regular unbounded set functions

R E G U L A R U N B O U N D E D S E T F U N C T I O N S

By

BRIAN JEFFERIES

Abstract. Set functions which are unbounded on an algebra of sets ari~ naturally by taking the products of bounded operators and spectral measures acting on a space of square integrable functions. The purpose of this note is to show that, provided a certain regularity condition is satisfied, there is a natural integration structure associated with such a set function and an auxiliary measure, so providing a complete space of integrable functions. Several examples illustrate the extent and limitations of the approach.

1. I n t r o d u c t i o n

The theory of the extension of an abstract measure takes as its starting point a

non-negative, a-addi t ive set function # : S -~ [0, ~ ) defined on an algebra S of

subsets o f a set fL There exists a unique a-addi t ive extension of # to the a-a lgebra

c~(S) generated by S. There is little difficulty adapting the techniques for a signed a-addit ive set function # : S ~ C, provided that the range of the set function # is a

bounded set of complex numbers.

There has been a growing interest in the situation where the assumption of

boundedness of the range of the set function # on the algebra S fails, and the

condition of a-addit ivi ty o f # is replaced by some weaker condition, as, for example,

with the case of bimeasures and polymeasures. The consideration of unbounded

set functions of this type is increasingly found in areas as diverse as non-stat ionary

processes [RC], [Y 1], harmonic analysis [GI], [GS 1 ], [GS2], [Y 1 ] and operator

theory [IS], [JR1 ],[JR2].

Originally, b imeasures were conceived in order to obtain a representation for

certain types of bil inear mappings [MT]. In the subsequent theory of integration

with respect to bimeasures and polymeasures , as developed by Ylinen [Y2] and

Dobrakov (see [D3] and the references therein), attention was restricted to the inte-

gration of n-tuples of functions - - quite adequate from the viewpoint o f obtaining

canonical extensions of multil inear mappings and for applications to non-stat ionary

processes. However, there are situations when this class of functions is too small.

The restricted notion of a Radon polymeasure, introduced in [J1], is well-

adapted to deal with problems where the only reason that the polymeasure is not

extendible to a measure "lies at infinity". There are many concrete examples of

this phenomenon. Since Radon polymeasures behave locally like measures, they

admit a more extensive class of integrable functions than arbitrary polymeasures .

125 JOURNAL D'ANALYSE MATH(:MATIQUE, VoL 65 (1995)

126 B. JEFFERIES

Integration with respect to vector-valued Radon polymeasures and its application

to functional calculi for non-commuting systems of operators is outlined in [JR 1 ].

If a polymeasure is not extendible to a measure because it possesses some set

of singularities in the interior of the underlying space, then it is not a Radon

polymeasure. However, it turns out that by making small modifications to the

techniques associated with integration with respect to Radon polymeasures, it

is possible to treat more general set functions and still obtain a rich integration

theory. The consideration of unbounded set functions in this class is needed to

represent solutions to the perturbed wave equation, in a manner analogous to the

representation of solutions of the Schr6dinger equation via path integrals [K3].

This note is primarily devoted to integration theory, by which we mean the theory

of integration structures developed in [K2]. It is by means of this integration theory

that an extensive functional calculus for non-commuting systems of operators, over

a complete function space, can be built along the lines suggested in [JRI]; it is

for this purpose that it is essential to have an adequate notion of integration with

respect to unbounded set functions. Part of the goal of the present note is to

make explicit the relationship between the notion of an integration structure, and

integration with respect to a Radon polymeasure as used in IJ 1 ]. Nevertheless, the

present work may be read independently of the monograph [K2].

The following example illustrates the features of the class of additive set func-

tions we wish to study. Denote the operator of multiplication by the characteristic

function Xa of a Borel set A C_ I~, acting on L2(R), by Q(A). The inner product of L2(II~) is denoted by ( . , �9 ). Let T : L2(i~) ~ L2(R) be a continuous linear operator

and let r E L2(~).

For all Borel subsets A,B of II~, set m(A • B) = (Q(B)TQ(A)r r There is no

harm in abusing the notation for the cartesian product by denoting the collection of

all products A x B of Borel sets by B(II~) x B(~). Then m is an additive set function

on the semi-algebra B(/I~) x B(i~) in the sense that if X, Y E B(/~) x B(R) are pairwise

disjoint sets such that X O Y E B(R) x B(I~) too, then m(X U Y) = m(X) + m(Y). Furthermore, [m(X)l < IITI[.IIr z for all X E B(~) x B(~), so that m is bounded on

the semi-algebra B(R) x B(R). Now m is additive, so there exists a unique additive

set function ffz defined on the algebra B(R) x aB(R) generated by B(~) x B(!~) such

that the restriction of ffz to the semi-algebra B(~) x B(R) is precisely m.

It may happen that ffz is unbounded on the algebra B(R) x aB(R), even though m

is bounded on the semi-algebra B(R) x B(R); this is what is meant by an unbounded set function in the present context. As soon as ffz is bounded on B(R) x ~B(R),

the set function m is the restriction to B(II~) x B(I~) of a signed measure defined on

the ~-algebra a(B(~) | B(R)) generated by B(R) x B(II~). This observation seems

to have been made first by J. Horowitz [H]; see [J1] and [KM] for more general

situations. It is easy to check that if the operator T : L2(II~) ~ L2(R) is pointwise

REGULAR UNBOUNDED SET FUNCTIONS 127

positive, which means to say Tf >_ 0 almost everywhere whenever f > 0 almost

everywhere, then ffz is bounded on/3(/1~) • aB(~).

For fixed B E/3(•), the set function A ~ m(A • B),A E /3(Ii~) is a-additive and

for each fixed A C/3(R), the set function B H m(A • B), B E 13(~) is a-additive; m

is separately a-additive, or a bimeasure. If the operator T has a locally bounded

integral kernel, then m is a Radon bimeasure, of the type considered in [J 11, [JR1 ].

The notion of an additive set function being regular with respect to some given

collection of compact sets is introduced in Section 2. In Example 2.3, we present

an example of a bimeasure m of the above type in which the operator T does not

have a locally bounded integral kernel, and m is not a Radon bimeasure. It does

however enjoy the regularity property just mentioned.

Integration with respect to a regular set function m is introduced in Section 3,

by analogy with the notion of integration with respect to a Radon polymeasure

considered in [J1]. For a certain type of auxiliary measure #, it is shown in

Proposition 3.5 how an integration structure in the sense of [K2] is achieved, using

the class of functions integrable with respect to m, that is, m is shown to be closable

with respect to # in a sense similar to the notion of a closed map used in the analysis

of unbounded operators in Hilbert space. We end Section 3 by showing that certain

schemes, which might be conjectured to provide an integration structure for a

regular set function by analogy with the techniques of integration with respect to

measures, do not actually work.

In Section 4, we give an example of a bimeasue m of the above type, with R

replaced by I1~ 3, so that m is not closable with respect to any measure #, thereby

illustrating the limitation of the notion of a set function being closable with respect

to a measure. The example is relevant to the representation of solutions to the

perturbed wave equation via path integrals [K3].

2. Regular set functions

Let S be a semi-algebra of subsets of a Hausdorff topological space fL Suppose

that for each compact set K belonging to the algebra a(S) generated by S, there

exists a dense subspace XK of the space C(K) of continuous functions such that each

f u n c t i o n f E XK is measurable with respect to the a-algebra cr(S) generated by S.

This condition is satisfied, for example, when S is the semi-algebra of all products

of Borel sets in the Cartesian product of finitely many Hausdorff topological spaces.

Take XK to be the vector space of all linear combinations of continuous product

functions defined on K; the algebra XK is dense in C(K) by the Stone-Weierstrass

theorem. This is the principal environment for the class of unbounded set functions

considered in this note.

Let m : S ~ C be an additive set function. The unique extension of m to the

algebra generated by S is denoted by the same symbol. The variation Iml : S --,

128 B. JEFFERIES

[0, oo] of m is defined for each A E S by

[m[(a) = sup l ~ lm(A n B)I } . kBEr

The supremum is taken over all finite partitions rr o f the set 9t by elements of the

semi-algebra S. A Radon measure on a Hausdorff topological space is taken in the

sense of Schwartz [S].

2.1. D e f i n i t i o n Let E be a non-empty family of compact subsets off~ belonging

to a(S). A bounded additive set function m : S ~ C is said to be E-regular if the

following two conditions are satisfied:

(i) for all K E E, the set function A H m(A), for A E S and A C_ K is the

restriction o f a complex valued Radon measure mK on K, and

(ii) for all A E S and E > 0, there exists K E E such that K c_ A and

Im(A) - m(K)l < ,. The complex Radon measure rnK in (i) is unique because it is determined by the

subspace Xtr mentioned above. Moreover, i fKl , K2 E E and A c_ K1 nK2 is a Borel

set, then mKj (A) = mr2 (A). Clearly, linear combinat ions of E-regular set functions

are E-regular. I f m is E-regular and m(K) = 0 for all K E E, then re(A) = 0 for all

A E S .

R e m a r k s (i) I f rn is a E-regular set function and m(S) C_ [0, oc), then m is

actually the restriction to S o f a non-negative measure defined on o-(S).

( i i ) If m is a E-regular set function and m is bounded on the algebra a(S) of

sets generated by the family S, then m is actually the restriction to S of a complex

measure defined on a(S) , for, if Iml denotes the variation of m on a(S), then

Iml(W < oo and for any set A E a(S) and any e > 0, there exist pairwise disjoint

subsets Bj E S of A such that

0 < Iml(A) - ~ Im(Bj)l < ~/2. j=l

Now choose Kj 6 E such that Kj c_ Bj f o r j = 1 , . . . ,n and ~ j " l lm(Bj) - m(Kj)l < t l ?/

e/2. Then 0 < Iml(A) - ~=1 Iml(gj) < c, so the set U~=l Kj is a compact set

approximating Iml(A) f rom the interior. Then Iml is a-addit ive IS, p. 51], so m is

also a-additive.

(iii) l fXl . . . . , Xn are locally compact Hausdorff spaces, S = 13(Xl ) x ... • 13(X~) and E is the family of all compac t product sets, then an additive set function

m : S ~ C with bounded range is E-regular if and only if it is a Radon polymeasure .

Let m be a E-regular set function and let ~2,~ be the collection of all points x E f t

for which there exists an open neighbourhood Vx of x such that

REGUI.AR UNBOUNDED SET FUNCTIONS 129

(i) for all compact C c C_ E~., there exists K E K7 such that C C_ K C_ V~, and

(ii) sup{[mxl(K): K E IC, K C_ V~} < ~o.

It follows that ~'~m is an open subset o f ~'l. The complement ~ \ ~],, o f ~m may be

viewed as the set of singularities of the additive set function m.

2.2. P r o p o s i t i o n It" ~,, r 0, then there exists a unique Radon measure # on

~m such that #(a) = ]ml(A) for all A E S such that A C_ ~m.

P r o o f For every K E /C such tha tK _c ~],~, letjK : K --+ ~ be the inclusion of K

in ~ and let/rE be the variation of the signed Radon measure mE. Then the image

#K ~ I of #X in ~ is a Radon measure. Let/~ be the least upper bound of the

family ~ of all Radon measures #K ~ I with K E E such that K C ~m- By virtue

of conditions (i) and (ii) and [S, Proposition 7, p. 56], R. is bounded above, so that

the least upper bound # exists.

I fA E S and A C_ 9t, then for all c > 0, there exists K E /C such that K C_ A and

[m(A) - m(K)] < e. Moreover, if ]ml(A) < oo, then there exist pairwise disjoint sets ?1

Kj E E , j = 1 , . . . , n such tha tK i c_ a and I m [ ( a ) - [ m l ( U i = t K i ) < e. I f lml(a ) = c~, �9 m n then the sets Kj E lC,j = 1 . . . . n may be chosen so that the number [ ] (Uj- i Ki)

is arbitrarily large. To prove the equality #(A) = Iml(A) for all A E S such that

A c_ 9tm, it is therefore sufficient to establish that #(K) = Iml(K) for all K E E such

that K _c 9tm.

Let K E /C. The variation #K of the signed Radon measure mE satisfies the

equality I~K(S) = Im[(S) for all S E S such that S c K. Moreover, sup{#j(K n J) :

J E IC} = #K(K) by virtue of the consistency of the Radon measures mK,K E lC.

By [S, Proposition 7, p. 56], #(K) = IrK(K), so #(K) = Iml(K).

To prove uniqueness, suppose that #~, #2 are two Radon measures on ~m such

that ~1 (A) = #2(A) = Iml(A) for all A E S such that A c ~m. It is enough to show

that for every compact subset K of f~m, #1 (K) = #2(K). Now each compact subset

K of ~,~ is contained in a finite union of open sets V I , . . . , V, satisfying (i) and

(ii), so by the inner-regularity of #1 and #z, for every e > 0, there exist compact

subsets CI _C V 1 , . . . , C , c V, such that #I(Vj\C)) ,~ E/(2n) and ~2(Vj\G) ,~ e/(2n) for all j = l , . . . , n . According to condition (i), there exist sets Kj E )U

- - - - . . K n such that Cj c Kj c Vj, j = 1,. ,n. It follows that I 1 1 ( \ ( U ~ I K j ) ) < e /2 and

/~2(K\(Uk=~ Kj)) < e/2. The quality #1 (A) = /z2(A ) holds for all A E S such that

A C_ Uj"=, Kj, so #~ (/") = #z(f) for all continuous func t ions f : U j ~ Kj ~ ~ and so

#1 (A) = #2(A) for all Borel sets A c U j ~ Kj. In particular,

j=~ j=t

Then I#l (K) - u2(K)] <_ #~ (K\(LJ~'=~ Kj)) + #2(K\(U~__~ Kj)) < e. It follows that

u, (K) = re (K) . []

130 B. JEFFERIES

The following example shows that a bimeasure can be/C-regular with respect to

some family/C of compact sets without being a Radon bimeasure.

2.3. E x a m p l e Let D = ( I / i ) d / d x be the selt-adjoint operator associated with

the group of translations in L2(R). Then sgn(D) is the operator defined by the

operational calculus for self-adjoint operators in L2(R) associated with the signum

function sgn. The operator sgn(D) is precisely the Hilbert transform H given by

i lim [ ~(Y) ay ( / / 0 ) ( x ) = ~ _ 0 . , x - y

I.v-y > �9

for all 0 E L2(N) and almost all x E R [St, p. 541.

Let B(R) be the family of all Borel sets in R. As noted previously the collection

of all sets A x B with A E B(R) and B E B(R) is denoted by B(R) x B(IR). The

spectral measure Q : B(R) ---, L2(R) is defined by Q(B)O = ;~B~ for all B E B(R)

and 0 E L2(R). Let 0 E Lz(R) be non-zero and define the map m : B(R) x B(R) ---* C

by m(A x B) = (Q(B)HQ(A)O, 6) for every A E B(R) and B E B(R). The map m

has a unique additive extension to the algebra generated by B(R) x B(IR). Then m

is separately a-additive but the variation Ira] of m is not the restriction of a Radon

measure on the product space R x R. h is not difficult to see that

Iml(A x B ) = _1 f IO(x)~(Y)ld~.d v I , r - y l " "

A xB

for all A, B E B(R); an explicit argument is given in [J3 ]. It follows that ifA and B are

closed intervals such that the interior ofA x B intersects diag = { (x, x) : x E R}, then

Iml(A x B) = oo and ifA x B is a positive distance from diag, then Iml(A x B) < oo.

The diagonal in R 2 is the set of singularities mentioned in Proposition 2.2.

Let/C be the collection of all finite unions of compact product sets disjoint from

diag. The variation of m on each set K E /C is the restriction of the indefinite

integral of the function

(x,y) ~ _1 I~(x)~(y)l/Ix - yl, (.v,~v) E K, 71"

with respect to Lebesgue measure, so condition (i) of Definition 2.1 is satisfied

by m.

To verify condition (ii), we need to show that for every A,B E B(R),e > 0,

there exists K E /C such that Irn(A x B) - re(K)] < e. The separate ~r-additivity of

m ensures that there exists some compact set K -- Kl x K2 such that m(Kl x K2)

approximates m(A x B), but the special nature of m ensures that K may be chosen

from the class/C.

R E G U L A R U N B O U N D E D S E T F U N C T I O N S 131

For each/5 > 0, set

i f x O(Y) dy (Hr~)(x) = ~ -yl>6 x - y

for all ~b E L2(I1~) and almost all x E /~, and set mr(A • B) = (Q(B)HrQ(A)O, ~) for

every A E /3(R) and B E B(/t~). The variation of rn6 is the restriction to/3(,~) x/3(R)

of a Radon measure on R 2 denoted by Imr[. Given A, B E /3(I~) and e > 0, choose

> 0 such that Im(A • B) - mr(A x B)l < E/3 and choose compact sets Kl C A and

K2 c B such that Imr(A • B) - mr(K1 • K2)I < e/3. Such a choice is possible by

the separate cr-additivity of mr. Now the set W = {(x,y) E Kl • K2 : I x - Yl -< 6} is compact with Imrl-measure zero. There exists a finite open cover H of W by

product sets, such that [mrl(uH ) < e/3. Then K = (Kl • K2)\ UL/E E and

Imr(Ki • K2) - mr(K)l = Imr((K1 • K2) A (UL/)) I < Imrl(t_~) < ~/3.

Because the set K is disjoint f rom W, mr(K) = m(K), so combining the estimates,

Im(A • B) - re(K) I < e. It follows that m is a E-regular set function on the semi-

algebra/3(R) x /3(~) . []

3. Integration with respect to regular set functions

The notion of an integration structure was introduced in [K2] to encompass

integration with respect to both measures and certain examples of unbounded set

functions. The slight generalisation of this scheme we need is as follows. Let

be a collection of functions defined on a non-empty set fL We suppose that ~ at

least contains the function 0 identically zero on f~. A gauge on ~ is a function

p : ~ ~ [0, oc) such that p(0) = 0. A family F of gauges on ~ is said to be

collectively integrating if the following condition holds: if f E ~ , c i E C and

E ~ , i = 1 ,2 , . . . have the property that

(3.1) ~_~lci[p(f, -) < ~ f o r a l l p E F , i=1

and for every w E f~ such that

(3.2) ~ Icillf,(w)l < i = l

oo it follows tha t f (w) = ~ i = l c/fi(w), then

(3.3) p ( f ) < ~ l c i l P ( f ) for a l l p E F . i=1

132 B. JEFFERIES

If F is just a singleton set {p}, then p is called an integrating gauge. Following the theory of [K2], suppose that I" is a collectively integrating family

of gauges. Let us denote by s F) the vector space o f all f unc t ions f : f2 ~ C for

which there exist numbers ci E C, and functions j5 E 7-t, i = 1 ,2 , . . . such that (3.1)

holds, and for every w E f~ such that (3.2) holds, it follows thatf(w) = ~i~x cir.(w). A function belonging to s I') is said to be (7-/, F)-integrable. If the collection of

functions 7-[ is understood from the context, we shall merely say that a function is

F-integrable. The vector space s i m ( ~ ) of all finite linear combinations of functions

belonging to 7-/is clearly contained in s F).

For each p E F, a n d f E s F), the number qp(f) is defined by

OO

(3.4) qp(f) = inf Z Icilp(jS); i=l

the infimum is taken over all numbers c i E C, and functions f. E 7-/, i = 1 , 2 , . . .

such that (3.1) holds, and with the property that for every w E f/ such that (3.2)

holds, the equality f(,~) = ~i~=l cif,(,~) is true. The condition that a family F

of gauges on 7-/ is collectively integrating may be reformulated in terms of the

condition qp(f) = p(f) for a l l f E 7-t. It follows that for each p E F, the functional

qp : s 1-') ~ [0, oo) is a seminorm on the vector space s F).

If p : 7-( ---, [0, oc) is any gauge on ~ , not necessarily an integrating gauge, then

the gauge qp defined by (3.4) is an integrating gauge defined on the vector space

s p) o f all f unc t ions f such that qp(f) < eo: i f f is a function defined on f/, and

if ci E C andfi E s p), i = 1 , 2 , . . . have the property that

CX~

(3.5) ~ Icilqp(f,) < ~ , i=l

and for every ~o E f / such that (3.2) holds, it follows thatf(aJ) = ~-~i~=l cifi(w), then

f E s p) and

(3.6) qv(f) < ~ Icilqp(fi) �9 i=1

A function f for which qp(f) = 0 for each p E I" is termed a I'-nullfunction. It is then possible to form the quotient space L l (~, F) of s F) with the vector

space of all null functions so that each of the seminorms qp, p E 1-" induces a

corresponding seminorm, also denoted by qp, on the quotient space L l ( ~ , F). The

collection of seminorms qp, p E I" defines a locally convex Hausdorff topology r r

on L ~ (~ , 1-'). It turns out that the image of s i m ( ~ ) via the quotient map is dense in

the locally convex space L 1 (7-/, I').


The following statement provides a convenient condition for guaranteeing that a

function is p-null for an integrating gauge p. The proof is given in [K2, Proposition

2.2], which is reproduced here.

3.1. P r o p o s i t i o n Let p be an integrating gauge on ~. A function f is p-null i f and only if there exist functions hj E s 7-(, p ) , j = 1 ,2 , . . . such that ~'~= l qp ( hj ) < oo and Y~=l Ihj(.01 -- oo for all w E f~ such that f (w) ~ O.

P r o o f First, suppose that qp(f) = 0. Let X = {~o E fl : f ( w ) ~ 0}. Then

condition (3.5) holds for the functions f = f and the numbers ci = 1, i = 1,2 . . . . ; OO

the sum is zero. Moreover Xx(~o) = ~ i= l f , (w) for all ~ E fl for which (3.2) holds,

namely, for all ,; ~ X. It follows by the inequality (3.6) that Xx E /2(7-/, p) and Xx is F-null. The functions hj = xx , j = 1 ,2 , . . . have the required properties.

Now let hj E s = 1 ,2 , . . . be functions with the property mentioned

above and setf2j = hj andf2j_l = -hj for a l l j = 1,2 . . . . Then ~ - l qp(fJ) < oo,

and for every n = 1 ,2 , . . . , f (w) = ( 1 / n ) ~ l f j ( w ) = 0 for all w E f~ such that

~ , IJ~(~)l < ~ . It follows from (3.6) that the inequality qp(f) < ( l / n ) ~ 1 q p ~ )

is valid. This is true for all n = 1 ,2 , . . . only if qp(f) = O. []

It follows that L l (~ , p) is complete. It turns out that the problem of the quasi-

completeness of the space (Ll(7-t, F ) , r r ) under more general conditions is more

subtle (see, for example, [K4]).

The collectively integrating family of gauges F is said to be integrating for a

linear map m : s i m ( ~ ) ---, C if there exists a number C > 0, and gauge p E s

such that for e v e r y f E s i m ( ~ ) , [m(f)l < Cqp(f). If 1-" is integrating for m, then

it follows that m is the restriction to sim(7-t) of a ~ -con t inuous linear functional

rh �9 Ll (7-t, 1") ---, C. When there is no danger of confusion, th( f ) will be denoted by

m(f) for a n y f E L 1 (~ , F). We will also write

~ f dm, ~ f ( w ) dm(~)

for m(f) . The triple (L 1 (~ , s F, m) is called an integration structure. An efficient method for constructing an integrating gauge for additive set func-

tions by using an auxiliary measure follows. Let S be a semi-algebra of subsets

of a set 9t and suppose that m : S ~ C is a bounded additive set function. Let

ba(S) be the family of bounded additive set functions on the semi-algebra S en-

dowed with the uniform norm over S. Let s i re (S) be the collection of all finite

linear combinations o f characteristic functions o f elements of S. Then the bounded

additive set function fm : S ~ C is defined by linearity, in the obvious way, for

e a c h f E sire(S) . The set function m is said to be closable with respect to a finite

measure # defined on the a-algebra a(S) generated by S if the closure of the graph

134 B. JEFFERIES

{( f , fm) : f E s im(S)} of the integration m a p f ~ f m , f E s i r e (S ) in the product

space L 1 (#) x ba(S) is the graph o f a function.

The proof o f the following statement is straightforward.

3.2, P r o p o s i t i o n Let m : S --. C be a bounded additive set function and

suppose that # : a(S) ---* [0, cxD) is a measure. The fol lowing conditions are

equivalent:

(i) for any sequence fn E s im(S) , n = 1 , 2 , . . . such that f , ~ 0 in LI(#)

and f~m, n = 1 ,2 , . . . converges in ba(S), for each A E S, the sequence

f ,m(A) , n = 1 ,2 , . . . converges to 0 as n ---, oo,

(ii) m is Iz-closable,

(iii) the gauge f ~ SUPAEs Ifm(A)l + ~(Ifl), f E s i r e (S ) is an integrating gauge

f o r m.

A E-regular set function admits a natural integration structure as follows.

3.3. D e f i n i t i o n Let E be a family of compact subsets o f ~ belonging to the

algebra a(S) of sets generated by the family S. Let m : S ---. C be a bounded

additive E-regular set function. A f u n c t i o n f : f~ ---, C is said to be integrable with

respect to m if for each K E E the restrictionfK o f f to the set K is mK-integrable

and there exists a E-regular set func t ionfm : S ~ C such that fm(K) = fKfrdmlr

for all K E E.

The set function fm is uniquely defined among the family of all E-regular set

functions. The numberfm(A) is sometimes denoted by fa fdm and m ( f ) is used

to denote fm(f~). Let L 1 (m) be the space of (equivalence classes of) m-integrable

functions with the family of norms

PK : f ~-* sup Ifm(A)l + [mKl(IfKI), AES

f E Ll(m),

defined for each K E E.

The set function m may actually be bounded on the algebra a(S) generated by

S so that it is the restriction to S o f a complex measure rh defined on the a-algebra

a(S) generated by S. In this case, L I (rh) is, in general, a proper subspace o f L I (m)

(see Example 3.7).

3.4. P r o p o s i t i o n Let m be a E-regular set function. The family P = {pK : K E

E} of gauges is integrating for the integration map f ~ m ( f ) , f E L I (m).

P r o o f Let f , f i E L 1 (m), i = 1 , 2 , . . . be functions such that (3.1) holds, and for

every ~ E f~ such that (3.2) is true, it follows that f (w) = ~i~lj~(w). Here we may


take ci ~--- 1 for all i = 1 , 2 , . . . , because each gauge PK, K E E is a s eminorm on the

vec tor space L l (m).

To prove that the inequali ty (3.3) holds, it is enough to show that for each K E E,

PK(f--~'~i=lf')n ---+ 0 a s n ~ ~ because PK( f ) <-- PK(f--~'~i=lf)+~'~.i=ln n PK(f ' ) fora l l

i = 1 , 2 , . . . . By the Beppo-l_~vi convergence theorem, ImKI(IfK - E i n = l (J~)K l) -----r 0

as n ---, ~ , so for each c om pa c t set K E E,

t l OO OO

Ifm(K) - ~--~j~m(K)l <_ ~ Ifim(K)l _< ~ Imxl(l(jS)KI) --+ 0 i=1 i=n+l i=n+l

as n ---, ~ . OO

Given e > 0, by (3.1) there exists N = 1 , 2 , . . . such that ~i=/1 [fim(A)l < e for all

n > N and all A E S. It fo l lows that Ifm(K) - ~N=lf, m(K)l < e for all K E E. The

regulari ty o f f m andfim, i = 1 , 2 , . . . shows that SUPA~S Ifm(A) - ~/N afim(A)[ < e,

-~ i=1J5) 0 n-- ,co . proving that for each K E E, Pr ( f /1 ~ as []

In practice, it is useful to have a single gauge which is integrating for m.

3 .5 . P r o p o s i t i o n Let m be a E-regular set function. Let # : or(S) ~ [0, ~ ) be a

measure such that f o r every K E E there exists br > 0 such that ImKI(A) ___ bK#(A )

for all A E a(S) such that A c_ K. Then the graph { ( f , fm) : f E Ll(m)} of the integration map f ~ fro, f E L 1 (m) n L 1 (#) is closed in the product space L 1 (#) x ba(S).

P r o o f Suppose that f~ ---, f in L l (#), f~ belongs to L 1 (m) N L l (#) and f/1m

converges in ba(S). Nowf/1XK ~ f X K in L l (#) as n ---, ~ for each K E E, so

ImKI(IfK -- (A)KI) ~ bK#(IAxK --f~KI) --< bK#(If - L I ) ~ 0 as n ~ ~ .

In particular, the r e s t r i c t i on f r o f f to K is mr- in tegrable .

Because the set func t ionsfnm, n = 1 , 2 , . . . converge uni formly on S, the limit r

offnm, n = 1 , 2 , . . . in ba(S) is an additive E- regu la r set function. Moreover ,

r(K) = l i m f n m ( K ) = limoo_ fr(f/1)KdmK = f r f x d m K ,

so it fol lows t h a t f is m-integrable and r = f m . []

3 .6 . C o r o l l a r y Let m, # be as above. Let puO c) = SUPAEs [fm(A)l + #(If l) for all f E L 1 (m) N L 1 (#). Then (L I (m) N L 1 (#), Pu, m) is an integration structure.

So, as above, a E - regu la r set funct ion is closable with respect to a suitable

measure. It can happen, however, that a b imeasure is not closable with respect

136 B. JEFFERIES

to any measure (see the example in Section 4) because the regularity assumption

fails.

The following example f rom [J2l shows that s i m ( S ) may not be dense in the

Banach space (L 1 (m) n L 1 (#), Pu), even when m is non-negative, in which case m

is the restriction to S of a measure.

3.7. E x a m p l e Let m be the Radon polymeasure on 9t = (0, 1] • [0, 1] • [0, 1]

obtained by taking the restriction of the product of the Lebesgue measure A on

[0, 1] to the semi-algebra S of subsets of f~ consisting of products of Borel sets.

Letfn : [0, 1] ~ C,n = 1 ,2 , . . . be an orthonormal basis of L2[0, 1], and set U~ =

(1/(n + 1), 1/n],n = 1 , 2 , . . . . Define the func t ion f : f~ ~ C by

f (x ,y , z ) = Z n(n + 1)Xu.(x)f,(y)fn(z), fora l l (x,y,z) E fL nEN

Let # be the measure A ~ fa x2dxdydz on the Borel ~r-algebra/3(f~) of f/. Then

f is m-integrable in the sense o f Definition 3.3 and integrable with respect to the

measure ~t. For every A E/3((0, 1]), B, C E/3([0, 1]) we have the equality

fm(A • B • C) = ~_,n(n + 1)(;~v~ nEN

The sum converges by an application of the Cauchy-Schwar tz inequality, because

for each n = 1 ,2 , . . . , (;~t;oA)(A) is bounded by A(U,) = 1/n(n + 1), and the

sequences {(xR,f ,)},~l and { ( f , , xc )} ,~ l are square summable. In particular,

fm((O, 1] • B • C) = A(B n C). The space M([0, 1]) of countably additive set

functions u :/3([0, 1]) ~ C is endowed with the variation norm.

Now suppose that there exist simple functions sn E s im(S) , n = 1 ,2 , . . . such

that Pu(f - s,) ~ 0 as n ~ cx~. Then fm would be the limit, uniformly on S,

of set functions s~m, n = 1 , 2 , . . . . For each n -- 1 , 2 , . . . , the range of the vector

measure u~ : B ~ s,m((O, 1] • B • �9 ), B E /3([0, 1]) with values in the Banach

space M([0, 1]) is contained in the absolutely convex hull of a finite set, so it is

compact; the uniform limit B ~ A(B A �9 ), b E /3([0, 1]) of the M([0, 1])-valued

measure u~, n = 1 ,2 , . . . would therefore have compact range. However, this would

contradict [DU, Example IX. 1.1 ]. Consequently, the function f cannot belong to

the closure of s im(S) in (L I (m) n L t (#), Pu). []

The closure of s i re(S) in (L I (m) nLI (#), Pu) is the space L 1 (s im(S) , p~,). Certain

other potential integration structures for a regular bimeasure can be seen to fail

as follows. For example, if the closure of the graph { ( f , m ( f ) ) ; f E s im(S)} of

the m a p f ~ m ( f ) , f E s im(S) in the product space Ll(~) • C is the graph of a

function, then there exists C > 0 such that ]m(f)] < C#(f) for a l l f E s im(S) . In

particular, m is the restriction to $ of a signed measure.


Why introduce an auxiliary measure at all? For example, the completion of

the quotient space of s im(S) in the norm f ~ SUPAcS Ifm(A)l,f E s im(S) is

norm-equivalent to the Banach space L I (m) when S is a cr-algebra or algebra and

m : S ~ C is a-additive and bounded. Proposition 3. I 0 shows what may happen if

S is merely a semi-algebra of product sets and m is a regular bimeasure.

Let A be the Lebesgue measure on [0,1]. The Lebesgue measure on f~ =

[0, 1] • [0, 1] is denoted by A | A. The integral of a A | A-integrable function

f : f~ --, C is denoted by A | .k(f). The collection of all/3([0, 1]) • 1])-simple

functions is denoted by ~ . It is convenient to denote the restriction of A | A to the

semi-algebra/3([0, 1]) • 1]) of products of Borel subsets of [0, 1] by A • A.

Thus, i fA and B are two Borel subsets of [0, 1], then (A • A)(A • B) = A(A)A(B).

The indefinite integral of a simple f u n c t i o n f E 7-[ with respect to A • A is written

asf . (A • A). The definite integral [f.(A • A)](gt) is written as (A • A)(f).

The semivariation seminorm ]1 �9 I1~• with respect to A • A is defined on ~ by

Ilfll;,• = sup{l[y.(A • A)](A • B)I : A,B Borel sets in [0, 1]}, f E 7-/.

The number I lfll~ • ~ is equivalent to the semivariation of the vector valued measure

A ~ [f.(A• A)](A • ),A E B([0, 1]) [DU, 1.1.11], the values o f which are understood

to be taken in the space of scalar measures on [0, 1] with the total variation norm.

Note that i f f is a non-negative /3([0, 1]) • /3([0, l])-simple function on 9t, then

Ilfll~• = (A • A)(f). For any f u n c t i o n f : ft ~ C, the number qA• is defined by applying formula

(3.4) to the gauge II Jl, that is, qA• = i n f { ~ i ~ 1 IIJ~ll~• where the infimum is

taken over all choices of functions3~ E 7-(, j = 1 , 2 , . . . such that f (w) = ~i~_lfi(0v) OG

for every w E f~ for which Ei=I IJS(~)l < ~ . In view of the terminology introduced earlier, the collection of all functions

f : f~ ~ C such that q;~• < cx~ is denoted by E(7-/, I1' I1~• A function belonging

to zz(7-t, t1" I1~• is said to be ll" 11~• ~-integrable. The space s I1" II~• is a vector

space and q;~• is a seminorm on it. Moreover, s II �9 lira• is q~•

and 7-/is dense in s ]l" []A• [K2, Theorem 2.4].

As noted above, the seminorm I1' I1~ • ~ is integrating on ~ if and only if q~ • --

Ilfll~• for a l l y E 7-[. The inequality I(A • A)(f)l _< IIfll~• holds for a l l y E ~ .

As we shall see, the gauge II �9 II~• is not integrating for the linear m a p f

(A • A)(f), f E 7-( because it is not actually an integrating gauge, a deficiency

related to the fact that the space of Pettis integrable functions with values in L l (A)

is not complete with respect to the semivariation norm, see [DU, Example VIII. 1.4].

The next two statements are obvious.

3.8. L e m m a For everyf E 7-t, IIf[IA• ~ A o A(Ifl).

138 B. JEFFERIES

3 .9 . C o r o l l a r y Every )~ | )~-integrable funct ion f : ~ ~ C is II . ]l x • x-integrable,

and qa• < A | ,X(ifr ).

3 . 10 . P r o p o s i t i o n The seminorm II �9 is not an integrating seminorm on

P r o o f For each n = I, 2 . . . . , let g,, be the funct ion x ~ sin(nTrx)/n, x E [0, 1].

Then the funct ions g,,, n = I, 2 , . . . are uncondi t ional ly summable in L I ()Q, but

not absolu te ly summable . We const ruct a sequence o f funct ions f,,, n = 1 , 2 , . . . ,

bounded in t t (A), such that the funct ions (x,y) ~ ~'~,=lf,,(x)g,,(y), k = 1 , 2 , . . . do

not converge in Lebesgue measure on [0, I] • [0, 1], that is, there exists 6 > 0 such

that nl

The cons t ruc t ion is model led on the argument in [Th, p. 65].

Identify the unit interval [0, I] with the circle o f c i r cumfe rence one by ident i fying

the endpoints . Let x0 denote the image on the circle o f the point 0. Start ing at x0,

let Jh be adjacent segments o f length I /k on the circle, for every k = I, 2 , . . . and

let I,,, n --- I, 2 , . . . be the cor responding adjacent intervals on [0, 1], where those

segments Jk, k = I, 2 , . . . for which x0 E Jk are omit ted. We may suppose that the

intervals I,, are c losed on the left and open on the right. Then 3~(1,,) < l / n for every

n = 1 , 2 , . . . and there exists an increasing sequence j , , ,n = 1 , 2 , . . . o f posi t ive

integers such that the intervals l t , k = j,,...j,,+l - 1 are pai rwise disjoint and the I I J , , + l - I ; length o f the interval L,, = k../t=j,, t& is greater that 1 - 2/j,,.

Let f,, = Xt,,/A(I,,) for every n = 1,2 . . . . . Then IIAII~ -< 1 for all k = 1 , 2 , . . . and

J , , ~ t - - 1 j . . I - I j , + + I - - I

]~(x)gt(y) = ~ X,, (x) lgt (Y) l /A( l t ) >_ ~ X,,(x)lsin(kTry)l t---.i,, t=i, , t - i , ,

for all x , y E [0, !]. If y belongs to the set A,, where I sin(nrr-)1 > l /x /2 , and if

x E L,,, then the value o f the funct ion def ined above is grea ter than 1/x/~. Because

)~(A,,) = 1/2, (A | A)(L,, • A,,) >_ (1 - 2/j , , ) /2. It fo l lows that

Ill

lim ()~| I ~--~Ji(.r)gi(y) I >_ 1/v~)) > 1/2. I'11,11~00

. j ~ l l

The sum ~'~,,~1 g,, converges uncondi t ional ly in LI()Q, so because ~,,},,~l is

bounded in LI()~), the sum ~ , ,~ l .~(f,,xA)g,, converges uncondi t ional ly in Lt()~),

un i formly for A E /3([0, l]) by virtue o f [DU, Corol lary 1.2.6]; that is,

m!,im s u p { s : A , B B o r e l s e t s i n [ O , l ] } = 0 . j~r /


Consequently, there exists numbers mi > hi, i = 1 ,2 , . . . such that

ml

(A | ~fj(x)gj(y) j=rt~

and mi 1

sup{ ~ ( f jA) (A) (g jA) (B) :A,BBorelsetsin[O,l]} < j=ni

for all i = 1 , 2 , . . . . Let

m i

A= N U {(x,y): ~-~fj(x)gj(y) >_ l /v~}. k= 1 i>k j=ni

Then (A | A)(A) _> 1/3, and if (x,y) E A then Eir I~d,,fj(x)gj(y)l = ~ . Each of the funct ions~ @ gj, j = 1 ,2 , . . . is A | A-integrable, so it is II �9 ][~•

~x~ ( E ~ ) . by Corollary 3.9. Moreover, ~--~i=l q~ x a =,, ( | gj) < 1 An application of

Proposition 3.1 ensures that qAxA(XA) = O. Nevertheless, the set A has positive A | A-measure.

Now there exist sets A,, n = 1 ,2 , . . . in the algebra generated by B([0, 1] •

/5([0, 1]) such that (A | A)(A~,t,) ~ 0 as n ~ ~ . According to Corollary 3.9,

q,kxA(XaAa,,) <~ (A| A)(AZ~4~n) for all n = 1 , 2 , . . . , so XAn ""+ 0 in z : ( ~ , II. IIm• and (A | A)(A,,) ~ (A @ A)(A) as n ~ ~ . Because A, can be expressed as the finite

union of pairwise disjoint sets from/5([0, 1]) • 1]), the function Xa, belongs

to ~ for each n = 1 ,2 , . . . . But II~a,, I1~• = (A | A)(A~), n = 1 , 2 , . . . , so for some

positive integer m, [IXA,,[I~• > 1/4 and qAxA(XA,,) < 1/4. Therefore II �9 I1~• is not integrating on ~ . []

According to Corollary 3.6, ]l" is turned into an integrating gauge by the

addition of the L l-norm of a suitable measure. Note that if II �9 I[~• is restricted to

the collection S of characteristic functions of product sets, then it is integrating on

this diminished family of functions, and the resulting space E(S, II �9 I1~• is the

collection of all A | A-integrable functions.

4. A non-regular bimeasure

The example considered in this section is of the same type as the bimeasure

in the introduction. The family of all products of Borel subsets of ~3 is denoted by B(R 3) • B(II~3). We consider a bimeasure m : B(~ 3) x B(R 3) ~ C

given by m(A x B) = (Q(B)TQ(A)0,6) for all A,B E B(/K3). Here 6 belongs to

L2(R 3), Q is the spectral measure of multiplication by characteristic functions and

T : L2(II~ 3 ) ~ L2(IK 3 ) is a bounded linear operator.

1 4 0 B. JEFFERIES

If T has a locally bounded integral kernel, then as mentioned previously, m is a

Radon bimeasure, that is, it is/C-regular with respect to the collection/C of compact

product sets. If T is a singular integral operator of the type considered in Example

2.3, then m is a E-regular set function, where/C is the family of finite unions of

compact product sets disjoint from the diagonal in ~3 • R3. In Proposition 4.4,

the operator T has a distributional kernel of order one and the bimeasure m is not

closable with respect to any measure ~,.

The Fourier transform of the uniform surface probability measure # on the unit

sphere Sl centred at 0 in R3 is (27r) -3/2 sin(Igl)/Igl [G-S, p. 364]. Because

sin(l~l ) d s inr - for r = I~1, ~ - v ~ I~1 r dr r

the equality ( . V~ sin(l~l)/l~l = cos(l~l) - sin(l~l)/l~l holds, so convolution with the

distribution V x . (x#) defines a bounded linear operator on L2(I~3). Let

T : L2(~ 3) ~ L2(R 3) be the operator defined by

sin(l(I)'~ ~(~), f o r a l l ~ C I~ 3 T(~)~(~)= cos(l~l) I~1 /

Then for smooth 6, (T6)(x) = fs, y. V6(x -y)d#(y). Let 6 be an element of S(R3),

the rapidly decreasing functions on iR 3, and define m : B(]R 3) • B(R 3) ---, C by

m(A • B) = (Q(B)TQ(A )6, 6) for all A,B E B(~3).

Let N + = {0, 1 , . . .} and for any c~ = (~1,~2,c~3) E (N+) 3 a n d x = (xx,x2,x3) E /R 3 , x '~ denotes the number x 1 '~:t 2-'~x 3 '~ . For any pair of bounded Borel measurable

f unc t i ons f : ~3 ---r C, g : ]~3 ~ C, the f unc t i o n f | g is defined by (f | g)(x,y) = f(x)g(y) for all x, y E/~3. The bimeasure (f | g)m is defined by

( ( f | g)m)(A • B) = (Q(B)TQ(A)f O,-~6), f o r a l l A , B E B(/R3).

The bimeasure ( f | g)m is what would be obtained from applying the theory of

integration of product functions with respect to bimeasures developed in [D3]. It is

straightforward to check that for some C > 0, the inequality I ( ( f | g)m) (A • B)I _

CIIfll~llgl[~ holds for all A,B E B(I~ 3) and all bounded func t ionsf , g. Let u be a finite non-negative measure on B(/R 6).

4.1. L e m m a Suppose t h a t f : ]~3 ~ C, g : ]R 3 ~ C are bounded Borel measurable functions. The pair ( f | g, ( f | g)m) belongs to the closure of the set {(s, s m ) : s E s im(B(R 3) x/3(R3))} in Ll(u) x b a ( B ( ~ 3) x B(R3)).

P r o o f Let r . ,& E s im(B(~3) ) be simple functions such that I[f - rnllo~ --* 0

and llg - s . l l~ ---' 0 as n ~ ~ . There exists a number C > 0 such that

II((f| x B) - ((r, | x B)II _< C(ILf - rnlt~llgllo~ + I[fnll~llg - s, llo~)


for all n = 1 ,2 , . . . andA,B E B(R3), so ( r , | ~ ( f | in ba(B(I~ 3) x B(~3))

as n ~ ~ . The simple functions r , | E B(/I~ 3) • B(R3), n = 1 ,2 , . . . converge

uniformly t o f | g = on R 3 x I1{ 3 , so r~ | s~ ---, f | g in L~(v) as n ---, c~. Consequently,

( f | g , ( f | g)m) is the limit in L~(v) • ba(B(R 3) • B(~3)) of the sequence

(r, | | = 1 ,2 , . . . . []

Now suppose that p : R ~ R is a polynomial. The function ff~p : ~3 X ]I~ 3 ~ ' 2

defined by ~bp(X,y) = P(lY - xl2) e-(Ixl'~- Yl ) for all x ,y E R 3, is a linear com-

bination ~ : cjfj | gj of bounded product functions ~ | gj, j = 1 , . . . ,k. Let

�9 pm = ~k_ l cj(f] | gj)m. If ~b, ~ are bounded Borel measurable functions on :]{3, then ~ p m ( ~ | "7) is defined by

k 69pm(~ | = Z cj(([~fj] | [TgjJ)m)(I~ 3 x ]I{3).

j = l

4.2. L e m m a Let Rp : S (~ 3) ---+ S(R 3) be the operator o f convolution with the distribution ~b ~ fa3 ~(y)p(lyl2)~Ty �9 (y#)(dy), ~ E 8(I/~3). Then

Rp~b =p(1)T~p - 2p'(1)# �9 ~ f o r a l l ~ C S(~3).

Consequently, Rp defines a bounded linear operator on L2(I~ 3 ).

Proot ' Now p(Ixl2)Vx. (xu) = Vx. ( xp ( Ix[2 )U) - ( x . Vxp(Ixl2))U in the sense of distributions. Let f~3 = 47r denote the surface area of the unit sphere in R 3. The

Fourier transform of the distribution Vx �9 (xp(lx]2)#) is

(e -i(;~'x) , Vx . (xp(]xl2)#)) = i f s e-i<x~) (I, x)p([xl2)d#(x) 1

= p ( 1 ) ~ 3 / fo~eil;~lc~176 cos0-sinOdO

= p ( 1 ) 2 r r i l ]-i~l eiUudu

= p(1) (cos(I,~l) sin(I,~l) ] I~1 J"

Because x . Vxp(Ixl 2) = 2p'(1) for Ixl = 1, the result is proved. []

The bounded linear operator on L2(~ 3) which Rp defines is denoted, again, by gp .

4.3. L e m m a Let p : ~ ~ R be a polynomial, let <hi(x) = e-iX'2 ga(x),x E •3. Then for all A, B ~ /~(~3), ~bpm(A • B) = (Q(B)RpQ(A )c~I , 4~1 ).

142 B. JEFFERIES

P r o o f Because bimeasures on B(/I~ 3) x ~(~3) are in one-to-one correspon-

dence with separately continuous bilinear maps on Co(/l~ 3) • Co(/l~ 3) and the set S(R 3) • S (R 3) separates such maps, it suffices to prove that ~pm(O x -~) =

(Q('y)Rp(t)Q(O)Oa,Ol) for all ~,3' E 8(~3). For the function �9 : ]I~ 3 • ~3 ...., ]I~

defined by ~ ( x , y ) = x~y;~ e (-('xl%iyl2)) for all x , y E/I~ 3,

kOm(~p | 3,)

= ..f~ _f, y~ e-~Y%Iy)-~)Iy - x)~ - x)~ly - x)/V~, x#)ldx)dy

= ~ [ ~ [ , y~ e-~Y :'~ly)-~ly)Ix" Vyl ly - x)~ - x)~ly - x))l#ld~)dy.

I fp( ly - x[ 2) = ~k=x c).xOJy~J, then it follows that

tbpm( r | -y ) k

= Z Cj fX 3 fS Y/3Je-ly'2"y(Y)O(Y)(Y--X)aJe-Y-X2~y(Y-x)O(y- x)(Vx . x # ) ( d x ) d y j = l 1

= )fR3 fS, e-ly12"y(Y)49(Y)e-iy-x12O(Y - x)O(y - x)p[(Ix[2)(Vx, x # ) ( d x ) d y

= (Q('7)RpQ(~)O1, q~l).

[]

4.4. P r o p o s i t i o n There exists a sequence s , E s i m ( B ( R 3) • B(It~3)),

n = 1 , 2 , . . . o f s imple func t ions such that s , m , n = 1 , 2 , . . . converges in

b a ( B ( R 3) • B(I~3)) to a non-zero bimeasure and s , ~ 0 in L 1 (v) as n ~ oo.

P r o o f If p : I1~ ~ R is a polynomial, then by Lemmas 4.2 and 4.3, for all A , B C B(/I~ 3)

[~pm(A • B)[ = I(Q(B)RRQ(A)O1,01)1 IIRRIIII~II~

_< lp(1)lllYllllr 2 + 2[p'(1)111r 2,

so choose polynomials pn ,n = 1 ,2 , . . . such that supxcR[pn(x)[e-lxff/2 ~ 0 as

n --* ~ and p ' (1 ) , n = 1 ,2 , . . . converges to a non-zero number. Then d~pm, n =

1 ,2 , . . . converges in ba( /3(R 3) x B(R3)) to a non-zero bimeasure b by Lemma

4.3 and the functions ~p., n = 1 ,2 , . . . converge to zero in L l(v). Consequently, (0, b) belongs to the closure of the set {(s, sm) : s E s i m ( B ( R 3) x B(/I~3))} in Ll ( v ) x ba(B(il~ 3) x B( ] ]~3 ) ) , by virtue of Lemma 4.1. []


REFERENCES

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SCHOOL OF MATHEMATICS UNIVERSITY OF NEW SOUTH WALES

NSW 2052, AUSTRALIA E-MArL: B.JEFFERIES@ UNSW.EDU.AU

(Received August 1, 1993 and in revised form July 3, 1994)

regular unbounded set functions

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