Lecture Nates in MathematicsEditors:A. Dold, HeidelbergF.Takens, Groningen

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Vasile Ene

Real Functions ­Current Topics

Springer

95-164CIP

Author

Vasile Ene23 August, 8717Judetul Constanta, Romania

LIbrary of Congress Cataloglng-ln-Publlcatlon Data

Ene. Vas] le. 1957-Real funct10ns contemporary aspects I Vasile Ene.

p. cm. -- (Lecture notes 1n mathematics· 1603)Includes b i b l iographical references and index:ISBN 0-387-60008-61. Functions of real variables. I. Title. II. Series: Lecture

notes In mathematlcs (Springer-Verlag) ; 1603.QA3.L28 no. 1603[QA331.5]515' .83--dc20

Mathematics Subject Classification (1991):Primary: 26A24Secondary: 26A21, 26A27, 26A30, 26A39, 26A45, 26A48, 26A51

ISBN 3-540-60008-6 Springer-Verlag Berlin Heidelberg New York

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Preface

The Lebesgue integral, introduced at the beginning of our century marked a turningpoint in the development of the mathematical analysis in general and for the evolutionof the notion of integral in particular. Most 20th century books devoted to the theoryof the integral are dominated by ideas having their source in Lebesgue's theory. Oneof these ideas is the property of absolute convergence of the Lebesgue integral (f isintegrable if and only if If I is integrable), very convenient in most fields of functionalanalysis.

However it is well-known that the theory of trigonometric series and other lines ofthinking in mathematical analysis lead to the introduction of some generalizations ofthe Lebesgue integral that are no longer absolutely convergent. The first monographgiving a clear account of these nonabsolute integrals (for instance, the Perron andDenjoy integrals) was due to Stanislaw Saks [S3J. But this remained an isolated case;most subsequent books devoted to the theory of the integral ignored nonabsolute in-tegrals, despite the fact that the research literature devoted to them became richerand richer. The aim of the present monograph is just to fill this gap.

We don't ignore the existence of several books concerned in the last decade, withthe Kurzweil-Henstock integral, one of the most important nonabsolute integrals; butas is well known, this integral is equivalent to the Perron integral, while our aim isto go further and to cover the next steps in the literature devoted to nonabsoluteintegrals. In this respect, an outstanding result is the Foran integral (1975), whichis a classical generalization of the Denjoy integral in the wide sense. In 1986, Isekigave another generalization of the Denjoy integral, and called it the sparse integral.However neither of the two generalizations is contained in the other. Our purpose isto give a descriptive definition of an integral which contains both, Foran's integral andIseki's integral. In order to do this, and also to classify the so many existing integrals,it seems natural to perform a very deep study on the large number of classes of realfunctions that have been introduced in this context (many of them by the authorhimself).

Since many examples and counterexamples (66) are necessary to illustrate theproperties of various classes of functions, we considered that it would be useful togather them in a separate chapter, at the end of the book.

To show the uniqueness of the integrations we need several monotonicity theo-rems, which are studied in Chapter IV. This part of the book is also interesting in itsown right, because it presents strong generalizations of many mono tonicity theorems,studied in the books of Saks (Theory of the Integral [S3]), Bruckner (Differentiationof Real Functions [Br2]) and Thomson (Real Functions [T5]) (often Lusin's condition(N) is replaced by Foran's condition (M)).

Thomson introduced, in 1982, the notion of local systems, trying to unify the the-

VI

ory of the derivation and integration processes, and to indicate the close connectionbetween them ([T4], [T5]). Since then many authors have used local systems in thestudy of continuity, derivability, monotonicity and integrability. Beside monotonicity,we use local systems to investigate complete Riemann integrals (Kurzweil-Henstockintegrals). We give Perron and descriptive type definitions for these integrals, andshow that they are contained in Denjoy type integrals (in the wide sense).

In Chapter V we extend the classical result of Nina Bary, that a continuousfunction on an interval is the sum of three superpositions of absolutely continuousfunctions ([BaD. We also show the close relationship between Nina Bary's wrinkledfunctions and Foran's condition (M).

The present monograph can be used as a textbook for a special course on math-ematical analysis, addressed to graduate students in mathematics. By its systematicnature and by the great attention given to examples, this monograph should be ap-propriate for all students who want to deepen their knowledge and understanding ofone of the most important chapters of mathematical analysis.

In writing this monograph, the author had the benefit of very useful discussionswith Professor J. Marik and Professor Clifford E. Weil, during a ten-month stay atMichigan State University, East Lansing, 1991-1992, and also, during the same visitwith Professor Paul Humke, who was the first to suggest writing this monograph,after an attempt to write a survey article on a similar topic.

The anonymous reviewers of the manuscript provided significant help.Professor Solomon Marcus, who guided me in my first steps in real analysis as

well as in my Ph.D. thesis, helped me in many respects in preparing this monograph.The Springer Publishing House provided useful help in improving the final pre-

sentation.To all these persons, to Springer Publishing House and last but not least to my

wife Gabriela Ene, who took care of technical aspects related to English and printing,I expend my warmest thanks.

Vasile Ene

Contents

1 Preliminaries 11.1 Notations .. . . . . . . . . . . . . . . . . . .. 11.2 The 8-decomposition of a set . . . . . . . . .. 21.3 Notions related to Hausdorff Measures. Conditions [L. and a.f.l. 21.4 Oscillations . . . . . . . . . . . . . . . . . . . . . 41.5 Borel sets F,,, G5 ; Borel Functions; Analytic sets 81.6 Densities; First Category sets . . 91.7 The Baire Category Theorem; Romanovski's Lemma 101.8 Vitali's Covering Theorem . . . . . . . . . . . 111.9 The generalized properties PG, [PG], PI; P2G 111.10 Extreme derivatives. . . . . . . . . . . . 131.11 Approximate continuity and derivability 141.12 Sharp derivatives D#F . 151.13 Local systems; examples . . . . . 161.14 S-open sets . . . . . . . . . . . . 191.15 Semicontinuity; S-semicontinuity 21

2 Classes of functions 252.1 Darboux conditions V, V_, V+ 252.2 Baire conditions B1, HI, 81 272.3 Conditions Ci ; C:; [CiG); [CG] . 312.4 Conditions internal, internal', Zi, uCM . 332.5 Conditions V-BI , VB I . . . . . . . . . . 362.6 Conditions B], wI'] , S:; and st-local systems 392.7 Conditions VB, VB, VBG . . . 412.8 Conditions VB', VB', VB'G . . . . . . . . . 442.9 Conditions monotone' and VB' . . . . . . . . 472.10 Conditions VB', VB'G and V, V_, [CG), [CiG), lower internal, internal 502.11 Conditions AC, ACG, . . .. . . . . . . 512.12 Conditions AC', AC', AC'G, AC'G . . 542.13 Conditions L, 1., LG,1.G . . . . . . . . . 582.14 Summability and conditions VB and AC 602.15 Differentiability and conditions VBG, VB'G 622.16 A fundamental lemma for monotonicity . 652.17 Krzyzewski's lemma and Foran's lemma 702.18 Conditions (N), T], T2 , (S), (+), (-) 712.19 Conditions wS, wN. . . . . . . . . . . . 78

Vlll

2.20 Condition (N) .2.21 Conditions NOO, N+oo, N-oo2.22 Conditions M*, M* .....2.23 Conditions (M), M, Ngoo, N:oo2.24 Derivation bases .2.25 Conditions ACD# , ACDo, ACD2.26 Condition YD# , YDo, YD ....2.27 Characterizations of AC*G nC, AC*G n c., AC and AC2.28 Conditions ACn , ACw , ACoo , F .2.29 Conditions Ve; Vn: VBoo, 8 .2.30 Variations v", Vw , Voo and the Banach Indicatrix2.31 Conditions So, wSo and ACoo , VBoo, (N)2.32 Conditions i.; u; Loo , .c ....2.33 Conditions hZ, hZ, f.l., (TIl. . ..2.34 Conditions SACm SACw , SA Coo , SF2.35 Conditions SVe., SVs.: SVBoo,;S82.36 Conditions DWn , DWw , DWoo, DW*2.37 Conditions En' Ew , Eoo , £ .2.38 Conditions SAC, SACG, SVB, SVBG, SY

CONTENTS

78798284878889909396

· 100· 103· 104· 108· 111.114.116.118· 121

3 Finite representations for continuous functions 1273.1 quasi-derivable < AC*; DW*G+AC*; DW*G and approximately quasi-

derivable <:::; AC; DW1G +AC; DW1G . . . . . . . . 1273.2 C <:::; DW1 + DW1 on a perfect nowhere dense set . 1303.3 Wrinkled functions (W) and condition (M) . . . . . 1313.4 C = quasi-derivable + quasi-derivable. . . . . . . . 1343.5 C = AC*; DW1G + AC*; DW1G + AC*; DW1G . .135

4 Monotonicity 1414.1 Monotonicity and conditions (-), V BwG, V_81 . . . 1414.2 Monotonicity and conditions M,uCM,AC,Ci, ci, Vi .1424.3 Monotonicity and conditions N:», N'" . 1454.4 Local monotonicity . . . . . . . . . . . . . . . 1494.5 S-derivatives and the Mean Value Theorem . 1494.6 Relative monotonicity . 1514.7 An application of Corollary 4.4.1 . 1514.8 A general monotonicity theorem . . 1524.9 Monotonicity in terms of extreme derivatives. . 158

5 Integrals 1615.1 Descriptive and Perron type definitions for the Lebesgue integral. . 1615.2 Ward type definitions for the Lebesgue integral. . . . . . .. . 1665.3 Henstock variational definitions for the Lebesgue integral .. . 1675.4 Riemann type definitions for the Lebesgue integral (The McShane

integral) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.5 Theorems of Marcinkiewicz type for the Lebesgue integral . 1725.6 Bounded Riemannf sums and locally small Riemanrif sums . 1735.7 Descriptive and Perron type definitions for the V*-integral . . .174

.219

. 219

.220

.220

.220

.221

.221

CONTENTS IX

5.8 An improvement of the Hake Theorem 1805.9 An improvement ofthe Looman-Alexandroff Theorem. The equivalence

of the V* -integral and the (Pj,k)-integral . . . . . . . 1845.10 Ward type definitions for the V*-integral . . . . . . . 1865.11 Henstock Variational definitions for the V* - integral . 1875.12 The Kurzweil-Henstock integral . . . . . . . . . . . . 1885.13 Cauchy and Harnak extensions of the V* - integral . 1895.14 A theorem of Marcinkiewicz type for the V*- integral . 1905.15 Bounded Riemann sums and locally small Riemann sums . 1925.16 Riemann type integrals and local systems. . . . . . . . . . . 1935.17 The < LPG> and < LDG > integrals . . . . . . . . . . . . . 1975.18 The chain rule for the derivative of a composite function . . 2005.19 The chain rule for the approximate derivative of a composite function . 2025.20 Change of variable formula for the Lebesgue integral . 2045.21 Change of variable formula for the Denjoy* integral . . 2055.22 Change of variable formula for the < LDG > integral . 2065.23 Integrals of Foran type . . . . . . . . . . . . . . . . . . 2075.24 Integrals which extend both, Foran's integral and Iseki's integral . 210

6 Examples 2136.1 The Cantor ternary set, a perfect nowhere dense set. . 2136.2 The Cantor ternary function 'P. . . . . . . . . . . . . . 2146.3 A real bounded S: closed set which is not of F,,-type . 2146.4 An S: lower semicontinuous function which is not 131 . 2156.5 A function FECi, F f/- Ct . . . . . . . . .2156.6 A function F E V, FE [CtG] , F f/- [CG] . . 2156.7 A function F E VB!> F f/- [CiG] . .2166.8 A function F E uCM; F f/- RCM .2166.9 A function concerning conditions V+, V_, CM, sCM, lower internal .2176.10 A function concerning conditions: V_, V, internal, IiI, 131 , B!> wB!>

[VBG], (-), TI , T2 (Bruckner) 2176.11 A function concerning conditions: 131 , lil' V_ , V+, lower internal,

internal, internal" (Dirichlet) 2186.12 A function concerning conditions: V, V_, BI , Ci , C:, lower internal,

internal", VB, VB*G, N:> .6.13 A function F E V, F Elil \ 131 ; -F E Zi \ c, -F E v_131 \ Zi6.14 A function F Elil \ 131 , F E lower internal, F f/- V_ .6.15 A function F E sCM, F f/- internal" .6.16 A function F E AC*G \ AC, FE c: \ V, F E sCM\ internal"6.17 A function F E (D.C.), FE BI , F f/- m2, F f/- V .6.18 A function F E (+) n (-); F f/- VB IT2 ••••••••.••••

6.19 A function G E V, G f/-lil' G f/-13I , exists n.e., 0 a.e.(Preiss) . 222

6.20 A function H E V, H f/-13I , H f/-li!> exists on (0,1) (Preiss) .. 2226.21 A function F E VB I , F(x) = 0 a.e., F is not identically zero (Croft) .. 2236.22 A function P E V, F E [CG], FE [VBG], F f/- VB*G, F f/- C (Bruckner)2236.23 A function F E AC*, F f/- VB* 224

x CONTENTS

6.24 A function FEe, F E Tl, FE VBG, F rt VB*G 2246.25 A function F E [bAC*G] n VWG n N-oo F rt lower internal 2256.26 A function FEe n (S) n LG, F rt AC*G, F'(x) does not exist on a

set of positive measure, F(x) + x E LG, F(x) + x rt TI 2256.27 A function F E (S) nC such that the sum of F and any linear noncon-

stant function does not satisfy (N) (Mazurkiewicz) 2266.28 A function F E (M), F rt T2 •••••••••.••••••••••••.. 2276.29 Functions concerning conditions (M), AC, TI , T2 , (S), (N), L, L2G,

VBG, SF, quasi-derivable 2296.30 A function G E NOO, F rt (M), F rt (+) 2376.31 Functions concerning conditions (S), (N), (M), TI , T2, ACG, ACn,

SACn, VB2, VBG, SVB, F, SF 2386.32 A function FE lower semicontinuous, F E AC2, F rtAC 2446.33 A function Fn E Ln+l on a perfect set, E; E VBn on no portion of this

set, F; E Ln+IG, F; rt ACnG on [0,1] 2456.34 Functions F E L2G, G, E (N), G: = F' a.e., Gs - F is not identically

zero, F rt SACG . . . . . . . . . . . . . .2476.35 A function F Ek, F rt T2, F rt B . . . . . . . . . . . . . . . . .. . 2506.36 A function F E VB2 on C, "2(F; C) :s: 1 ... . . . . . . . . . .. . 2526.37 A function Fp E L 21', Fp rt AC2P-I, r; E VB2 on C; V2(Fp;C):S: 1 .2526.38 A function G E VB2, G rt ACn on C, G E F on [0,1] . . . . . .. .2546.39 A function FI E VB2 on C, "2(FI ; [0,x] n C) = <p(x) (G. Ene) .. .2546.40 A function r, E (N) on [0,1], r; rt Vs, on C, r; E Vs; on C (G. Ene)2556.41 A function G I E L2G, GI E F, G I rt SVBG, G1 rt SACG, does

not exist on a set of positive measure . . . . 2566.42 A function F E SACG, F rt F, F rt ACG . . . . 2586.43 A function F E DWI, F rt DW* . . . . . . . . . .2616.44 A function F E AC*; DWIG, F rt AC"; DW"G . .2616.45 Functions FI ,F2 E en AC*; DW*G, FI,F2 are derivable a.e., F{ =

a.e., F I and F2 do not differ by a constant 2626.46 Functions FI, F2E en AC*; DWIG, FI, F2are approximately derivable

a.e., FI + F2 rt quasi-derivable . . . . . . . . . . . . . . . . . . . . 2626.47 Functions FI ,F2 E En B, F}, F2 rt F, FI +F2 rt £ . . . . . . . . .2646.48 A function Gn E En+! , Gn rt En, Gn E Ln2+2n+}, Gn rt VBn2+2n . 2656.49 Functions concerning conditions !o,E,F, VB2G, B, EIG .2686.50 A function FE £ n VBwG, F rt B . . . . . . 2706.51 A function F E (N), F rt AZ (Foran) . . . . . 2716.52 A function F E AC 0 AZ, F rt AZ (Foran) . .2726.53 A function H E AC +AZ, H rt AZ (Foran) .2726.54 A function G E AC· AZ, G rt AZ (Foran) . .2726.55 A function FI E AC, FI rt Ln, FI rt L . . . . .2736.56 Functions FI E AC2G, F2E AZ, FI + F2 rt (M), F{ = a.e. . . 2736.57 A function F E AZ, F rt [£] . . . . . . . . . . . . . . . . . . . . . 2746.58 Functions FI E (S), FI E AC 0 a.fl, FI rt a.f.l., F2E L, FI +F2 rt T2 2766.59 Functions GI E a.fl., G2 E AC, GI + G2 rt a.f.l, . 2796.60 Functions HI E a.fL, H 2 E AC, HI .H2rt a.f.l 280

CONTENTS XI

6.61 A function F E a.fL, F E Til F fj. B, F is nowhere approximatelyderivable, (Foran) 280

6.62 A function G E a.fl., G E TI , G is nowhere derivable, G:p(x) = 0 a.e.,G fj. W, G E W' (Foran) 284

6.63 A function FEW on a perfect nowhere dense set of positive measure,with each level set perfect, F is nowhere approximately derivable .... 287

6.64 A function G I E DWI nC, G I is not approximately derivable a.e. on aset of positive measure 291

6.65 A function FEe, F is quasi-derivable, F fj. AC a AC + AC 2916.66 Examples concerning the chain rule for the approximate derivative of

a composite function 291

Bibliography

Index

293

305

Introduction

Beyond advanced calculus, the prerequisite for understanding this book is the basictheory of functions of a real variable including Lebesgue integral.

We shall be dealing with real functions of a real variable and throughout the bookwe state explicitly the exact domains of definition and the ranges of them.

We overlap with other books very few, but frequently we shall need results of theclassical books of Saks [83], Natanson [N], Bruckner [Br2] and Thomson [T5], whichwe take over without proofs (see Chapter 1). On occasion, when several theoremshave similar proofs, we prove only one or two of these theorems. Where we do notgive a complete proof, however, we provide references.

Chapters 3, 4, 5 and 6 are dependent of Chapters 1 and 2. However, it is notnecessary to read all the sections of them, since we give detailed references in all theproofs of what is needed. A reader interested in a particular result from a certainpage of the book is not expected to read the entire material until that page, becausehe is sent back very accurate to the exact notions which he needs.

All results contain short historical remarks, unless they are due to the authorhimself.

Numerous examples will be needed, so we have gathered them in Chapter 6. Weconsidered appropriate to illustrate some of these examples with pictures, althoughpictures does not replace proofs (however the author's proofs are independent frompictures).

We have also included a long list of bibliography in order to mention backgroundreading, to give credit for original discoveries or to indicate directions for furtherstudy, but we make no claims of the completeness of this list.

We tried to make an index as rich as possible, so that it might be useful. If thereader is interested in a notion or a class of functions, he should look up the Contentsor the Index for it.

We think the present book will serve the following purposes: to report on somerecent advances in the theory of real functions; to serve as a textbook for a course in thesubject; to serve as a reference work for persons studying real analysis independently;and to stimulate further research in this exciting field.