈ B if it is continuous relative to g at the points in I0

where g is unilaterally discontinuous.

27.4. Theorem. Given f, g, u ∈ B, suppose u is nondecreasing , and let

s = sf,g.

(a) If f ≪ u and g ≪ u, then

(5) L(f, g) =b∫

a

[(D∗uf)

2 + (D∗ug)

2]1/2du ,

and the converse holds provided f and g are partially continuous relative to u.(b) If f ≪s u and g ≪s u, then

(6) s(t) =t∫

a

[(D∗uf)

2 + (D∗ug)

2]1/2du, t ∈ I .

P r o o f. First, suppose f ≪ u and g ≪ u. Then s≪ u by Theorem 27.2, andsince u is internal, it follows from Theorem 14.4 that s∗ ≪ u∗. Hence (5) followsfrom Theorems 17.1 and 27.3.

Now, to prove the converse, suppose (5) holds and f and g are partially con-tinuous relative to u. Then the same holds for s by Theorem 27.2. Further, sinces is nondecreasing, it is obvious that s∗ ≪− u∗. Hence it follows from (5) andTheorems 23.5 and 27.3 that s∗ ≪ u∗. Consequently s ≪ u by Theorem 14.4,and so by Theorem 27.2, f ≪ u and g ≪ u.

Finally, if f ≪s u and g ≪s u, then it follows from Theorem 27.2 that s≪s u,and so (6) follows from Lemma 18.6 and Theorems 18.2 and 27.3.

We include here two consequences of the last two theorems which indicate therelationship of these theorems with the existing results in the direction.

On choosing u = τ in Theorems 27.3 and 27.4 we obtain, with the help ofTheorem 16.4, the following theorem of Tonelli (see [35; 36], or [34], p. 123).

27.5. Corollary. Let f, g ∈ B and s = sf,g. Then

(a) s′(t) = [{f ′(t)}2 + {g′(t)}2]1/2 for almost every t;

(b) L(f, g) ≥∫ b

a[{f ′(t)}2 + {g′(t)}2]1/2 dt; and

(c) f and g are AC iff equality holds in (b).

Now, on choosing g also to be τ , we obtain the following known characteriza-tion of AC (see [29], pp. 227, 228).

27.6. Corollary. A function f ∈ B is AC iff

L(f) =b∫

a

[1 + (f ′(t))2]1/2 dt .

Next, we will obtain a relativized version of the last corollary by choosing u tobe g in Theorems 27.3 and 27.4. For this purpose we need the following lemma.

90 K. M. Garg

27.7. Lemma. If f, g ∈ B, then L(f, g) = L(f, g) and sf,g = sf,g.

P r o o f. We need to verify here only the inequality L(f, g) ≤ L(f, g), for thereverse inequality is obvious. Given ε > 0, since g+ ⊥ g− (see Corollary 5.6),there exists a partition a = t0 < t1 < . . . < tn = b of I for which there is adecomposition (S−, S+) of Sn such that

∑

i∈S+

{g+(ti)− g+(ti−1)}+∑

i∈S−

{g−(ti)− g−(ti−1)} < ε .

By refining this partition if necessary we can further assume that

L(f, g) <

n∑

i=1

σ(f, g; ti−1, ti) + ε .

Now, for each i ∈ Sn, it is easy to see that

g(ti)− g(ti−1) ≤ |g(ti)− g(ti−1)|

+ 2min{g+(ti)− g+(ti−1), g−(ti)− g−(ti−1)} .

Next, using the fact that the length of any side of a triangle is less than thesum of the lengths of the other two sides, we obtain from the above inequalities,

L(f, g) <∑

i∈S+

[σ(f, g; ti−1, ti) + 2{g+(ti)− g+(ti−1)}

+∑

i∈S−

[σ(f, g; ti−1, ti) + 2{g−(ti)− g−(ti−1)}+ ε

<

n∑

i=1

σ(f, g; ti−1, ti) + 3ε ≤ L(f, g) + 3ε .

The required inequality is now obtained on making ε→ 0.

This establishes the relation L(f, g) = L(f, g). Further, on applying the aboveargument to the interval [a, t], t ∈ I, we obtain the relation sf,g(t) = sf,g(t) forevery t ∈ I.

27.8. Theorem. Given f, g ∈ B, suppose g is internal , and let s = sf,g. Then

(a) |D∗gs| = [1 + (D∗

gf)2]1/2 µg-a.e.;

(b) L(f, g) ≥∫ b

a[1 + (D∗

gf)2]1/2 dg;

(c) if f ≪ g, then equality holds in (b), and the converse holds provided f is

partially continuous relative to g; and

(d) if f ≪s g, then s(t) =∫ t

a[1 + (D∗

gf)2]1/2 dg, t ∈ I.

P r o o f. Since g is internal, µg = µg by Lemma 7.1, and by Lemma 20.4,|D∗

gg| = 1 µg-a.e. Hence |D∗gf | = |D∗

gf · D∗gg| = |D∗

gf | µg-a.e., and similarlyD∗

gs = |D∗gs| µg-a.e. Using these facts and Lemma 27.7, the parts (a) and (b)

follow from Theorem 27.3 on choosing u to be g.


Next, since f ≪ g iff f ≪ g, g ≪ g, and f is due to Lemma 2.1 continuous atany point relative to g iff it is so relative to g, the part (c) follows similarly fromTheorem 27.4.

Further, if f ≪s g, then since g is internal, it is easy to see that g ≪s g, andso f ≪s g. Consequently, (d) also follows from Theorem 27.4 on choosing u to beg.

28. A general formula for arc length and a problem of Denjoy. In thissection we first obtain in Theorem 28.2 a general formula for arc length whichholds without any hypothesis, and is based on the relative Lebesgue decomposi-tion. Then in Theorem 28.4 we obtain a characterization of mutual singularityin terms of arc length. Finally, in Theorem 28.5, we obtain a solution of an oldproblem of Denjoy [7] on arc length in higher dimensions.

To obtain the general formula for arc length we need the following lemma.

28.1. Lemma. Let ϕ,ψ, g ∈ B. If ϕ ⊥− ψ and ψ ⊥ g, then

L(ϕ+ ψ, g) = L(ϕ, g) + V ψ .

P r o o f. Given a ≤ t1 < t2 ≤ b, since the length of any side of a triangle is lessthan the sum of the lengths of the other two sides, we have

(1) |σ(ϕ+ ψ, g; t1, t2)− σ(ϕ, g; t1, t2)| ≤ |ψ(t2)− ψ(t1)| .

Hence it is clear that

(2) L(ϕ+ ψ, g) ≤ L(ϕ, g) + V ψ .

Now suppose ϕ ⊥− ψ and ψ ⊥ g. Then given ε > 0, there exists a partitiona = t0 < t1 < . . . < tn = b of I for which there is a decomposition (S−, S+) of Sn

such that

(3)∑

i∈S+

{ψ(ti)− ψ(ti−1)}+∑

i∈S−

{

g(ti)− g(ti−1)}

< ε .

By refining this partition if necessary we can assume further that

(4) L(ϕ, g) <

n∑

i=1

σ(ϕ, g; ti−1, ti) + ε .

Also, since ϕ ⊥− ψ, by partitioning the intervals [ti−1, ti], i ∈ S− if necessary, wecan also assume (see the proof of Theorem 5.3) that

(5)∑

i∈S−

|ϕ(ti) + ψ(ti)− ϕ(ti−1)− ψ(ti−1)|

>∑

i∈S−

|ϕ(ti)− ϕ(ti−1)|+∑

i∈S−

|ψ(ti)− ψ(ti−1)| − ε ,

92 K. M. Garg

and that, due to (3),

(6)∑

i∈S−

|ψ(ti)− ψ(ti−1)| >∑

i∈S−

{ψ(ti)− ψ(ti−1)}

− ε > V ψ − 2ε .

Now from (1) and (3) we obtain∑

i∈S+

σ(ϕ+ ψ, g; ti−1, ti) ≥∑

i∈S+

σ(ϕ, g; ti−1, ti)−∑

i∈S+

|ψ(ti)− ψ(ti−1)|

>∑

i∈S+

σ(ϕ, g; ti−1, ti)− ε .

Further, from (3), (5) and (6) we obtain∑

i∈S−

σ(ϕ+ ψ, g; ti−1, ti)

≥∑

i∈S−

|ϕ(ti) + ψ(ti)− ϕ(ti−1)− ψ(ti−1)|

>∑

i∈S−

|ϕ(ti)− ϕ(ti−1)|+∑

i∈S−

|ψ(ti)− ψ(ti−1)| − ε

>∑

i∈S−

σ(ϕ, g; ti−1ti)−∑

i∈S−

|g(ti)− g(ti−1)|+ V ψ − 3ε

>∑

i∈S−

σ(ϕ, g; ti−1, ti) + V ψ − 4ε .

Now on combining the last two inequalities we obtain, with the help of (4),

L(ϕ+ ψ, g) ≥n∑

i=1

σ(ϕ+ ψ, g; ti−1, ti)

>

n∑

i=1

σ(ϕ, g; ti−1, ti) + V ψ − 5ε > L(ϕ, g) + V ψ − 6ε .

Thus on making ε → 0 we obtain L(ϕ + ψ, g) ≥ L(ϕ, g) + V ψ. The desiredequation is obtained on combining this inequality with (2).

28.2. Theorem. Given f, g ∈ B, suppose g is internal , and let ϕ and ψ be the

AC and singular components respectively of f relative to g. Then

(7) L(f, g) =b∫

a

[1 + (D∗gϕ)

2]1/2 dg + V ψ .

Moreover , if f satisfies the condition (Lg), then

(8) s(t) =t∫

a

[1 + (D∗gϕ)

2]1/2 dg + Va,tψ, t ∈ I .

Furthermore, if f is partially continuous relative to g, then (7) and (8) also


hold with D∗gf in place of D∗

gϕ.

P r o o f. Since ϕ ⊥ ψ by Theorem 13.4, it follows from the above lemma that

(9) L(f, g) = L(ϕ, g) + V ψ .

Now since ϕ≪ g, (7) follows from Theorem 27.8.

Next, if f satisfies the condition (Lg), then so does ϕ by Theorem 23.3. Thusϕ ≪s g, and since g ≪s g, we have indeed ϕ ≪s g. Hence (8) follows fromTheorem 27.8 and an equation similar to (9) for [a, t].

Further, if f is partially continuous relative to g, then it follows easily fromTheorem 23.3 that ψ and g are nowhere unilaterally discontinuous from oppositesides. Hence by Corollary 20.2, D∗

gψ = 0 µg-a.e., and so D∗gf = D∗

gϕ µg-a.e.Now since g is internal, µg = µg, and so (7) and (8) also hold with D∗

gf in placeof D∗

gϕ.

On choosing g = τ in the above theorem we obtain the following generalformula for the arc length of the graph of any function of bounded variation. Thisformula seems to be new and may be compared with the known result stated inCorollary 27.6.

28.3. Corollary. Let f ∈ B. Then

L(f) =b∫

a

[1 + (f ′(t))2]1/2 dt+ Vfs .

The following theorem deals with the characterization of mutual singularityin terms of arc length.

28.4. Theorem. Let f, g ∈ B. Then f ⊥ g iff

(10) L(f, g) = Vf + Vg .

P r o o f. The necessity part follows directly from Lemma 28.1 on choosingϕ ≡ 0 and ψ = f .

To prove the sufficiency, suppose (10) holds, and let L = L(f, g). Now, givenε > 0, choose a partition a = t0 < t1 < . . . < tn = b of I such that

L <

n∑

i=1

σ(f, g; ti−1, ti) + ε

≤n∑

i=1

[{f(ti)− f(ti−1)}2 + {g(ti)− g(ti−1)}

2]1/2 + ε .

Let S+ denote the set of indices i ∈ Sn for which

f(ti)− f(ti−1) ≥ g(ti)− g(ti−1) ,

and set S− = Sn ∼ S+. When 0 ≤ α ≤ β, it is easy to see that (α2 + β2)1/2 ≤

94 K. M. Garg

β + α/2. Hence we have

L <∑

i∈S+

[{f(ti)− f(ti−1)}+12{g(ti)− g(ti−1)}]

+∑

i∈S−

[12{f(ti)− f(ti−1)}+ {g(ti)− g(ti−1)}] + ε .

But since L = Vf + Vg =∑

i∈Sn[{f(ti)− f(ti−1)}+ {g(ti)− g(ti−1)}], it is thus

clear that∑

i∈S−

{f(ti)− f(ti−1)}+∑

i∈S+

{g(ti)− g(ti−1)} < 2ε .

This proves that f ⊥ g.

Next we consider the Denjoy’s problem. Given f1, . . . , fn ∈ B, n > 1, the arclength of the curve x1 = f1(t), . . . , xn = fn(t), t ∈ I, in n-dimensions is definedin an analogous manner. We will denote it by L(f1, . . . , fn), and La,t(f1, . . . , fn),t ∈ I, is also defined as before.

Denjoy [7] proposed the following problem for nondecreasing functions fi: Tofind conditions which are necessary and sufficient in order that La,t(f1, . . . , fn) =f1(t) + . . .+ fn(t) for t ∈ I.

In the following theorem we present a more general result on functions ofbounded variation which is similar to the above theorem. The solution of Denjoy’sproblem follows from this theorem on choosing fi’s to be nondecreasing functionsfor which fi(a) = 0 for each i. In the case when n = 2 this solution has beenobtained earlier by Kober [23] in terms of his notion of “contravariance” (seeRemark 9.8).

28.5. Theorem. Let f1, . . . , fn ∈ B, n > 1. Then

La,t(f1, . . . , fn) = Va,tf1 + . . .+ Va,tfn, t ∈ I ,

iff f1, . . . , fn are pairwise mutually singular.

P r o o f. It is enough to prove the result for t = b, for if fi and fj are mutuallysingular, then so are their restrictions to [a, t] for any t ∈ I.

We will prove the result by induction. For n = 2 the result has already beenestablished in the last theorem.

Given n > 2, suppose the result holds for n− 1. Define

s(t) = La,t(f1, . . . , fn−1), t ∈ I .

It is then clear that L(f1, . . . , fn) = L(s, fn) and s(b) = L(f1, . . . , fn−1). Further,by Theorem 28.4, L(s, fn) = s(b) + Vfn iff s ⊥ fn.

We will next prove that s ⊥ fn iff fi ⊥ fn for each i < n. Given a ≤ t1 <t2 ≤ b, since

maxi<n

|fi(t2)− fi(t1)| ≤[

n−1∑

i=1

{fi(t2)− fi(t1)}2]1/2

≤n−1∑

i=1

|fi(t2)− fi(t1)| ,


it is clear that

(11) maxi<n

{f i(t2)− f i(t1)} ≤ s(t2)− s(t1) ≤n−1∑

i=1

{f i(t2)− f i(t1)} .

Hence if s ⊥ fn, it follows from the definition of mutual singularity that fi ⊥ fnfor each i < n.

Next, to prove the converse, suppose fi ⊥ fn for each i < n. Then, givenε > 0, by choosing successive refinements of partitions we can find a partitiona = t0 < t1 < . . . < tk = b of I with decompositions (Si−, Si+), i = 1, . . . , n − 1,of Sk such that for each i < n,

(12)∑

j∈Si+

{f i(tj)− f i(tj−1)}+∑

j∈Si−

{fn(tj)− fn(tj−1)} < ε .

Now set S+ =⋂n−1

i=1 Si+ and S− =⋃n−1

i=1 Si−. Then it follows from (11) that

∑

j∈S+

{s(tj)− s(tj−1)} ≤∑

j∈S+

n−1∑

i=1

{f i(tj)− f i(tj−1)} <n−1∑

i=1

ε = (n− 1)ε ,

∑

j∈S−

{fn(tj)− fn(tj−1)} ≤n−1∑

i=1

∑

j∈Si−

{fn(tj)− fn(tj−1)} <n−1∑

i=1

ε = (n − 1)ε .

Hence on replacing ε by ε/2(n − 1) in (12) it follows that s ⊥ fn.

We have thus proved that L(f1, . . . , fn) = L(f1, . . . , fn−1) + Vfn iff fi ⊥ fnfor each i < n. Hence it follows from the induction hypothesis that the resultholds for n.

VI. Convergence in B

29. Stability of variations and components under norm convergence.

In this section we first introduce a commonly used norm on B under which Bis known to be a Banach space. Then we obtain some theorems determiningthe variations (viz. the variation functions) and components of norm limits ofsequences and series of functions in B as norm limits of corresponding variationsand components of the elements of the given sequence or series. Componentsrelative to other functions in B are also considered.

The linear space B is known to be a Banach space under the following normwhich is sometimes called the variation norm:

‖f‖ = |f(a)|+ Vf, f ∈ B ,

(see e.g. [8], p. 337, or [12]). When it is necessary to distinguish this norm fromother norms, it will be denoted by ‖f‖v .

96 K. M. Garg

It is clear that a sequence {fn} in B converges in this norm to f ∈ B ifffn(a) → f(a) and V (fn − f)→ 0 as n→ ∞. Further, this norm is clearly stron-ger than the uniform norm, and so convergence in this norm implies pointwiseconvergence.

To avoid repetition, we assume in this section {fn} to be a sequence of elementsof B and f, g ∈ B. Also, we will use fn

v−→f to denote the convergence of the

sequence {fn} in the (variation) norm to f . Similarly,∑

n fnv

−→f will denotethe convergence of the series

∑

n fn in the norm to f , i.e. the convergence of thesequence of partial sums of the series, viz. sn =

∑ni=1 fi, n = 1, 2, . . . , in the

norm to f .We begin with the variations of the norm limit of a sequence in B.

29.1. Theorem. Given {fn} and f in B, then fnv

−→f iff

(1) fn(a) −→ f(a), f+n

v−→f+ and f−

nv

−→f− .

Consequently , if fnv

−→f , then fnv

−→f .

P r o o f. First, suppose fnv

−→f . Then, of course, fn(a) → f(a). To prove theother two relations of (1), let n be fixed, and let a ≤ x < y ≤ b. Then

V +x,yf ≤ V +

x,y(f − fn) + V +x,yfn ≤ Vx,y(f − fn) + V +

x,yfn .

Hence,

(f+ − f+n )(y)− (f+ − f+

n )(x)

= {f+(y)− f+(x)} − {f+n (y)− f+

n (x)} ≤ Vx,y(f − fn) .

On combining this inequality with the one obtained on reversing the roles of fand fn, we thus obtain

|(f+n − f+)(y)− (f+

n − f+)(x)| ≤ Vx,y(fn − f) .

It is now clear that

‖f+n − f+‖ = V (f+

n − f+) ≤ V (fn − f) ≤ ‖fn − f‖ ,

and so f+n

v−→f+. The proof of the other relation is similar.

Next, to prove the converse, suppose the relations in (1) hold. Then sinceV (fn − f) ≤ V (f+

n − f+) + V (f−n − f−), it is clear that ‖fn − f‖ → 0 as n→ ∞,

i.e. fnv

−→f .The last part also follows from the inequality ‖fn−f‖ ≤ ‖f+

n −f+‖+‖f−n −f−‖

for each n.

The following theorem on the variations of the limit of a series is based onTheorem 5.3 since it involves additivity of various variations, and is indeed anextension of that theorem from finite sums to infinite sums. This theorem hasbeen quoted and used earlier in [14] in the construction of certain classes of ACand continuous singular functions.

The functions in a series∑

n fn are called here “pairwise mutually LS” iffi ⊥− fj whenever i 6= j.


29.2. Theorem. Suppose∑

n fnv

−→f . Then fn’s are pairwise mutually LS iff

any of the following equivalent conditions holds:

(a) Vf =∑

n

Vfn, (a′) f =∑

n

fn ,

(b) V +f =∑

n

V +fn, (b′) f+ =∑

n

f+n ,

(c) V −f =∑

n

V −fn, (c′) f− =∑

n

f−n ,

where the convergence in (a′), (b′) and (c′) may be considered to be pointwise, or ,equivalently , in the norm.

P r o o f. It is enough to prove the result for pointwise convergence in (a′), (b′)and (c′). For, as the series in each of these parts consists of nondecreasing func-tions, its pointwise convergence automatically implies convergence in the norm.

Set sn =∑n

i=1 fi for each n. We will first prove the equivalence of pairwisemutual LS of fn’s with (a).

First, suppose fn’s are pairwise mutually LS. Then if n > 1, it follows by arepeated application of Theorem 6.1 that fn ⊥− sn−1. Hence, by Theorem 5.3,

Vsn = Vsn−1 + Vfn = Vsn−2 + Vfn−1 + Vfn = . . . =n∑

i=1

Vfi .

Consequently, by Theorem 29.1, Vf = limn Vsn =∑

n Vfn.Next, suppose (a) holds but there exists some pair of functions in

∑

n fn, sayf1, f2, which are not mutually LS. Since the series

∑

n>2 fn also converges inthe norm, it follows once again from Theorem 29.1 that

V(

∑

n>2

fn

)

= limnV(

n∑

i=3

fi

)

≤ limn

n∑

i=3

Vfi =∑

n>2

Vfn <∞ .

Hence, by Theorem 5.3,

Vf ≤ V (f1 + f2) + V(

∑

n>2

fn

)

< Vf1 + Vf2 +∑

n>2

Vfn ,

which contradicts (a). This proves the equivalence of (a).Now since (a′)⇒(a), it is enough to prove the implications

(a) ⇒ (b) ⇒ (c) ⇒ (c′) ⇒ (b

′) ⇒ (a

′) .

(a)⇒(b). If (a) holds, then

V +f =1

2{Vf + f(b)− f(a)} =

1

2

∑

n

{Vfn + fn(b)− fn(a)} =∑

n

V +fn .

(b)⇒(c). If (b) holds, then

V −f = V +f − f(b) + f(a) =∑

n

{V +fn − fn(b) + fn(a)} =∑

n

V −fn .

98 K. M. Garg

(c)⇒(c′). Suppose (c) holds but (c′) does not hold. Then there exists somepoint x ∈ I such that f−(x) <

∑

n f−n (x). Now since

|V −x,bf − V −

x,bsn| ≤ V −x,b(f − sn) ≤ ‖f − sn‖ → 0 ,

we have

V −x,bf = lim

nV −x,bsn ≤ lim

n

n∑

i=1

V −x,bfi =

∑

n

V −x,bfn ≤

∑

n

V −fn <∞ .

Hence,

V −f = f−(x) + V −x,bf <

∑

n

f−n (x) +

∑

n

V −x,bfn =

∑

n

V −fn ,

which contradicts (c).(c′)⇒(b′). If (c′) holds, then for each x ∈ I,

f+(x) = f−(x) + f(x)− f(a) =∑

n

{f−n (x) + fn(x)− fn(a)} =

∑

n

f+n (x) .

(b′)⇒(a′). If (b′) holds, then for each x ∈ I,

f(x) = 2f+(x)− f(x) + f(a) =∑

n

{2f+n (x)− fn(x) + fn(a)} =

∑

n

fn(x) .

The next two theorems deal with the stability of various components undernorm convergence.

29.3. Theorem. (a) fnv

−→f iff (fn)cv

−→fc and (fn)dv

−→fd.

(b)∑

n fnv

−→f iff∑

n(fn)cv

−→fc and∑

n(fn)dv

−→fd.

P r o o f. To prove (a), let n be given. Then

fn − f = {(fn)c − fc}+ {(fn)d − fd} ,

where (fn)c − fc and (fn)d − fd are mutually singular by Lemma 7.4. Hence, byTheorem 5.3,

V (fn − f) = V {(fn)c − fc}+ V {(fn)d − fd} .

Consequently, we have

‖fn − f‖ = |fn(a)− f(a)|+ V (fn − f)

= |(fn)c(a)− fc(a)|+ V {(fn)c − fc}+ V {(fn)d − fd}

= ‖(fn)c − fc‖+ ‖(fn)d − fd‖ .

The part (a) follows clearly from this relation.Next, to prove (b), set for each n, sn =

∑ni=1 fi. Then since sn =

∑ni=1(fi)c+

∑ni=1(fi)d, where

∑ni=1(fi)c is continuous and

∑ni=1(fi)d is a jump function, it

is clear that

(sn)c =

n∑

i=1

(fi)c and (sn)d =

n∑

i=1

(fi)d .

Hence (b) follows from (a).


29.4. Theorem. Given {fn} and f , g in B, let ϕn +ψn be for each n the Le-

besgue decomposition of fn relative to g, and ϕ+ψ be the Lebesgue decomposition

of f relative to g. Then

(a) fnv

−→f iff ϕnv

−→ϕ and ψnv

−→ψ; and

(b)∑

n fnv

−→f iff∑

n ϕnv

−→ϕ and∑

n ψnv

−→ψ.

P r o o f. Given n, we have (ϕn − ϕ) ≪ g by Theorems 10.1 and 10.3, and(ψn − ψ) ⊥ g by Theorems 5.1 and 6.1. Hence it follows from Theorem 13.4 thatϕn − ϕ and ψn − ψ are mutually singular, and so by Theorem 5.3, V (fn − f) =V (ϕn − ϕ) + V (ψn − ψ). It follows now from the definition of relative Lebesguedecomposition that

‖fn − f‖ = |fn(a)− f(a)|+ V (fn − f)

= |ϕn(a)− ϕ(a)| + V (ϕn − ϕ) + V (ψn − ψ)

= ‖ϕn − ϕ‖+ ‖ψn − ψ‖ .

The part (a) follows clearly from this relation.Next, to prove (b), set for each n, sn =

∑ni=1 fi, un =

∑ni=1 ϕi and vn =

∑ni=1 ψi. Then sn = un + vn, where un ≪ g by Theorem 10.3 and vn ⊥ g by

Theorem6.1.Also, according to the definition of relative Lebesgue decomposition,

un(a) =

n∑

i=1

ϕi(a) =

n∑

i=1

fi(a) = sn(a) .

Hence it follows from the uniqueness of the relative Lebesgue decomposition (seeTheorem 23.1) that un + vn is the Lebesgue decomposition of sn relative to g. Itis now clear that (b) follows from (a).

Finally, on choosing g = τ in the above theorem we obtain the following resulton the ordinary AC and singular components.

29.5. Corollary. (a) fnv

−→f iff (fn)av

−→fa and (fn)sv

−→fs.

(b)∑

n fnv

−→f iff∑

n(fn)av

−→fa and∑

n(fn)sv

−→fs.

30. Norm closed sets and subspaces of B. In this section we first obtainclosedness in the norm of the sets of elements of B which are LC, continuous,LAC, AC, LS or singular relative to any given element of B (see Theorem 30.1).Some of these sets are thus closed subspaces of B.

Further, analogous to the decomposition of continuity, AC and singularity inChapters II and III, we introduce here a decomposition of the property of jumpfunctions into lower and upper jump functions. The sets of such elements alsoturn out to be norm closed.

30.1. Theorem.Given g ∈ B, the sets of elements of B which are LC, LAC or

LS relative to g are closed convex cones in B. Consequently , the sets of elements

of B which are continuous, AC or singular relative to g are closed subspaces of B.

100 K. M. Garg

P r o o f. Let A, B and C denote the sets of elements of B which are LC, LACor LS respectively relative to g. The fact that A is a convex cone follows easilyfrom the definition of relative LC (see §12), and this property of B follows fromTheorems 10.1 and 10.3, and that of C from Theorems 5.1 and 6.1.

To see that A is closed, suppose fn ∈ A for each n and fnv

−→f . To prove theLC of f relative to g, first suppose g is right continuous at some point x ∈ I.Then fn(x) ≤ fn(x + 0) for each n, and since {fn} converges uniformly to f , itis easy to see that f(x+ 0) = limn fn(x+ 0). Hence

f(x) = limnfn(x) ≤ lim

nfn(x+ 0) = f(x+ 0) .

When g is left continuous at some point x ∈ I, it is proved similarly that f(x−0) ≤f(x). Hence f ∈ A, which proves that A is closed.

We will next consider C. Suppose fn ∈ C for each n and fnv

−→f . Then fn+gv

−→f+g, and so, by Theorem 29.1, Vf = limn Vfn and V (f+g) = limn V (fn+g).But for any n, since fn ⊥− g, V (fn + g) = V fn + V g by Theorem 5.3. Hence,

V (f + g) = limnVfn + Vg = Vf + Vg .

Thus it follows again from Theorem 5.3 that f ⊥− g, i.e. f ∈ C.Next, we will obtain the closedness of B with the help of Theorem 13.4. Set

H = {h ∈ B+ : h ⊥ g}, and for each h ∈ H let Eh = {f ∈ B : f ⊥− h}.Then according to Theorem 13.4, f ∈ B iff f ∈ Eh for every h ∈ H. ThusB =

⋂

h∈H Eh. But it is clear from the closedness of C that Eh is closed for eachh ∈ H. Hence B is closed.

Next, let A0, B0 and C0 denote the sets of elements of B which are continuous,AC or singular respectively relative to g. Then it is clear that A0 = A ∩ (−A),and so A0 is a closed (linear) subspace of B. A similar argument holds for B0

and C0.

Now on choosing g = τ in the above theorem we obtain the following resulton the sets of LC, continuous, LAC, AC, LS and singular elements of B. Thelast part of this corollary is known (see e.g. [12]).

30.2. Corollary. The sets of LC, LAC and LS elements of B are closed con-

vex cones in B. Consequently , the sets of continuous , AC and singular elements

of B are closed subspaces of B.

We now come to the decomposition of the property of jump functions.Given f ∈ B, let f be called a lower or upper jump function if for each ε > 0

there exists a finite subset F of I such that for every finite set of nonoverlappingintervals {[ai, bi] : i = 1, . . . , n} included in I ∼ F ,

n∑

i=1

{f(bi)− f(ai)} > −ε or < ε respectively .

In the following theorem we obtain a characterization of the property of lowerjump functions. It follows easily from this theorem as seen in Corollary 30.4 that


f is a jump function iff it is a lower and upper jump function.

30.3. Theorem. A function f ∈ B is a lower jump function iff f− is a jump

function. Consequently , f is nondecreasing iff it is a LC, lower jump function.

P r o o f. We will prove the necessity part by contradiction. Hence suppose fis a lower jump function but f− is not a jump function. Then f−

c (b) > 0. Setε = 1

4f−c (b), and let F = {xi : i = 1, . . . , p} be any given finite subset of I.

Let {yi} be the points of discontinuity of f . Now choose an integer q such that∑

i>q ωf (yi) < ε, and a δ > 0 such that

(1) |fc(x)− fc(y)| <ε

p+ qwhenever x, y ∈ I and |x− y| ≤ 2δ .

Also, since f−c (b) = 4ε, we can find a finite set of nonoverlapping intervals {[ai, bi]:

i = 1, . . . , n} in I such that

(2)

n∑

i=1

{fc(bi)− fc(ai)} < −3ε .

Now let {[ci, di]: i = 1, . . . , k} be the set of closed intervals that are obtainedon deleting {

⋃pi=1(xi−δ, xi+δ)}∪{

⋃qi=1(yi−δ, yi+δ)} from the intervals {[ai, bi]:

i = 1, . . . , n}. Then it follows clearly from (1) and (2) that

k∑

i=1

{f(di)− f(ci)} <k

∑

i=1

{fc(di)− fc(ci)}+∑

i>q

ωf (yi)

<

n∑

i=1

{fc(bi)− fc(ai)}+ ε+ ε < −ε .

But this clearly contradicts the hypothesis that f is a lower jump function.Next, to prove the sufficiency, suppose f− is a jump function. Let {xi} be the

points of discontinuity of f−. Given ε > 0, choose k such that∑

i>k ωf−(xi) < ε.Now set F = {xi : i = 1, . . . , k}. Then if {[ai, bi] : i = 1, . . . , n} is any finite set ofnonoverlapping intervals included in I ∼ F , we have

n∑

i=1

{f(bi)− f(ai)} ≥ −n∑

i=1

{f−(bi)− f−(ai)} ≥ −∑

i>k

ωf−(xi) > −ε .

This proves that f is a lower jump function.The last part follows on the other hand from the first with the help of Corol-

lary 12.5.

Since f is clearly a jump function iff f− and f+ are jump functions, the abovetheorem leads to the following new characterization of jump functions.

30.4. Corollary. A function f ∈ B is a jump function iff it is a lower and

upper jump function, or , equivalently , iff for each ε > 0 there exists a finite

subset F of I such that for every finite set of nonoverlapping intervals {[ai, bi] :i = 1, . . . , n} included in I ∼ F ,

∑ni=1 |f(bi)− f(ai)| < ε.

102 K. M. Garg

Finally, in the following theorem we deal with the properties of some of theremaining sets in B.

30.5. Theorem. The sets of nondecreasing and lower jump functions are

closed convex cones in B. Consequently , the set of jump functions is a closed

subspace of B.

P r o o f. Let A and B denote the sets of elements ofB which are nondecreasingor lower jump functions respectively. Then it is clear that A and B are convexcones. Also, it is clear that A is closed under pointwise convergence, and so it isalso norm closed.

To see that B is norm closed, let C denote the set of continuous nondecreasingelements of B, and set for each g ∈ C, Eg = {f ∈ B : f ⊥− g}. We claim thatf ∈ B iff f ∈ Eg for every g ∈ C.

First, suppose f ∈ B, and let g be any element of C. Then f− is a jumpfunction by Theorem 30.3, and so f− ⊥ g by Lemma 7.4. Hence f ∈ Eg byCorollary 6.4. Next, to prove the converse, suppose f 6∈B. Then by Theorem 30.3,f− is not a jump function, so that f−

c is not constant. Now set g = f−c . Then it

is clear that f− and g are not mutually singular, and so f 6∈ Eg by Corollary 6.4.The claim is thus established.

We have thus proved that B =⋂

g∈C Eg. Hence it follows from Theorem 30.1that B is norm closed.

Now, if B0 is the set of jump functions in B, then B0 = B ∩ (−B) by Corol-lary 30.4, and so B0 is a closed subspace of B.

31. Strong convergence and term-by-term differentiation. In this sec-tion we first obtain a relativized version of Fubini’s theorem on term-by-termdifferentiation (see Theorem 31.1), and then we obtain an extension of this ver-sion under a notion of convergence in B which is stronger than norm convergence(see Theorem 31.2). This leads to another generalization of Fubini’s theorem froma series of nondecreasing functions to a norm convergent series in B whose ele-ments are pairwise mutually LS (see Corollary 31.3).

Also, similar results are obtained on term-by-term differentiation of stronglyconvergent and norm convergent sequences in B (see Corollaries 31.4 and 31.6).

Given a sequence {fn} in B, we will call {fn} strongly Cauchy if the sequence{fn(a)} and the series

∑

n V (fn+1 − fn) are convergent in R. A series∑

n fn inB will in turn be called strongly Cauchy if its sequence of partial sums is so, i.e. ifthe series

∑

n fn(a) and∑

n V fn are convergent in R.Every strongly Cauchy sequence {fn} in B is clearly also Cauchy in the norm,

and so, due to the completeness of B, it converges in the norm to some elementf of B. We will then call f the strong limit of {fn}, and write fn

s−→f .

It is interesting to note here that strong convergence in B is in a sense noless common than norm convergence, for as we see in Theorem 31.5, every normconvergent sequence in B admits a strongly convergent subsequence. The com-


pleteness of B is thus obtained in that theorem as a consequence of the extendedversion of Fubini’s theorem.

Further, when a series∑

n fn in B consists of nondecreasing functions, itspointwise convergence to f ∈ B is clearly equivalent to its strong convergenceto f . If, on the other hand,

∑

n fn converges in the norm to f and fn’s arepairwise mutually LS, then it follows from Theorem 29.2 that

∑

n fns

−→f .When a series of nondecreasing functions

∑

n fn in B converges pointwise tosome function f , Fubini proved that f ′(x) =

∑

n f′n(x) for almost every x (see

[9]; or [19], p. 267).We begin with the following relativized version of this theorem.

31.1. Theorem. Let∑

n fn be a series of nondecreasing functions in B which

converges pointwise to f ∈ B. Then for every internal function g ∈ B,

D∗gf(x) =

∑

n

D∗gfn(x) for µg-almost every x .

P r o o f. Let us first assume that g is nondecreasing. Also, by replacing eachfn by fn − fn(a), we can assume that fn’s are nonnegative.

Define, for each n, sn =∑n

i=1 fi and rn = f−sn. According to Theorem 16.4,there exists a set A ∈ B such that µg(I ∼ A) = 0 and each of the functions fn(n = 1, 2, . . .) and f has a finite normalized derivative relative to g at the pointsof A.

Now, given n and x ∈ A, suppose x+h ∈ I and g∗(x+h) 6= g∗(x). Then sincef∗n+1 and r∗n+1 are nondecreasing, it is clear that

0 ≤s∗n(x+ h)− s∗n(x)

g∗(x+ h)− g∗(x)≤s∗n+1(x+ h)− s∗n+1(x)

g∗(x+ h)− g∗(x)≤f∗(x+ h)− f∗(x)

g∗(x+ h)− g∗(x).

Hence 0 ≤ D∗gsn(x) ≤ D∗

gsn+1(x) ≤ D∗gf(x). Thus {D

∗gsn(x)} is a bounded non-

decreasing sequence, and so is convergent. Consequently, the series∑

nD∗gfn(x)

converges for every x ∈ A.Next, since rn(b) → 0 as n → ∞, we can choose an increasing sequence of

positive integers {ni} such that∑

i rni(b) < ∞. Then, for each i, since rni

isnondecreasing, we have 0 ≤ rni

(x) ≤ rni(b) for every x ∈ I. Hence it follows

from the comparison test that∑

i rni(x) converges for every x ∈ I. Also, rni

isnonnegative and nondecreasing for each i. Hence on applying the above argumentto this series it follows that there is a set B ∈ B such that µg(I ∼ B) = 0and

∑

iD∗grni

(x) is convergent for every x ∈ B. Consequently, for each x ∈ B,D∗

grni(x) → 0 as i→ ∞. Thus we obtain

D∗gf(x) = lim

iD∗

gsni(x) for x ∈ B .

Now for each x ∈ A ∩B, since {D∗gsn(x)} is convergent, it follows that

D∗gf(x) = lim

nD∗

gsn(x) = limn

n∑

i=1

D∗gfi(x) =

∑

n

D∗gfn(x) .

104 K. M. Garg

This proves the result in the case when g is nondecreasing, for µg(I ∼ A∩B) = 0.

In the general case, on applying the above result to g we obtain a set E ∈ Bsuch that µg(I ∼ E) = 0 and

(1) D∗gf(x) =

∑

n

D∗gfn(x) for x ∈ E .

Further, by Lemma 20.4, there exists a set F ∈ B such that µg(I ∼ F ) = 0 and|D∗

gg(x)| = 1 for x ∈ F . Now let x ∈ E ∩ F . Then if D∗gg(x) = 1, we have

D∗gf(x) = D∗

gf(x) ·D∗gg(x) = D∗

gf(x),

and similarly D∗gfn(x) = D∗

gfn(x) for each n, so that we obtain from (1),

D∗gf(x) =

∑

n

D∗gfn(x) .

A similar argument holds whenD∗gg(x) = −1. This proves the result since µg(I ∼

E ∩ F ) = 0 (see Lemma 7.1).

We now obtain an extension of the above theorem to any strongly Cauchyseries in B. As we see later in Remark 31.7, such an extension does not holdunder norm convergence in general.

31.2. Theorem. If∑

n fn is a strongly Cauchy series in B, then it converges

in the norm to some f ∈ B, and for every internal g ∈ B,

D∗gf(x) =

∑

n


Consequently , f ′(x) =∑

n f′n(x) for almost every x.

P r o o f. Suppose∑

n fn is strongly Cauchy. Then the series∑

n fn(a) and∑

n Vfn converge to some real numbers α and β respectively. Now since 0 ≤

f±n (x) ≤ f±

n (b) ≤ Vfn for each n and x, it follows from a theorem of Weierstrass([20], p. 115) that the series

∑

n f+n and

∑

n f−n converge uniformly to some

functions u and v respectively on I. Clearly, u and v are nondecreasing. Nowdefine

(2) f(x) = α+ u(x)− v(x), x ∈ I .

Then f ∈ B. Further, since

(3) fn(x) = fn(a) + f+n (x)− f−

n (x)

for each n and x, it follows that

f(x) =∑

n

fn(x), x ∈ I .

Now define sn =∑n

i=1 fi for each n. Then the sequence {sn} converges uni-formly to f . Given n, if a = x0 < x1 < . . . < xk = b is any partition of I,


thenk∑

j=1

|(f − sn)(xj)− (f − sn)(xj−1)| ≤k

∑

j=1

∑

i>n

|fi(xj)− fi(xj−1)|

=∑

i>n

k∑

j=1

|fi(xj)− fi(xj−1)| ≤∑

i>n

Vfi .

Hence V (f − sn) ≤∑

i>n Vfi for each n. Thus it follows from the convergence of∑

n Vfn that V (f − sn) → 0 as n → ∞. Consequently,∑

nfn converges in thenorm to f .

Next, by the above theorem, there exists a set E ⊂ I with µg(I ∼ E) = 0such that for each x ∈ E the functions f+

n , f−n (n = 1, 2, . . .), u and v have finite

normalized derivatives relative to g at x and

D∗gu(x) =

∑

n

D∗gf

+n (x) and D∗

gv(x) =∑

n

D∗gf

−n (x) .

Hence when x ∈ E, it follows from (2), (3) and Lemma 3.1 that the functions fn(n = 1, 2, . . .) and f also have finite normalized derivatives relative to g at x and

D∗gf(x) = D∗

gu(x)−D∗gv(x) =

∑

n

{D∗gf

+n (x)−D∗

gf−n (x)} =

∑

n

D∗gfn(x) .

This proves the first part of the theorem.Now, on choosing g = τ , the last part follows clearly from the first with the

help of Theorem 16.4.

In the case of a series∑

n fn whose elements are pairwise mutually LS, we ob-tain from the above theorem the following version of Fubini’s theorem under normconvergence, for such a series is strongly Cauchy by Theorem 29.2. This versionalso generalizes the Fubini’s theorem since any pair of nondecreasing functionsare automatically mutually LS.

31.3. Corollary. Let∑

n fnbe a series in B whose elements are pairwise

mutually LS. Suppose∑

n fnv

−→f and g ∈ B is internal. Then

D∗gf(x) =

∑

n


Consequently , f ′(x) =∑

n f′n(x) for almost every x.

Further, if {fn} is a sequence in B, on applying the above theorem to the series∑

n gn, where g1 = f1 and gn = fn − fn−1 for n > 1, we obtain the followingresult on strongly Cauchy sequences.

31.4. Corollary. If {fn} is a strongly Cauchy sequence in B, then it conver-

ges in the norm to some f ∈ B, and for every internal g ∈ B,

D∗gf(x) = lim

nD∗

gfn(x) for µg-almost every x .

106 K. M. Garg

Consequently , f ′(x) = limn f′n(x) for almost every x.

We next obtain the completeness of B from the last corollary.

31.5. Theorem. Every Cauchy sequence in B admits a strongly Cauchy sub-

sequence. Consequently , B is complete.

P r o o f. Given a Cauchy sequence {fn} in B, choose a subsequence {fni} such

that ‖fni−fnj

‖ < 1/2i whenever i < j. Then since |fni(a)−fnj

(a)| ≤ ‖fni−fnj

‖,{fni

(a)} is a Cauchy sequence in R, and so is convergent. Also,

∑

i

V (fni+1− fni

) ≤∑

i

‖fni+1− fni

‖ <∑

i

1

2i= 1 .

Hence {fni} is strongly Cauchy.

Now, by the last corollary, {fni} converges in the norm to some f ∈ B. But

since {fn} is Cauchy in the norm, this implies that fnv

−→f .

Finally, on combining the above theorem with Corollary 31.4 we obtain

31.6. Corollary. If a sequence {fn} in B converges in the norm to f ∈ B,then it admits a subsequence {fni

} such that for every internal g ∈ B, D∗gf(x) =

limiD∗gfni

(x) for µg-almost every x. Consequently , f ′(x) = limi f′ni(x) for almost

every x.

31.7. R ema r k. We include here an example to show that the term-by-termdifferentiation in Theorem 31.2 and Corollary 31.4 do not hold under norm co-nvergence in general.

Let I = [0, 1], and let us recall here that χE denotes, for each set E ⊂ I, thecharacteristic function of E on I. Define

ϕ1 = χ[0,1/2], ϕ2 = χ[1/2,1],

ϕ3 = χ[0,1/4], ϕ4 = χ[1/4,1/2], ϕ5 = χ[1/2,3/4], ϕ6 = χ[3/4,1] ,

and so on. Now define, for each n,

fn(x) =x∫

0

ϕn(t) dt, x ∈ I .

Then {fn} is a sequence of nondecreasing, AC functions in B.

Now, since fn(0) = 0 and Vfn =∫ 1

0ϕn(t) dt for each n, it is clear that

Vfn → 0 as n → ∞. Hence {fn} converges in the norm to f ≡ 0. However,since f ′

n(x) = ϕn(x) for almost every x, the sequence {f ′n(x)} clearly has no limit

for almost every x. The same holds for the relative derivative as it follows onchoosing g = τ . Thus Corollary 31.4 does not hold under norm convergence.

Further, on setting g1 = f1 and gn = fn − fn−1 for n > 1, we obtain a normconvergent series

∑

n gn in B for which∑

n g′n(x) has no limit for almost every

x. Hence Theorem 31.2 also does not hold under norm convergence.


32. Stability of arc length under strong convergence. With the helpof the general formula for arc length obtained earlier in Theorem 28.2, we obtainhere a theorem dealing with stability of arc length under strong convergence.

For this purpose we need the following lemma.

32.1. Lemma. If {fn} is a strongly Cauchy sequence in B, then there exists a

normalized nondecreasing function u on I such that u− f∗n is nondecreasing for

each n.

P r o o f. Let us first observe that the sequence {f∗n} also is strongly Cauchy.

Since f∗n(a) = fn(a) for each n, the sequence {f∗

n(a)} is obviously convergent inR. Further, for each n we have by Lemma 3.1,

V (f∗n+1 − f∗

n) = V (fn+1 − fn)∗ ≤ V (fn+1 − fn) ,

so that∑

n

V (f∗n+1 − f∗

n) ≤∑

n

V (fn+1 − fn) <∞ .

Consequently, {f∗n} is strongly Cauchy, and hence the elements of the given se-

quence {fn} can be assumed to be normalized without any loss of generality.Now define

(1) u(x) = f1(x) +

∞∑

n=1

( fn+1 − fn )(x), x ∈ I .

Since∑

n V (fn+1 − fn) < ∞, u is clearly a well defined nondecreasing functionon I. Let sn denote for each n the partial sum of the first n terms in the serieson the right side of (1). Then sn is normalized for each n by Lemma 3.1 andTheorem 8.1, and sn

v−→u. Hence, given x ∈ I0, we have, as observed earlier,

u(x+0) = limn sn(x+0) and u(x− 0) = limn sn(x− 0). And so it follows clearlyfrom the normalizedness of sn’s that u is normalized.

Now, given n and a ≤ x < y ≤ b, since

fn = f1 + (f2 − f1) + . . . + (fn − fn−1) ,

we have

Vx,yfn ≤ Vx,yf1 + Vx,y(f2 − f1) + . . .+ Vx,y(fn − fn−1) ≤ u(y)− u(x) .

Thus fn(y) − fn(x) ≤ u(y) − u(x), so that u(x) − fn(x) ≤ u(y) − fn(y). Thisproves that u− fn is nondecreasing.

32.2. Theorem. Suppose a sequence {fn} in B converges strongly to f , andg ∈ B is internal. Then

(2) L(f, g) = limnL(fn, g) .

Consequently , L(f) = limn L(fn).

P r o o f. Let ϕn +ψn be for each n the Lebesgue decomposition of fn relativeto g, and ϕ + ψ be the Lebesgue decomposition of f relative to g. Then, by

108 K. M. Garg

Theorem 29.4, ϕnv

−→ϕ and ψnv

−→ψ. Also, for each n, ϕn(a) = fn(a), and since(ϕn+1 − ϕn) ⊥ (ψn+1 − ψn) by Theorem 13.4, it follows from Theorem 5.3 thatV (ϕn+1 − ϕn) ≤ V (fn+1 − fn). Hence it follows from the strong convergence of

{fn} to f that ϕns

−→ϕ. Consequently, we have by Corollary 31.4,

(3) D∗gϕ(x) = lim

nD∗

gϕn(x) for µg-almost every x .

Next, since g is internal, µg = µg by Lemma 7.1. Hence by Theorem 28.2 wehave

L(f, g) =∫

I

[1 + (D∗gϕ)

2]1/2 dµg + V ψ ,

and, for each n,

L(fn, g) =∫

I

[1 + (D∗gϕn)

2]1/2 dµg + V ψn .

But since ψnv

−→ψ, we have V ψ = limn V ψn by Theorem 29.1. Hence to obtain(2) it is enough to show that

(4)∫

I

[1 + (D∗gϕ)

2]1/2 dµg = limn

∫

I

[1 + (D∗gϕn)

2]1/2 dµg .

Let u be a normalized nondecreasing function on I as determined by the abovelemma such that u− f∗

n is nondecreasing for each n. Now define

v(x) = [1 + {D∗gu(x)}

2]1/2

for each x ∈ I for which D∗gu(x) exists. Then v is µg-summable by Theorem 27.8.

We claim that for each n,

(5) [1 + {D∗gϕn(x)}

2]1/2 ≤ v(x) for µg-almost every x .

Given n, since u − f∗n is nondecreasing, it is clear that u− ϕ∗

n also is nonde-creasing. Hence if a ≤ x < y ≤ b, then

|ϕ∗n(y)− ϕ∗

n(x)| ≤ ϕ∗n(y)− ϕ∗

n(x) ≤ u(y)− u(x) ,

and so if g∗(y) 6= g∗(x), we have∣

∣

∣

∣

ϕ∗n(y)− ϕ∗

n(x)

g∗(y)− g∗(x)

∣

∣

∣

∣

≤

∣

∣

∣

∣

u(y)− u(x)

g∗(y)− g∗(x)

∣

∣

∣

∣

.

Hence |D∗gϕn(x)| ≤ |D∗

gu(x)| at each point x where the two normalized derivativesexist. Consequently, we obtain (5) with the help of Theorem 16.4, and hence (4)follows from (3) with the help of Lebesgue’s dominated convergence theorem.

This establishes (2). The last part follows of course from (2) on choosingg = τ .

In the case of norm convergence we obtain, on the other hand, the followingresult from the above theorem with the help of Theorem 31.5.


32.3. Corollary. Suppose fnv

−→f and g ∈ B is internal. Then there is a sub-

sequence {fni} of {fn} such that L(f, g) = limi L(fni

, g) and L(f) = limi L(fni).

33. Approximation in some subspaces of B by elementary functions.

In this section we obtain some results dealing with the approximation of functionsin some closed subspaces of B by elementary functions in those subspaces relativeto the variation norm.

To be specific, the jump functions can be approximated by step functionsin general (see Theorem 33.1), and under certain hypotheses, the AC functionsrelative to some u ∈ B can be approximated by piecewise linear functions relativeto u (Theorem 33.3). As it will be seen in the next section, the latter class offunctions can also be approximated in the variation norm by polynomials in u orin its components.

Also, the functions singular relative to u can be approximated by generalizedstep functions as defined later (Theorem 33.6), and every normalized function inB can be approximated by a generalized linear function relative to any normalizedfunction in B (Theorem 33.7).

In the particular case when u = τ , some of these results were indicated orproved earlier in [12] (see Remark 33.8).

We will use here Bd, Bc, Ba and Bs to denote the sets of purely discontinuous(or jump functions), continuous, AC or singular elements respectively of B. Eachof these sets was seen in §30 to be a closed subspace of B. (A similar notationhas been used in ([18], p. 269) for the spaces of LS-measures.)

Also, we will use B∗ to denote the set of normalized elements of B. It is clearfrom Lemma 3.1 that B∗ is a linear subspace of B, and it is easy to see that B∗

is closed under uniform convergence. Hence B∗ also is a closed subspace of B.Further, given any element u of B, we will use Ba(u) and Bs(u) to denote the

sets of elements f of B such that f ≪ u or f ⊥ u respectively. These sets werealso seen in §30 to be closed subspaces of B.

Now notations like B∗s , Bcs, Bcs(u) have obvious meanings, viz. B∗

s = B∗∩Bs,Bcs = Bc ∩Bs and Bcs(u) = Bc ∩Bs(u). All such sets are of course again closedsubspaces of B.

We will begin with the subspace Bd.Let us recall here that a function f : I → R is called a step function if there

exists a partition a = x0 < x1 . . . < xn = b of I such that f is constant on each ofthe open intervals (xi−1, xi), i = 1, . . . , n. Clearly, every step function is a jumpfunction.

33.1. Theorem. Given f ∈ Bd and ε > 0, there exists a step function g which

is continuous relative to f such that ‖f−g‖ < ε. Consequently , the step functions

constitute a dense subset of Bd.

P r o o f. Given f ∈ Bd, let {xn} be the points where f is discontinuous, andfor each n let ωn denote the oscillation of f at xn. If {xn} is finite, then f itself is a

110 K. M. Garg

step function, and so g can be chosen to be f . Hence suppose {xn} is infinite, andlet ε > 0. Then since

∑

n ωn = Vf <∞, there exists an n such that∑

i>n ωi < ε.Set A = {xi : i = 1, . . . , n}, and if a ∈ A, let f(a− 0) denote f(a).

Now define g(a) = f(a), and if x ∈ I, x > a, then

g(x) =

{

∑′i≤n{f(xi + 0)− f(xi − 0)}+ f(x)− f(x− 0) if x ∈ A,

∑′i≤n{f(xi + 0)− f(xi − 0)} otherwise,

where the summation∑

′ is taken only over those values of i ≤ n for which xi < x.Then it is clear that g is a step function which is continuous relative to f , andthat

‖f − g‖ = V (f − g) =∑

i>n

ωi < ε .

This proves the first part, and the second part follows directly from the first.

Next we will consider the subspace B∗a(u) where u ∈ B.

Given u ∈ B, let a partition a = x0 < x1 < . . . < xn = b of I be called au-partition of I if u is continuous at xi for i = 1, . . . , n− 1. A function f : I → R

will in turn be called a u-step function if there exists such a u-partition of I sothat f is constant on (xi−1, xi) for i = 1, . . . , n.

Next, f will be called linear relative to u, or simply u-linear, if f = α+βu forsome α, β ∈ R. Further, if there exists a u-partition a = x0 < x1 < . . . < xn = bof I such that f is u-linear on [xi−1, xi] for i = 1, . . . , n, then f will be calledpiecewise u-linear.

33.2. Lemma. Suppose u ∈ B and f : I → R is µu-summable. Then for every

ε > 0 there exists a u-step function ϕ on I such that

∫

I

|f − ϕ| dµu < ε .

P r o o f. Choose a µu-measurable simple function ψ =∑n

i=1 ciχAisuch that

∫

I

|f − ψ| dµu < ε/3 .

Let M = maxi≤n |ci| and δ = ε/(3nM). Now, given i ≤ n, there clearly existsan open subset Gi of I with finitely many (connected) components such thatµu(Gi∆Ai) < δ, where ∆ denotes the symmetric difference. Also, since Cu isdense in I, by shrinking the components of Gi slightly if necessary, we can assumethat the endpoints of all the components of Gi, except possibly a and b, are in Cu.

Now define ϕ =∑n

i=1 ciχGi. Then ϕ is clearly a u-step function. Set E =

{x ∈ I : ϕ(x) 6= ψ(x)}. Then

µu(E) ≤∑

i≤n

µu{x ∈ I : χGi(x) 6= χAi

(x)} =∑

i≤n

µu(Gi∆Ai) < nδ =ε

3M.


Consequently,∫

I

|f − ϕ| dµu ≤∫

I

|f − ψ| dµu +∫

I

|ψ − ϕ| dµu <ε

3+

∫

E

2M dµu < ε .

33.3. Theorem. Given u ∈ B∗, the piecewise u-linear functions in B∗ consti-

tute a dense subset of B∗a(u). Consequently , the piecewise linear functions on I

constitute a dense subset of Ba.

P r o o f. Let A denote the set of all piecewise u-linear functions in B∗. Thenit is clear that A ⊂ B∗

a(u).

To prove the denseness of A in B∗a(u), let f ∈ B∗

a(u) and ε > 0. Then byTheorem 16.4, f ′

u exists and is finite µu-a.e., and it is µu-summable on I. Henceby the above lemma there exists a u-step function ϕ on I such that

(1)∫

I

|f ′u − ϕ| dµu < ε .

Now define

g(x) = f(a) +x∫

a

ϕdu, x ∈ I .

Then it is clear that g ∈ A. Also, since u is normalized, so is clearly g, and sof − g is normalized by Lemma 3.1. Further, f − g ≪ u by Theorems 10.1 and10.3, and g(a) = f(a). Hence it follows from Corollary 17.3 that

‖f − g‖ = V (f − g) =∫

I

|f ′u − g′u| dµu .

But according to the definition of g we have by Corollary 18.4, g′u = ϕ µu-a.e.Hence it follows from (1) that ‖f − g‖ < ε. This proves that A is dense in B∗

a(u).

The last part follows clearly from above on choosing u = τ , for the piecewiseτ -linear functions are obviously AC and B∗

a(τ) = Ba.

To obtain an extension of the above theorem to the subspace Ba(u), let afunction f ∈ B be called weakly piecewise u-linear if there exists a partitiona = x0 < x1 < . . . < xn = b of I such that f is u-linear on each of the openintervals (xi−1, xi), i = 1, . . . , n.

33.4. Corollary. If u ∈ B∗, then there is a dense set of elements in Ba(u)which are weakly piecewise u-linear.

For, let f ∈ Ba(u) and ε > 0. Then by Corollary 10.4, fc ≪ u and fd ≪ u.Hence by the above theorem there exists a piecewise u-linear function ϕ ∈ B∗

a(u)such that ‖fc−ϕ‖ < ε/2. Also, by Theorem 33.1 there exists a step function ψ inB which is continuous relative to fd such that ‖fd−ψ‖ < ε/2. Now let g = ϕ+ψ.Then it is clear that g is weakly piecewise u-linear. Also, since fd is continuousrelative to u by Lemma 13.2, the same holds for ψ, and so ψ ≪ u by the same

112 K. M. Garg

lemma. Consequently, g ≪ u by Theorem 10.3, i.e. g ∈ Ba(u). This proves theresult since ‖f − g‖ ≤ ‖fc − ϕ‖+ ‖fd − ψ‖ < ε.

Next we consider the subspace Bs(u).Given u ∈ B, we will call f ∈ B a generalized u-step function if there exists

a sequence of open intervals {Un} in I such that f is constant on each Un andµu(I

0 ∼⋃

n Un) = 0.A function f ∈ B was called in [12] a generalized step function if there is a

sequence of open intervals {Un} in I such that f is constant on each Un and|I ∼

⋃

n Un| = 0.It is then clear that every u-step function is a generalized u-step function,

and since the identity function τ is continuous, a function is a generalized stepfunction iff it is a generalized τ -step function.

33.5. Lemma. Let u ∈ B.

(a) If f is a generalized u-step function, and each of f and u is continuous at

the points where the other has a removable discontinuity , then f ⊥ u.(b) If u is normalized and f ∈ B∗

s(u), then for every ε > 0 there exists a

sequence of nonoverlapping closed intervals {In} in I such that µu(I ∼⋃

n I0n) = 0

and∑

n VInf < ε.

P r o o f. To prove (a), suppose its hypothesis holds. Then it is clear thatD∗

uf = 0 µu-a.e. Hence it follows from Theorem 20.1 that f∗ ⊥ u∗, and so fromTheorem 8.2 that f ⊥ u.

Next, to prove (b), suppose u is normalized, f ∈ B∗s(u) and ε > 0. Set A =

{x ∈ I : f ′u(x) = 0}. Then it follows from Theorem 20.1 that µu(I ∼ A) =

0, and from Theorem 16.4 that µf (A) = 0. But since f is internal, we haveindeed µf (A) = 0. Consequently, there exists a sequence of closed intervals {In≡[an, bn] : n = 1, 2, . . .} in I such that A ⊂

⋃

n I0n and

∑

n

VInf =∑

n

{f(bn)− f(an)} < ε .

Clearly, the intervals in {In} may be assumed to be nonoverlapping, and we have

µu

(

I ∼⋃

n

I0n

)

≤ µu(I ∼ A) = 0 .

33.6. Theorem. Given u ∈ B, let A denote the set of generalized u-stepfunctions in B.

(a) If u is normalized , then A∩B∗ is a dense subset of B∗s(u), and A∩Bc is

a dense subset of Bcs(u).(b) If u is continuous, then A is a dense subset of Bs(u).

Consequently , the generalized step functions constitute a dense subset of Bs.

P r o o f. To prove (a), suppose u is normalized. Then it is clear from the abovelemma that A ∩B∗ ⊂ B∗

s(u) and A ∩Bc ⊂ Bcs(u).


Now to prove the denseness of A∩B∗ in B∗s(u), let f ∈ B∗

s(u) and ε > 0. Thenby the above lemma there exists a sequence of nonoverlapping closed intervals{In ≡ [an, bn] : n = 1, 2, . . .} in I such that µu(I ∼

⋃

n I0n) = 0 and

∑

n VInf < ε.Now define

ϕ(x) =

0 if x < an for every n,∑′

n{f(bn)− f(an)}+ f(x)− f(ak) if x ∈ Ik (k = 1, 2, . . .),∑′

n{f(bn)− f(an)} if x 6∈⋃

n In and x > bn for some n,

where the summation∑′

is taken only over those values of n for which x > bn.

Next, define g = f − ϕ∗. Then it is clear that g ∈ B∗, and since g is clearlyconstant on each I0n, g ∈ A. Further, since ϕ(a) = 0, g(a) = f(a), and so we have

‖f − g‖ = V (f − g) = V ϕ∗ ≤ V ϕ =∑

n

VInf < ε .

Hence A ∩B∗ is dense in B∗s(u).

Further, in the case when f ∈ Bcs(u), the above functions ϕ and g are clearlycontinuous, and so g ∈ A ∩Bc. Hence A ∩Bc is dense in Bcs(u).

Next, to prove (b), suppose u is continuous. Then it is clear from the abovelemma that A ⊂ Bs(u). Also, for each f ∈ B it follows from Theorem 8.2 thatf ⊥ u iff f∗ ⊥ u. Hence (b) follows from (a).

The last part follows clearly from (b) on choosing u = τ .

Finally, we obtain from Theorems 33.3 and 33.6 a result on the approximationof general functions in B∗.

We will call here a function f ∈ B generalized u-linear if there exists a se-quence of open intervals {Un} in I such that f is u-linear on each Un andµu(I

0 ∼⋃

n Un) = 0.

Further, f will be called generalized linear if it is linear on a sequence of openintervals {Un} in I such that |I ∼

⋃

n Un| = 0. Clearly, f is so iff it is generalizedτ -linear.

33.7. Theorem. If u ∈ B∗, then B∗ contains a dense set of elements which

are generalized u-linear. Moreover , the generalized linear functions constitute a

dense subset of B.

P r o o f. Suppose u ∈ B∗, and let f ∈ B∗ and ε > 0. Let ϕ + ψ be theLebesgue decomposition of f relative to u. Then ϕ ≪ u, ψ ⊥ u and ϕ and ψare normalized by Theorem 23.3. Hence by Theorem 33.3 there exists a piecewiseu-linear function ϕ1 in B∗ such that ‖ϕ−ϕ1‖ < ε/2. Also, by Theorem 33.6 thereexists a generalized u-step function ψ1 in B∗ such that ‖ψ − ψ1‖ < ε/2.

Now define g = ϕ1 +ψ1. Then g ∈ B∗ by Lemma 3.1, and it is clear that g isa generalized u-linear function. Also,

‖f − g‖ ≤ ‖ϕ − ϕ1‖+ ‖ψ − ψ1‖ < ε ,

which proves the first part.

114 K. M. Garg

The second part is obtained by a similar argument from the last parts ofTheorems 33.3 and 33.6 on choosing u = τ .

33.8. R ema r k. It is interesting to observe here that although the set of allstep functions on I generates in the uniform norm the Banach space of all re-gulated functions on I (see Choquet [3], p. 152), in the variation norm this setgenerates only Bd (see Theorem 33.1). Similarly, the set of all piecewise linearfunctions on I generates in the uniform norm the Banach space of all continuousfunctions on I, but in the variation norm it generates only Ba (see Theorem 33.3).The same is true of the set of all polynomials on I as it will be clear from The-orem 34.4 of the next section.

It may be noted here further that the last part of Theorem 33.6 has beenobtained earlier in [12] the hard way. Also, the last parts of Theorems 33.3, 33.7and 34.4 were pointed out there in an addendum without proof.

34. Approximation by relative polynomials. Given u ∈ B and f ∈Ba(u), in this final section we investigate the problem of approximation of f inthe variation norm by a polynomial in u. We obtain here an affirmative solutionof this problem in the cases when u is nondecreasing and either continuous or ajump function (see Theorems 34.1 and 34.4). Consequently, every AC functionin the ordinary sense can be approximated in the variation norm by a polynomialin x.

In the general case f is found to be approximable in the variation norm onlyby a sum of two polynomials in ud and uc (see Theorem 34.5); and this turns outto be the best possible result in its direction for a general u (see Remark 34.6).

We begin with the case when u is a jump function.

34.1. Theorem. If u ∈ Bd, then the polynomials in u constitute a dense

subset of Ba(u).

P r o o f. Given u ∈ Bd, since f ≪ u iff f ≪ u, we can assume here without lossof generality that u is nondecreasing. Let P denote the set of all polynomials inu. Then it is clear that P ⊂ Bd, and since u≪ u, it follows from Theorems 10.1,10.3 and 10.5 that P ⊂ Ba(u).

Now, to prove the denseness of P in Ba(u), let f ∈ Ba(u) and ε > 0. Thensince f ≪ u, f is continuous relative to u by Theorem 12.2, and it follows easilyfrom Theorem 13.3 that f is a jump function.

Let {xn} be the sequence of points where u is discontinuous.Given any positiveinteger n, set En = {xi : i = 1, . . . , n}. Now, define

fn(x) =

{

f(a) +∑′

i≤n{f(xi + 0)− f(xi − 0)} + f(x)− f(x− 0) if x ∈ En,

f(a) +∑′

i≤n{f(xi + 0)− f(xi − 0)} otherwise,

where the summation∑′ is taken only over those values of i ≤ n for which xi < x.


Next, let S−n and S+

n denote the sets of indeces i ∈ Sn for which u is discon-tinuous from the left or right respectively at xi. Set

An = {(u(xi), fn(xi)) : i = 1, . . . , n} ,

Bn = {(u(xi − 0), fn(xi − 0)) : i ∈ S−n } ,

Cn = {(u(xi + 0), fn(xi + 0)) : i ∈ S+n } ,

and let

Gn = An ∪Bn ∪ Cn ∪ {(u(a), f(a))} .

Then Gn is a finite subset of R2. Also, since f ≪ u, f is clearly constant on everyinterval (closed or open) on which u is constant, and so Gn does not containdistinct points with the same abscissa. Hence we can find a polynomial pn whosegraph includes Gn. It is then clear that fn(a) = pn(u(a)), and for each i ≤ n,since f is continuous relative to u, we have

fn(xi) = pn(u(xi)) and fn(xi ± 0) = pn(u(xi ± 0)) .

Now, define µ = µu, ν = µf and νn = µpn(u) for each n. Then for each n, sincepn(u) ≪ u, νn ≪ µ by Theorems 14.4 and 13.1. Also, since fn

v−→f , and µfn = νn

on every subset of En, it is clear that νn −→ ν pointwise. Hence it follows fromthe Vitali–Hahn–Saks theorem (see [17], pp. 169, 170 and [8], p. 158) that thesigned measures {νn : n = 1, 2, . . .} are uniformly AC relative to µ. Thus, givenε > 0, there exists a δ > 0 such that νn(E) < ε/2 for each n whenever µ(E) < δ.

Next, choose an integer n such that∑

i>n ωu(xi) < δ and∑

i>n ωf(xi) < ε/2.Set E = I ∼ En. Then since µ(E) < δ, we have νn(E) < ε/2, and since νn(A)=µfn(A) for each A ⊂ En, we obtain

‖f − pn(u)‖ ≤ ‖f − fn‖+ ‖fn − pn(u)‖

≤∑

i>n

ωf(xi) + (µfn − νn )(E) <ε

2+ νn(E) < ε .

Next, we consider the case when u is continuous.

Let K denote the space of all compact (or closed) subsets of I equipped withthe Hausdorff metric (see [24], pp. 160, 214 and [25], p. 47). The exponentialmetric space K is known to be compact (see [25], p. 45).

Now, given any continuous nondecreasing function u on I, set

Ku = {u−1(α) : α ∈ u(I)} .

It is then easy to see that Ku is a closed subset of K, and hence Ku is compact.We will use here Cu(I) to denote the space of all continuous real valued

functions on I which are constant on the level set u−1(α) for every α ∈ u(I).Clearly, Cu(I) is a closed (linear) subspace of the space C(I) of all continuousreal valued functions on I equipped with the uniform norm.

It may be noted here that Ba(u) ⊂ Cu(I). For if f ∈ Ba(u), it is clear fromTheorem 12.2 and the definition of relative AC that f ∈ Cu(I).

116 K. M. Garg

We claim that Cu(I) is isometrically homeomorphic to C(Ku), viz. the spaceof continuous real valued functions on Ku equipped with the uniform norm. For,let η be the map from Cu(I) to C(Ku) defined as follows: Given any f ∈ Cu(I),let ηf denote the element of C(Ku) for which

ηf (u−1(α)) = f(u−1(α)), α ∈ u(I) .

It is then easy to see that the map η : f → ηf is a norm preserving homeomor-phism of Cu(I) onto C(Ku).

We will need here the following modified version of the Weierstrass approxi-mation theorem.

34.2. Lemma. Suppose u ∈ B+c . Then the set P of all polynomials in u is a

dense subset of Cu(I) under the uniform norm.

P r o o f. Let η be the map as defined above. Then ηu ∈ C(Ku), and this func-tion clearly separates the points of Ku. Hence, since Ku is a compact metric space,it follows from the Stone–Weierstrass theorem that the set Q of all polynomialsin ηu is dense in C(Ku) (see e.g. [19], p. 95).

Now, if p(u) is any polynomial in u, it is clear that ηp(u)=p(ηu)∈Q. Similarly,if p(ηu) is any polynomial in ηu, then η

−1(p(ηu)) = p(u) ∈ P . Hence P = η−1(Q),and since η−1 is a homeomorphism of C(Ku) onto Cu(I), it follows that P isdense in Cu(I).

We will need here further the following lemma which is related with Lusin’stheorem.

34.3. Lemma. Suppose u ∈ B+c , and let ϕ be any µu-measurable simple func-

tion on I. Then for every ε > 0 there exists a function g ∈ Cu(I) such that

supx∈I |g(x)| ≤ supx∈I |ϕ(x)| and

µu{x ∈ I : g(x) 6= ϕ(x)} < ε .

P r o o f. According to the hypothesis, ϕ =∑n

i=1 ciχAiwhere the sets Ai are

µu-measurable. Since µu is a metric outer measure, for each i ≤ n there cle-arly exists an open subset Ui of I with finitely many components such thatµu(Ai∆Ui) < ε/2n, where ∆ denotes the symmetric difference.

Now define ψ =∑n

i=1 ciχUi. Then ψ is clearly a step function such that

µu{x : ψ(x) 6= ϕ(x)} ≤∑

i≤n

µu{x : χUi(x) 6= χAi

(x)}(1)

=∑

i≤n

µu(Ai∆Ui) < ε/2 .

Now let {xi : i = 1, . . . , k} be the finite set of points where ψ is discontinuous.Let δ0 denote the minimum distance between any pair of points in {a, b, x1, x2, . . .. . . , xk}. Choose δ > 0 such that δ < 1

2δ0 and

(2) |u(y)− u(x)| < ε/2k whenever x, y ∈ I and |x− y| < 3δ .


Now set Ui = I ∩ (xi − δ, xi + δ) for i = 1, . . . , k, and let U =⋃

i≤k Ui.

Now define g(x) = ψ(x) if x ∈ I ∼ U ; otherwise x ∈ Ui for some i ≤ k, in casexi ∈ I0, define

g(x) = ψ(xi − δ) +ψ(xi + δ)− ψ(xi − δ)

u(xi + δ)− u(xi − δ){u(x) − u(xi − δ)} ,

and when xi=a or b, define g similarly by choosing xi−δ = a or xi+ δ=b respec-tively. It is then easy to see that g ∈ Cu(I). Also, supx∈I |g(x)| ≤ supx∈I |ψ(x)| ≤supx∈I |ϕ(x)|, and if E = {x ∈ I : g(x) 6= ϕ(x)}, we obtain from (1) and (2),

µu(E) ≤ µu{x : g(x) 6= ψ(x)} + µu{x : ψ(x) 6= ϕ(x)}

< µu(U) +ε

2≤

∑

i≤k

ε

2k+ε

2= ε .

34.4. Theorem. If u ∈ Bc, then the polynomials in u constitute a dense subset

of Ba(u). Consequently , the polynomials in x constitute a dense subset of Ba.

P r o o f. There is as before no loss of generality in assuming u to be non-decreasing. Let P denote the set of all polynomials in u. Then P ⊂ Ba(u) asbefore.

Now, to prove the denseness of P in Ba(u), let f ∈ Ba(u) and ε > 0. Thenf ∈ Cu(I) by Theorem 12.2, and by Theorem 16.4, f ′

u exists and is finite µu-a.e.,and it is µu-summable on I. Hence there exists a µu-measurable simple functionϕ on I such that

(3)∫

I

|f ′u − ϕ| dµu < ε/3 .

Then M≡sup{|ϕ(x)| : x∈I} <∞. Now, by Lemma 34.3, there exists a functionψ ∈ Cu(I) such that |ψ(x)| ≤ M for every x ∈ I and, if E = {x ∈ I : ψ(x) 6=ϕ(x)}, then

(4) µu(E) < ε/(6M) .

Next, by Lemma 34.2, there exists a polynomial p in u, say p(u) =∑n

i=0 aiui,

such that

(5) |ψ(x)− p(u(x))| <ε

3µu(I), x ∈ I .

Now define

g(x) = c+n∑

i=0

aii+ 1

ui+1(x), x ∈ I ,

where the constant c is so chosen that g(a) = f(a). Then g ∈P . Further, usingstandard arguments of elementary calculus, viz. the binomial theorem and induc-tion, it is easy to see that (uk)′u(x) = kuk−1(x) for every positive integer k andx∈I. Hence g′u(x) = p(u(x)) for every x ∈ I. Now since g is continuous and p(u)

118 K. M. Garg

is obviously µu-summable, it follows from Theorem 21.4 that

g(x) = f(a) +x∫

a

p(u) du, x ∈ I .

Now since g ≪ u, f − g ≪ u by Theorems 10.1 and 10.3. Consequently, sincef(a) = g(a) and f , g and u are continuous, it follows from Corollary 17.3 that

‖f − g‖ = V (f − g) =∫

I

|f ′u − g′u| dµu .

But since g′u(x) = p(u(x)) for every x, we thus obtain with the help of (3), (4)and (5),

‖f − g‖ =∫

I

|f ′u − p(u)| dµu

≤∫

I

|f ′u − ϕ| dµu +

∫

I

|ϕ− ψ| dµu +∫

I

|ψ − p(u)| dµu

<ε

3+ 2Mµu(E) +

ε

3µu(I)· µu(I) < ε .

This proves that P is dense in Ba(u).

The last part follows clearly from the first on choosing u = τ .

Finally, on combining Theorems 34.1 and 34.4 we obtain the following theoremfor a general function u.

34.5. Theorem. Suppose f, u ∈ B and f ≪ u. Then for every ε > 0 there

exist two polynomials p and q such that

‖f − p(ud)− q(uc)‖ < ε .

P r o o f. Since f ≪ u, according to Theorem 13.3 we have fd ≪ ud andfc ≪ uc. Hence, given ε > 0, there exists by Theorem 34.1 a polynomial p in udsuch that ‖fd − p(ud)‖ < ε/2, and by Theorem 34.4 a polynomial q in uc suchthat ‖fc − q(uc)‖ < ε/2. Consequently,

‖f − p(ud)− q(uc)‖ ≤ ‖fd − p(ud)‖+ ‖fc − q(uc)‖ < ε .

34.6. R em a r k. We present here two simple examples to show that in The-orems 34.1 and 34.4 the polynomial in u can not be replaced in general by apolynomial in u, and in Theorem 34.5 the two polynomials can not be replacedin general by a single polynomial in u or u.

Let I = [0, 1], and define u(x) = x or 1 − x according as 0 ≤ x < 12 or

12 ≤ x ≤ 1 respectively, and let f(x) = x for every x ∈ I. Then u is continuousand f ≪ u. However, if p(u) is any polynomial in u, then since u(0) = u(1) = 0,we have p(u(0)) = p(u(1)), and hence

|{f(1)− p(u(1))} − {f(0)− p(u(0))}| = |f(1)− f(0)| = 1 ,


so that ‖f − p(u)‖ ≥ 1. A similar argument holds in the case when u is a jumpfunction.

Next, define u(x) = f(x) = 0 for 0 ≤ x < 12, and u(x) = x and f(x) = x+1 for

12 ≤ x ≤ 1. Then u ∈ B+, f ≪ u, and clearly f(x) = u(x) or u(x) + 1 accordingas 0 ≤ x < 1

2 or 12 ≤ x ≤ 1. Hence if p is any polynomial in u, it is clear that

|f(x)− p(u(x))| ≥ 12for either x = 0 or x = 1, and so ‖f − p(u)‖ > 1

2.

34.7. R ema r k (C o n c l u d i n g ). It should be clear from the present workthat relativization of many other results in the theory of functions of boundedvariation, particularly the ones which involve Lebesgue’s AC or singularity, canbe obtained with the help of the results presented here.

In a subsequent paper [15] we present decompositions of mutual singularityand relative AC of signed measures on any arbitrary measurable space, and obtainextensions of Theorems 7.2 and 13.1 to lower and upper singularities and ACs.Also, in [16], we investigate properties of typical functions (i.e. functions with theexception of a set of functions of the first category) in some closed subspaces of B.

References

[1] P. S. Bul len, Non-absolute integrals: a survey , Real Anal. Exchange 5 (1979-80), 195–259.

[2] M. C. Chakrabarty, Some results on AC-ω functions, Fund. Math. 64 (1969), 219–230.

[3] G. Choquet, Topology , Academic Press, New York 1966.

[4] P. J. Danie l l, Differentiation with respect to a function of limited variation, Trans. Amer.Math. Soc. 19 (1918), 353–362.

[5] Ch. J. de La Val l e e Pouss in, Sur l’integrale de Lebesgue, Trans. Amer. Math. Soc.16 (1915), 435–501.

[6] —, Integrales de Lebesgue, Fonctions d’ensemble, Classes de Baire, 2e ed., Gauthier-Villars, Paris 1950.

[7] A. Denjoy, L’additivite metrique vectorielle des ensembles et les discontinuites tangen-tielles sur les courbes rectifiables, Bull. Math. Soc. Roumaine Sci. 35 (1933), 83–105.

[8] N. Dunford and J. T. Schwartz, Linear Operators, Part I , Interscience, New York1964.

[9] G. Fubin i, Sulla derivazione per serie, Atti Accad. Naz. Lincei Rend. 24 (1915), 204–206.

[10] K. M. Garg, On nowhere monotone functions, Thesis, Lucknow University, Lucknow(India), 1962.

[11] —, On singular functions, Rev. Roumaine Math. Pures Appl. 14 (1969), 1441–1452.

[12] —, Characterizations of absolutely continuous and singular functions, in: Proc. of Con-ference on Constructive Theory of Functions (Approximation Theory) (Budapest 1969),Akademiai Kiado, Budapest 1972, 183–188.

[13] —, On bilateral derivates and the derivative, Trans. Amer. Math. Soc. 210 (1975), 295–329.

[14] —, Construction of absolutely continuous and singular functions that are nowhere of mono-tonic type, in: Contemp. Math. 42, Amer. Math. Soc., Providence, R.I., 1985, 61–79.

120 K. M. Garg

[15] K. M. Garg, Decompositions of mutual singularity and relative absolute continuity ofsigned measures , to appear.

[16] —, Properties of typical functions in some subspaces of the space of functions of boundedvariation, to appear.

[17] P. R. Halmos, Measure Theory , Van Nostrand, New York 1950.[18] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer, New York

1963.[19] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York 1965.[20] E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s

Series, Vol. II, Dover, New York 1958.[21] R. L. Je f fery, Non-absolutely convergent integrals with respect to functions of bounded

variation, Trans. Amer. Math. Soc. 34 (1932), 645–675.[22] —, Generalized integrals with respect to functions of bounded variation, Canad. J. Math.

10 (1958), 617–626.[23] H. Kober, On decompositions and transformations of functions of bounded variation,

Ann. of Math. 53 (1951), 565–580.[24] K. Kuratowsk i, Topology , Vol. I, Academic Press, New York 1966.[25] —, Topology , Vol. II, Academic Press, New York 1968.[26] H. Lebesgue, Lecons sur l’integration et la recherche des fonctions primitives, 3rd ed.,

Chelsea, New York 1973.[27] N. Lus in, Integral and Trigonometric Series, Moscow 1915 (in Russian).[28] I. P. Natanson, Theory of Functions of a Real Variable, Vol. I, Ungar, New York 1955.[29] —, Theory of Functions of a Real Variable, Vol. II, Ungar, New York 1960.[30] H. Rademacher, Eineindeutige Abbildungen und Messbarkeit , Monatsh. Math. Phys. 27

(1916), 183–291.[31] J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, S.B. Akad.

Wiss. Wien 122 (1913), 1295–1438.[32] J. Ridder, Uber den Perronschen Integralbegriff und seine Beziehung zu den R-, L- und

D-Integralen, Math. Z. 34 (1931), 234–269.[33] H. L. Royden, Real Analysis, 2nd ed., Macmillan, New York 1968.[34] S. Saks, Theory of the Integral , Monografie Mat. 7, PWN, Warszawa 1937.[35] L. Tonel l i, Sulla rettificazione delle curve, Atti Accad. Sci. Torino 43 (1908), 399–416.[36] —, Sul differenziale dell’arco di curva, Atti Accad. Naz. Lincei (5) 25 (1) (1916), 207–213.[37] W. H. Young, On integrals and derivates with respect to a function, Proc. London Math.

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Soc. Math. France 74 (1946), 147–178.

Index of symbols

General:

I compact interval [a, b] 5I0 open interval (a, b) 8B σ-algebra of Borel subsets of I 8Sn index set {1, . . . , n} 14τ identity function on I 9|E| Lebesgue outer measure of E 8χE characteristic function of E on I 9fnv−→f fn converges in variation norm to f 96

fns−→f fn converges strongly to f 102∑nfnv−→f series

∑nfn converges in variation norm to f 96∑

nfns−→f series

∑nfn converges strongly to f 103

µ+, µ−, µ upper, lower and absolute variations of the signed measure µ 9

Entities related with f :

Cf set of points where f is continuous 9Rf set of points where f has a removable discontinuity 10∆f set of points where f is derivable in the wider sense 9∆∞f , ∆+∞

f, ∆−∞f

sets of points in ∆f where f ′ is infinite, +∞ or −∞ 9‖f‖ or ‖f‖v variation norm of f 95µf signed measure (or LS-measure) induced by f 9L(f) arc length of the graph of f 86L(f, g) arc length of the curve {(f(t), g(t)) : t ∈ I} 86sf,g arc length function of the curve {(f(t), g(t)) : t ∈ I} 86σ(f, g; t1, t2) length of the linear segment from (f(t1), g(t1)) to (f(t2), g(t2)) 86

Functions related with f :

ωf oscillation function of f 8f∗ normalization of f 10f+, f−, f positive, negative and total variation functions of f 8V +x,yf , V −x,yf , Vx,yf positive, negative and total variations of f on [x, y] 8V +f , V −f , Vf positive, negative and total variations of f (on I) 8fd, fc, fa, fs discontinuous, continuous, AC and singular components of f 8fcs continuous singular component of f 8fi, fe internal and external parts of f 57

122 K. M. Garg

Properties of functions:

AC absolutely continuous 34LC, UC lower or upper continuous 39LAC, UAC lower or upper AC 34LS, US lower or upper singular 14, 15�, �−, �− AC, LAC or UAC relative to 34�i internally AC relative to 58�s strongly AC relative to 61⊥, ⊥−, ⊥− mutually singular, LS or US 15

Relative derivates and derivatives:

Dgf , Dgf lower or upper derivate of f relative to g 50D+fg , D+fg right lower or upper derivate of f relative to g 71D−fg , D−fg left lower or upper derivate of f relative to g 71f ′g derivative of f relative to g 50D∗gf normalized derivative of f relative to g 50Dsgf symmetric derivative of f relative to g 56Dgf extended normalized derivative of f relative to g 58

Function spaces:

B functions of bounded variation on I 5B+ nondecreasing functions in B 8B∗ normalized functions in B 109Bd, Bc purely discontinuous and continuous functions in B 109Ba, Bs AC and singular functions in B 109Ba(u), Bs(u) functions in B which are AC or singular relative to u 109

Index of terms

AC (absolutely continuous), 33 normalized derivative relative to, 50AC component relative to, 75 normalized function, 10AC relative to 33 nowhere simultaneously discontinuousarc length, 86 from the same side, 24arc length function, 86 nowhere unilaterally discontinuousBorel decomposition, 8 from opposite sides, 26condition (Lg), 58 partially continuous, LC, UCcondition (L∗g), 65 relative to, 46continuous relative to, 39 piecewise u-linear, 110derivative relative to, 50 property (Ng), 78extended normalized derivative regulated, 10

relative to, 58 relative derivative, 50external function, 57 relative Lebesgue decomposition, 75external part, 57 relative Lebesgue point, 84generalized linear function, 113 relative lower, upper derivates, 50, 71generalized step function, 112 relative normalized derivative, 50generalized u-linear function, 113 removable discontinuity, 10generalized u-step function, 112 singular component relative to, 75indefinite LS-integral strong limit, 102

(Lebesgue–Stieltjes integral), 57 strongly AC relative to, 61internal function, 22 strongly Cauchy, 102internal part, 57 symmetric derivative relative to, 56internally AC relative to, 58 UAC (upper absolutely continuous), 34LAC (lower absolutely continuous), 34 UAC relative to, 34LAC relative to, 34 UC (upper continuous), 39LC (lower continuous), 39 UC relative to, 39LC relative to, 39 US (upper singular), 32LS (lower singular), 32 US relative to, 15LS relative to, 15 u-partition, 110LS-measure (signed measure) induced by, 9 u-step function, 110Lebesgue decomposition relative to, 75 uniformly LC, UC, 38Lebesgue point relative to, 84 uniformly continuous, LC, UCLebesgue’s condition relative to, 58 relative to, 38lower jump function, 100 unilaterally discontinuous, 26lower regulated, 36 upper jump function, 100Lusin’s property (N) relative to, 78 upper regulated, 36mutually singular, LS, US, 14, 15 variation norm, 95normalization, 10 weakly piecewise u-linear, 111

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