Title: Tensor products of spaces of measures and vector integraion in tensor product spaces
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Title: Tensor products of spaces of measures and vector integraion in tensor product spaces
Physical Description: v, 114 leaves. : ; 28 cm.
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Creator: Story, Donald P., 1946-
Publication Date: 1974
Copyright Date: 1974
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Subject: Tensor products   ( lcsh )
Banach spaces   ( lcsh )
Integrals, Generalized   ( lcsh )
Mathematics thesis Ph. D
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 112-113.
Statement of Responsibility: by Donald P. Story.
General Note: Typescript.
General Note: Vita.
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TENSOR PRODUCTS OF SPACES OF MEASURES AND
VECTOR INTEGRATION IN TENSOR PRODUCT SPACES











by

DonaJld P. Story


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE
UNIVERSITY OF FLOR.IDA =IN PARTIAL FULFILLMENT OF THE
REQUIR~EMENTS' FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA

19741



























To my Parents

Whose faith in me never waivered.














ACKNOWLEDGMENTS


I would like to thank Kira whose encouragement during

the past two years has given me strength. I am indebted to

each member of my committee; special thanks are due to Dr .

James K. Brooks, who directed my research and guided my studies

in measure and integration theory, and to Dr. Steve Saxon for

his summer seminars on topological vector spaces. Finally,

I would like to thank Brenda Hobby for her excellent: typing

job.


















TABLE OF CONTENTS



Page

Acknowledgments ....................................... iii

Abstract ........................................... v

Introduction ......................................... 1

Chapter

I. Tensor Products of Vector Measures ........... 4

II. Tensor Products of Spaces of Measures ........ 16

III. Pettis and Lebesgue Type Spaces and
Vector Integration ........................... 46

IV. The Fubini Theorem ........................... 88

Bibliography ......................................... 112

Biographical Sketch ................................... 114












Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

TENSOR PRODUCTS OF SPACES OF MEASURES AND
VECTOR INTEGRATION IN TENSOR PRODUCT SPACES

By

Donald P. Story

August, 1974

Chairman: J. K. Brooks
Major Department: Department of Mathematics

This dissertation investigates the concepts of measure

and integration within the framework of the topological tensor

product of two Banach spaces. In Chapter I, basic existence

theorems are given for the tensor product of two vector measures.

The topological tensor product of certain spaces of measures

is studied in Chapter II, where the space of all measures with

the Radon-Nikodym Property and the space of all measures with

relatively norm compact range are identified in terms of tensor

products. Chapter III discusses the theory of integration of

vector valued functions with respect to a vector measure; the

value of the integral is in the inductive tensor product of

the range spaces, and the integral is a generalization of B. J.

Pettis' weak integral. Normed Pettis and Bochner-Lebesque

spaces are considered and the Vitali and Lebesgue Dominated

Convergence theorems are proved. Finally, in Chapter IV, the

integration theory of Chapter III is used with the product

measures discussed in Chapter I and II, to prove some Pubini

theorems for product integration.














INTRODUCTION

This dissertation concerns the topological tensor product

of certain spaces of measures, and integration theory for vector

valued functions with respect to a vector-valued measure.

The tensor product of spaces of scalar measures with

arbitrary Banach spaces has been studied by Gil de Lamadrid

[13] and most recently by D. R. Lewis [16]; however, with the

introduction of the notions of the inductive product measure

in 1967 by Duchon and Kluvinek {ll], and the projective product

measure in 1969 by Duchon [9], it is possible to study the

topological tensor product of two spaces of measures. The

existence of the inductive and projective product measures

was shown in [9] and [ll] by essentially two different methods.

In Chapter I, we generalize a lemma of Duchon and Kluvinek

from which we obtain those product measures directly. In

Chapter II, we then study various tensor products of spaces

of measures; using tensor products we obtain various isometric

embeddings into natural spaces of vector measures. Character-

izations of certain spaces of vector measures are obtained

as a consequence of this study; for example, we identify

in Chapter II the space of all X-valued measures, where X is

a Banach space, on which the vector form of the Radon-Nikedym

theorem is valid. Aside from its own intrinsic value, the

study of tensor products of spaces of measures can be used

to attack the very important problems of establishing criteria








for weak and norm compactness of sets of vector measures;

this method is exemplified by Lewis' paper on weak compactness

[161.

For X and Y Banach spaces, probably the most natural

integration theory for X-valued functions with respect to

a Y-valued measure is developed in Chapter III; in this chapter,

we define the strong, the weak, and the "Pettis" integrals,

which are successively inclusive. Each of these integrals

takes its values in the inductive tensor product space X 0, Y.

On the space of strongly integrable functions, a norm is defined

which makes it into a Banach space and integral convergence

is characterized by norm convergence; the strong integral

reduces to the Bochner integral when the measure is scalar

valued, and is a special case of the Brooks-Dinculeanu integral

defined in 15]. The weak integral is proven to be a particular

case of Bartle's bilinear integral [1]. In his general theory,

Bartle defines an integral which is Z-valued, where Z is a

Banach space, where he presupposes the existence of a fixed

bilinear map from XxY into Z. In our context, the bilinear

map is the canonical one from XxY into X 0E Y, and we obtain

Bartle's theory; however, more can be said. A norm can be

defined on the space of weakly integrable functions which

characterizes integral convergence, and the Lebesgue Dominated

Convergence theorem is obtained as well as the Vitali Convergence

theorem. Finally, we define the Pettis integral for weakly

measurable X-valued functions with respect to a Y-valued

measure. In case the measure is scalar valued, the Pettis








integral is precisely Pettis' weak integral defined in [171,

and for strongly measurable functions, reduces to the weak

integral.

In Chapter IV, the notion of tensor product measure

as discussed in Chapters I and II, and the integration theory

of Chapter III, are combined to obtain vector forms of the

Fubini theorem. In order to obtain the main result (Theorem

IV.3.6) it was necessary to assume that one of the two measures

has the Beppo Levi Property, a property analogous to the

Beppo Levi theorem. This property seems essential in proving

a general Fubini theorem, and it avoids making the even stronger

assumption that both measures have finite variation.

Throughout the dissertation, some related topics in

Operator Theory are discussed.














CHAPTER I
TENSOR PRODUCTS OF VECTOR MEASURES

1. Basic Notions.

We shall begin by establishing notation and basic concepts

used throughout this dissertation.

X, Y, and Z will always denote abstract Banach spaces

over the same scalar field (real or complex). The norm of

a vector x e X is the number Ixl. X* is the continuous dual

of X and X1* denotes the unit sphere of X*, that is, X1"

{~x*EX* : |x* =1}. If x E X and x* E X*, then the action of

x* on x is denoted by x*(x), , or . The scalar

field is denoted by 4, unless otherwise specified. R is the

set of real numbers, R' the nonnegative real numbers, R# =

R u~m}, and o is the collection of all natural numbers.

An algebra A of subsets of a pointset S is a family of

subsets of S closed under finite unions and complements. n

is a G-algebra of subsets of S if n is an algebra of subsets

of S and is closed under countable unions. The orderedpair

(S,R) consisting of a pointset S and a >-algebra of subsets

of S form a measurable space. Any function y:A -t X is called

a set function on A. A set function y is countably additive

(>-additive) if for every disjoint sequence (A ) A, with

u.A. E A implies

Fi(u.Ai) = EC A)
where the convergence of the infinite series is unconditional.








The set function p is finitely additive if the above equality

holds for every finite disjoint family (A.)n= A. A set

function Ii:A -+ X is a measure if the algebra A is a o-algebra

and 9 is a-additive. A measure 0 will sometimes be referred

to as a vector valued measure, an X-valued measure, or simply

a vector measure. A measure which takes its values in the

scalar field Q is called a scalar measure; if the range of

a measure is R it is a positive measure.

For A E A, let H(A) denote the collection of all measurable

partitions of A, that is, the collection of all finite disjoint

families (A.)n cA such that A = .u Ai.

The set function 9 has a variety of associated R -valued

set functions: the semivariation of 9, the quasivariation of

p, and the total variation of p. They are defined for A EA

as follows:

(1) Semivariation:
n n
(lllA) = sup~i ~la() y(A):" 60, ai i (A )i=1st(A)}

(2) Quasivariation:

~(A) = sup(ju(IB) :BEA, BcA}

(3) Total variation:
n n nA.
j0 (A) = sup{ 1 y([A ) :(A ) CHA)}

Sometimes it is convenient to extend the definition of

the semivariation of y from the algebra A to the power set

of S as follows: for E c S,

l1 9 | E) = inf { ( p (I(A) :AEA, EcA).








We remark that if y is an X-valued set function on the

algebra A, and x* E X*, then we can define a scalar set function

x*p by xXU(A) = , A E A. We now state a proposition

which will help establish a relationship between the three

variation set functions and which is of vital importance

throughout this dissertation.


1.1 Proposition. (Dinculeanu [7, p.55]) For A s A,

lIp) (A) = xsU 1 |x~I (A).

It is well known that if A is a scalar set function on

A, then X(A) < lA (A) I 4X(A) for all A E A, from this we see

that xfp(A) < Ix*p (A) 5 4x7U(A), for all x* E X*. Taking

the supremum over X,* we get

~(A) 4 Ilv (A) 4TT(A), A E A.

Thus, the semivariation and the quasivariation are equivalent

in the sense that U;(A) = 0 if and only if Illll (A) = 0. If

A is a o-algebra and p is a measure, then 9 has bounded

semivariation: sup |y| (A) < +m. In the same situation, we
AEA
may still have ly)(S) = +m; it is for this reason that the

semivariation of a vector measure is used as a "control" set

function. If [II(S) < +m, y is said to have bounded variation

or finite total variation. If y is scalar set function, then

|9 (A) = II| (A) for all A E A.

Let I-:A -+ X and X:A -+ R+ be set functions. We write

u << X and say u is absolutely continuous with respect to X if

lim y(A) = 0, A E A,
h(A)I0







that is, given E > 0, there exists a 6 > 0 such that for all

A EA such that X(A) < 6 we have |p(A) < E.

If (pa ted is a family of set functions on A, then (pa)
is uniformly bounded provided

sup{ pa (A) I:AEAla E A} <+m

The family (ve) is pointwise bounded if for each A EA

sup{ pa(A) :aMA} < +m.

It is a result of Nikodym's [12,p. 309] that if (ya) is a

pointwise bounded family of scalar measures, then (pe) is
uniformly bounded.

We say that the scalar measure X:n -t R+ is a control

measure for the vector measure u:n --* X, where R is a o-algebra,

if 94< A and A (A) < l(A) for all As 0 7 consequently, we have

1.(A) -+ 0 if and only if X(A) -+ 0. We state the following

theorem taken from Dunford and Schwartz [12].


1.2 Theorem. Let 9:R -+ X be a vector measure. Then

(1) There exists a control measure A for 9;

(2) There exists a sequence (xn*) X1* such that Ixn*M((A) = 0
for every ne W if and only if 1 JI(A) = 0, where As E .


Proof. Part (1) follows from Corollary IV.9.3 and Lemma IV.10.5

of (12]. Part (2) is derived from the proof of Theorem IV.9.2

of [12]. O

Finally, the end of a proof and the end of a numbered

remark will be denoted by 0.








2. Tenisor Products.

This dissertation is mainly interested in the topological

tensor products of Banach spaces, and the tensor product of

vector measures. We shall state the definitions of the former

concept, and give basic existence theorems for the latter.

A standard reference for topological tensor products is Treves

[18].

We state here in the form of a theorem, the definition

of the algebraic tensor product of X and Y.


2.1 Theorem. A tensor product of X and Y is a pair (M,a)

consisting of a vector space M and a bilinear mapping # of

X x Y into M such that the following conditions be satisfied.

(1) The image of X x Y spans the whole of M;

(2) X and Y are $-linearly disjoint, that is, if {x ) i=1
an y}= Ysc ht(nx, = 0, then the linear

independence of one set of vectors implies that each

member of the other set is the zero vector.

There are many equivalent definitions for the tensor

product of two spaces as well as constructions available.

The map 4 is called cannonical, and the space M is unique up

to vector space isomorphism. This follows from the universal

mapping property of M; namely, if G is a vector space and

b:XxY -+ G is a bilinear map, then there exists a unique linear

map E:M -t G such that b = I;. The space M is usually denoted

by X 8 Y, and the elements of the cannonical image of X x Y

by O(x,y) = xey; consequently, any element may be written in

the form i2 x.0y. for xi EX and yi E Y.








X B0 Y is the tensor product of X and Y endowed with

the e-norm (least crossnorm): for 6= .1 x.0y ,

6)el = sup{ il|:(x*,y"X*)XY1*xy

The completion of the normed linear space X 0_ Y is the Banach

space X 0_ Y and is called the inductive (or weak) tensor

product of X and Y. X 8_ Y is X 0 Y equipped with the w-norm

(greatest crossnorm): for 6 E X@ Y,

n n
lei = infQi~llx |* y :6 i lx Sy ).

The space X d& Y is the completion of X 0~ Y and is called

the projective (or strong) tensor product of X and Y. Obviously,

8 5 6, 6 eT e XOY.
Let (S,n) and (T,A) be two measurable spaces, and

u:R -+ X and v:A --* Y measures. R 0A12will denote the algebra

of finite disjoint unions of measurable rectangles of the

set S x T; R @aA~ is the a-algebra generated by R 8 A and is
called the product a-al~gebra. We are concerned with the

existence of "product" measures, U 0 v, on RCO A with values

in X 0E Y Or X 80 Y subject to the identity u~v(ExF) = p(E)BV(F),
for E E and Fe A .

We begin by making the following definition.

The semivariation of p with respect to 7 (for y=E or n)

and Y is the R -valued set function ||pII on R defined by

p~ y(A) = sup{ yil(A )"Yiy :y Y, y 5 1,(A ) =1sE(A)}

2.2 Lemma. For each As E II ('fA) = I (1A).







Proof. I(( :(A) = sup{ Ely(A )Byi :(A i)n EH(A() y Y,' 51Y~ )



= sup{ i 1x*p(i j(Ai i :(A EX1)e6Ay,*X


= sup{ x*y( (A):x*EX *}

= ||p| (A).
The last equality is due to Proposition 1.1. Thus ((vi (A)


Now let E > 0 be given, there exists (A )~= E n(A) and

scalars (cr) with lai 1 such that


Choose y E Y with ly = 1 and y* E Y1* such that y*(y) = 1;
this is possible by the Hahn-Banach theorem. Then if yi = a y
for 1 < i s n, then y*yi = ai and Iyi| 1. Thus,

(lilA) o E+ 1 y*[i)v(A )

s E + li 1 i~y(A )l



Since E > 0 was arbitrary, it follows that II I(A) 5

III (A). Consequently, II~1 j(A) = Ill /A).

We now state a generalization of a lemma due to Duchon
and Kluvinek which appears in [1l].
(S,R) and (T,A) are measurable spaces. For vector measures
p:R --* X and v:A -+ Y, define
M~v:RCOA XdY by 90V(u E xF ) = E y(E )Bv{F ), where







U.Ejxp. is a finite disjoint union, E. E n and F. E Ai. Then
p ev is a finitely additive set function on thie algebra n 0 A.

2.3 Lemma. (Duchon and Kluvinek) Let y = E or n and suppose

(1) (Ua) is a family of X-valued measures on n and (V ) is
a family on Y-valued measures on A;

(2) sup a c~l (S) < +oo and sup | v BI(T ) < +m;
(3) X:n -- R+ and #:Al -t R are positive measures such that

j~a (*) << X uniformly in a and

VB < 4 uniformly in B.
Then for the family ("aVB Of xy Y-valued finitely additive
set functions on R 0 A, we have pa8v 6 < B on n 0 A, where XxO:ngeo -+ Rt is the usual product measure
of A and 0.

Proof. We must show to every E > 0), there exists a 6 > 0

such that whenever Ge E @A1 and Xx)(G) < 6, we have

|ua Bv(G)l < E, for all a and 3.
To that end, let E > 0 be given, there exists 6 > 0 such that

1(E) 6 iplie pa (E) < E uniformly in a, and O(F) < 6

implies IV (F)I < E uniformly in 3.
Suppose G = iglE xFi E nO and hx)(G) < 6~ where (E ) 0
is disjoint and (Fi ) n A
Recall that for s E S, the s-section of G is
Gs = (teT:(s,t)EG}.

Write D = (siliEI:k ~ E~.:(s<.
We then have

62 > hx4(G) = /S4(Gs~dX(s)








= jE.(Gs~dX(s) > U. -D(Gs~dX(s)

S6 X(u E -D).

From this we obtain X(v E -D) < 6 and so

Ila l~(UE -D) < E fOr all a. (#)

We may suppose 4 (F.) < 6 for i = 1,2,...,p, hence

Iv (F) < ~E for all 8 and 1 < i < p,
that is, Iv(F ) < 1 for all B and 1 < i p.

Therefore, for i = p+1,...,k, we have 4(F ) 6 and so

D =idlE To see this, suppose se D the 1(gE xFi~ )<.

If se E.E we must have (u.E~x.s=F n o4(. u
then Is then ((F ) < 6 which implies s eD since F =(u Ex
Notethat|pa i p+1E ) < E uniformly in a because of (#).

By assumptions (2), there exists a positive number N such

that Ia (S) < N and II ~(T) 6 N for all a and 8, it follows
that lu() 1 for all B and Fe E .


Then,

IU, V(G)I =


k
1 ~ia(E )@vBFi (F


kZ
i 1 a(E )@v (P ) +yN( p+1 a(E )BU (F )l
E N E) N aE)

E* i ( +.9 i p+1E )


E'N + N*E = 2EN.








We have Ig(pa B(G < 2cN regardless of a or B whenever

Axf(G) < 6 that is, vv as <
As a corollary of Lemma 2.3, we prove the existence

theorem of Duchon and Kluvinek {ll].


2.4 Theorem. Let u:R -+ X and v:A -+ Y be measures. Then the

set function u~v:nOA -+ XB Y can be extended uniquely to a

measure 1ye v:R0 A -+ X@ EY and



Proof. There exists control measures A and 4 of u and v,

respectively, by Theorem 1.2. We then have u << and v << .

Since vector measures are bounded we have II9| (S) < +m and

II I(T) < +m. Regarding Lemma 2.2, Ilu (~E) = li l(E) so that

u (IY S) < +m and |llY () << Thus the hypothesis of Lemma

2.3 is satisfied for the singleton families (u) and (v).

So we have p~v << Ax$ on ROA when X8Y is endowed with its

E-norm.

Because yev << Xx# on n 0 A and X x 4 is a positive measure

on R 8, A, we may extend u 0 v uniquely to a measure U QE v

defined on R 0a A with values in X BE Y by [7], p. 507. O


Lemma 2.3 suggests the following definition. A vector

measure u:D -+ X is dominated (with respect to Y) if there

exists a positive measure X on n such that Il l(E) --+ 0

whenever X(E) -+t 0, that is, ri~ii (*) << A.


2.5 Theorem. Let u:R -+ X and v:A -+ Y be vector measures,

and suppose y is dominated (with respect to Y) by a positive








measure A. Then there exists a unique measure ye v:R0 A --+

XBuY which extends 9 B v; consequently, 90 U(ExF) = p(E)QV(F)
for Ee E and Fe A .

Proof. Choose a control measure 4 of v. Then v << O and

ll9i (-) < X. According to Lemma 2.3, v< xo 0A

provied y (S) < +m. If this is shown, the theorem i

proven because the extension is guaranteed by (7], p. 507.
To show Ill i(S) < +m, there exists 6 > 0 such that

A (E) < 6 implies Il l(E) < 1. By Saks lemma ([12], IV.9.7),

there exists E 2,E2' 'En R disjoint such that S = u Ei

and each Ei is either an atom or 1 (E ) < 6 Since |p (ltS) <

i1 ()an p (E ) < 1 for all those ifowhc
X(E ) < 6, to show p -II(S) < +co, it suffices to prove that
if E is an atom, then II1 I(E) < +m.

Le-t E E be an atom of A, that is, if G c E and G E t

then X(G) = 0 or X(G) = X(E). Because u is a-additive, it is

bounded, so we can find a number N such tha-t lu(A) < N*A (E),

for all As E Now for G c E, Ge E either X(G) = 0 (in

which case II|9 l(G) = 0, hence I1(G)j = 0) or A(G) = X(E);

in either case, we have Iy(G) 5 NX(G).
Now

Ill9 (E) =sup fi (10(G )By .ny eY, yi 1,(G )EHi(E)}


< sup {.E y~(G.) :(G.)EH(E)}

< N.2 X(G.)


= N*A (E) < +mr








where H(E) is the collection of all measurable partitions

(G ) of E. O

Theorem 2.5 was first proved by M. Duchon in (91. The

measures p 0E v and v 0~ v are called the inductive and pro-

jective tensor products of p and v, respectively.


2. 6 Corollary. If either U or v have finite variation, then

9 8_ v exists.


Proof. If p or v has finite variation, say y, then il l(A) <

IU (A). But then U is dominated by the positive pleasure |p ,

by Theorem 2.5, u 8_ v exists. O


In the next chapter, we shall study various tensor products

of spaces of measures and give some structure theorems.














CHAPTER II
TENSOR PRODUCTS OF SPACES OF MEASURES

1. Algebraic Tensor Products of Spaces of Measures.

M. Duchon seemed to have developed the theory of product

measures primarily for the study of Borel and Bairemeasures

on locally compact Hausdorff spaces [8] and for the study of

convolutions of Borel measures defined on a compact Hausdorff

topological semigroup with values in a Banach algebra. Here,

however, we develop the study of tensor products of abstract

spaces of measures.

Throughout this chapter, (S,R) and (T,A) will denote

fixed but arbitrary measurable spaces; X and Y are Banach

spaces.

The space ca(S,R;X), or simply ca(R;X), is the space

of all measures FI:R -+t X. ca(G;X) is a Banach space when

equipped with the semivariation norm |* |(5S). When X = Q4,

we write ca(R) instead of ca(R;4). In this case, the semi-

variation norm is identical with the total variation norm

- I(5).

When various subspaces of ca(n;X) are under consideration,

descriptive letters are placed in juxtaposition with "ca," for

example: cabv(n;X) is the subspace of ca(R;X) consisting of

all those measures with finite total variation, Ccabv(G;X)

is the subspace of all measures of finite total variation

and with relatively norm compact range. Any subspace of
16







ca(R;X) consisting of measures with finite total variation

will have as its norm, the total variation norm *| (S)

rather than the semivariation norm. Since (/js) < [(S),

the total vairation norm defines on this subspace a topology

which is, ingeneral, strictly finer than the topology induced

by the semivariation norm.
Recall that from the universal mapping property of tensor

products, any bilinear map from the Cartesian product of two
Banach spaces into a third Banach space induces a unique

linear map from the algebraic tensor product of the first two

spaces into the third (see the remarks following Theorem

I.2.1). The following theorem establishes the basic algebraic

structure in which we shall be working throughout this chapter.


1.1 Theorem. (a) The bilinear map pE:(I,V) e lilV induces
an algebraic isomorphism which embeds ca(R;X) 8 ca(A;Y) into

ca(SxT,000A;X~E "

(b) The bilinear map $ :(y,v) -+ pe v induces an algebraic
isomorphism which embeds cabv(R;S) 8 ca(A;Y) into ca(SxT,

RDGA;XB Y).

Proof. Since y~E always exists whenever y and v are measures,

the map 4~ is defined on ca(G;X) x ca(AZ;Y) and takes its

values in ca(R0 D;XBqY);pe v exists whenever p has fnt

total variation, so that 4n is defined on cabv(R;X) x ca(A;Y)

and has its range in ca(0qAn;XB Y). It is not difficult to

see that 4E and $~ are bilinear.
In order to prove that the unique linear maps induced

by 4E and Q are isomorphisms, it suffices, according to







Theorem I.2.1 (b) to prove that the coordinate spaces of

4 ,W = and n, are @a-linearly disjoint. To this end, let

w = E or n be fixed. Suppose {pl' 2'"'^'~ n} ca(R;X) is a

linearly independent set (9 ,lsicn, is assumed to have

bounded variation if a = a), and {v ,V2'" "'n) c ca(A;Y)
Sc h n n
suc tht l / "i' i) = 0, that is, l i wvi = 0. We
want to show vl 2 "" n .
n n
.1 lp.0~ v. 0mens E 0V.(G} = 0 for all G E ,B~
in particular
n n
0 =i@iai(ExF) = Ep (E)0U (F), (1)

for all Ee E and Fe A .

Fix F E and choose x* E X1* and y* e Y1* arbitrarily,

and apply the functional x*0 y*, to both sides of equation (1):

0 = x*0 y*(0) = v()

=i lx*pi(E)-y*V (F)

= .

x* X arbitrary implies i2 p (E)'*yVi (F) = 0 for all

E E D. But y*V.(F) are scalar quantities which appear in

linear combination with the measures pl' 2 "" n,' and since

they form an independent set and i(1 ~i F i(*) 0 this

implies y*vi(F) = 0, i = 1,2,...,n. y* E YI* was choose
arbitrarily also so that V.(F) = 0 for i = 1,2,...,n; this

then implies v. 0 for all i.

This only proves half the condition for being w-linearly

disjoint, we must also prove that if {vl' 2"'"' n} 5 ca(A;Y)







forms a linearly independent set and {pl' 2"'"'~n n} c(R;X)

such that Ui.v= 0, the O . The
i= 1 2 n
proof of this is analogous to the above proof.

Thus the linear maps induced by 4E and (n are isomorphisms,
which proves (a) and (b). O


1.2 Corollary. cabv(R;X) 8 cabv(A;Y) c cabv(62OOA1;X@/Y
algebraically.

Proof. In view of Theorem 1.1, we have

cabv(R;X) 8 cabv(A;Y) c cabv(R;X) B ca(A;Y)

c ca(R00A;XQ Y).
It suffices therefore to prove that all measures in the

space on the left have finite variation. Let p E cabv(R;X)

and v E cabv(A;Y) and take disjoint sets G = 1uEinxF~ni
R0A, n = 1,2,...,p. Then

n=~1IUn(, x n ( n= 1 if=1 I(i )( i" I
Sk"
n= ln iC(E n) *IV(F n)
n=1 i=1 ~ ~i ilv ~Fn
k n
E A~ln y (E n)*jFn
n=1 =1 i




It follows that for any Ge E OA we have lyB v (G)

(9 xv/(G), hence for all G c 90 A.
Thus Iu@ v((SxT) 5 I9 xIV (SxT < +m, so that cabv(R;X) 0
cabv(A;Y) consists of measures with finite variation and

therefore lies in cabv(DOAn;Xs Y). O








1.3 Remark. Duchon [9] has shown that j@0 @ (G) =/y xlvi(G)

for all Ge OO A~ whenever both y and v have finite variation. O


Topological embheddings of (a) in Theorem 1.1 will be

considered later in this chapter; first, however, we prove

the following theorem.


1.4 Theorem. (a) The bilinear map qE:(u,x) -* xy induces

an isometric algebraic isomorphism on ca(R)BEX into ca(R;X).

(b) The bilinear map i9 :(p1'x) xy induces an isometric

algebraic isomorphism on ca(n)q~X into cabv(R;X).

Thus ca(n)4 X c ca(O:X) and ca(n)@ X c cabv(R;X)

isometrically.


Proof. The proof of that ca(R)@ X c cabv(R;X) isometrically

will be postponed until Theorem 2.3 infra, where we shall.

characterize this space; we state (b) now only for completeness.

It is clear that ca(R)@X c ca(R;X) by considering the

bilinear map (u,x) --+ xy, where xF! E ca(R;X) is defined by

xp) (E) = x*M (E). Consequently, ca(n)BX consists of all X-valued

"step-measures" on D of the form ~.- o .e n

vi E ca(R).
The step measure ilx qi has finite total variation since

U. has finite variation for each i = 1,2,...,n. We can therefore
consider ca(R)@X as an algebraic subspace of cabv(9;X). Part

(a) claims that when we consider ca(R)BX as a subspace of

ca(O;X), the e-norm is exactly the norm induced on ca(G)@X

by ca(G;X), namely, the semivariation norm. Part (]b) claims

that the n-norm is the total variation norm. Here we prove

the isometry of part (a).







To that end, let lxZ a0@.Fo h eea

theory, the E-norm can be defined as the norm of Ejx q

when it is considered as a linear map from X* into ca(R)

defined by = Ejx*(x.)y.. Thus

lijlll x |E xsu E ~~ix*(x )p (S)




= [Mixi~i||(s) (2)

In going from (1) to (2), we have invoked the Dinculeanu

result, Proposition I.1.1. Thus the E-nOrm is equal to the

semivariation norm. O


We now prove two technical lemmas followed by a theorem

which gives insight into the algebraic structure of vector

measures defined on product o-algebras and taking their values

in a Banach space X, that is, measures of the form X:R0 A -X.

This situation is, of course, a bit more general than measures

X of the form ye v, w = E or w, which take their values in

the tensor product of two Banach spaces.


1,5 Lemma. Let Al' 2,.' n be n linearly independent scalar

measures defined on R. Then there exists sets E ,E2'..,,E

in 0 such that the determinant of the n x n matrix (Ai(E ))nxn
is non-zero. We write

@(Al' 'XZ"'' n;E'E2'...,En) = det(Xi(E ))nxn < 0.

Proof. The proof is by induction on n.

Case n=2. Suppose X11'2 form a linearly independent set of

scalar measures on R such that @(Al' 2;E1'E2) = 0 for all








choices of E ,E2 E a. This means that

X1(E )*A2(E2) 1 (E2 X 2(El) = 0

for all E ,E2 E 51. Fix E2 E 51 and let El vary over 51. h1

and h2 independent implies X2(E2) = 0 and X1(E2) = 0. Since

E2 was arbitrary we have h1 = 2 = 0, a contradiction.

Case n = k+1. Suppose the lemma is true whenever n c k, and

that {11' 2"-' k' k+1} is a linearly independent set of k + 1

scalar measures such that for all choices of ElE2'...Ek+1

*(h1' 2'"' k' k+l;1;E,2,...,EkEk+1) = 0. (1)
Writing the determinent in (1) in terms of its first row

expanslan:
kpl i+1
i=1(-1) X (E )*4(XL ",.1 .,A~k+1;E2'E3'..,EkE+)0 (2)

where h. means that h. is deleted from the list of entries.

Since the measures {L}.i= are linearly independent and

(2) is valid as E1 varies over R, we obtain

~(hl,..'i' "' Xk+1;E2,E3,..,Ek'Ek+1) = 0 (3)
k+1
for any i and any choice of (E ) =2 .()iacotdcio
of our induction hypothesis since we are back to the case

n = k. O7


By Theorem 1.1, the spaces cabv(R;X) O ca(A) and ca(R) b

cabv(A;X) lie algebraically in ca(GoAh;X) and, in fact, lie

in cabv(GeoA;X); consequently, we may consider the set-theoretic
intersection of these two subspaces:
I(R,A;X) = cabv(GRX)@ca(A)nca(R)@cabv(A;X)





23

1.6 Lemma. If 6 E I(n,A;X), then there exists an integer

n r 1 and vectors xl,x2,..,xn, scalar measures pl ~2'" '"n

ca(R) and V1' 2'"'"'V nE ca(A) such that G = ilx (9 xv ).

Proof. Without loss of generality, we may assume 6 t 0.
P-
6 E cabv(R;X)Bca(A) implies 6 = il~i i


for 9i E cabv(0;X) and vi E ca(R). 6 E cafQ)8cabv(A;X)

Lmplies 6 = .Ev 0v for uj E ca(R) and v. E cabv(Ai;X).

We assume henceforth that p < n, and that each family

{9i) },i~ (q},' {4 9 is a linearly independent family of
measures.

By Lemma 1.5, there exists sets Fl' 2'"'",F pEA such

that @(Vl' 2'".' p F1' 2" p) a 0.O Write as simply 4 z 0.
With this observation, we use Cramer's rule to solve

the system


V (F )yl )+ 2(F2) 2 (.+' +p 1 )p' j 1 j (Fl



vl(F )"l .f2(Fp)"2 (Fp )#2 (.' + p Fp p(' =j j Fp Uj(

Define for i = 1,2,...,p and j = 1,2,...,n the vector


x = E ^oi-/ ~-10) u j (i) ui+1(Foji+1) '

where S_ is the symmetric group on p-letters; obviously,
xi E x for all i and j.
n
Thus by Cramer's rule 9 (*) = jC1x p (*) for i = 1,2,...,p.

Substituting this into 6 = .1 p.@v. We get











the lemma is proved upon re-indexing this representation. O;


1.7 Theorem. I(R,A;X) = X~ca(R)@ca(A).


Proof. From Lemma 1.6 we have I(R,A;X) c X~c~J~ca()ca)

since any member of I(R,A;X) can be represented in the form

E .xi(yixy.) where xi E X, Mi E ca(n) and v. E ca(A), which

clearly puts it in X~ca(n)@ca(A).

Conversely, if 6 E X~ca(n)~ca(A), then we can write

6as 6 = Cixim. where x. E X and m. E ca(R)0Ca(A). Let i
1 1
be fixed, we can write m. in the form mi = E 41xy where
1 i
uj E ca(0) and v. E ca(A). Now we have that

xi ~jxy ) =(xi j) Vj cabv(n;X) a ca(A), but

x (u xvj) = p 0(xiv ) e ca(n) a cabv(A;X). Thus x (ujxvj) f
i~~~~ ]
I(G,A;X) and therefore xim. = xjxi(pixvj) E I(GA;X), and in

turn 6 = Ejxjm. E I(GA;X). O

2. The Radon-Nikodym Property

We now introduce a notion which has not appeared in

the literature -- that of the Radon-Nikodym property of a

measure.

A vector valued measure r:n -+, X which has finite total

variation is said to have the Radon-Nikodym property, or

simply the R-N property, if whenever X:n -+ R+ is a positive

measure such that r << h, then there exists a Bochner integrable

function f:S --+ X such that








T(E) = jEf dX

for all E E n. We say that f is the Radon-Nikodym derivative

of T with respect to A and write f = or dp= f dh.

Recall that a Banach space X has the Radon-Nikodym

property if for every measurable space (S,R) and any vector

measure at: -t X of finite variation, r can be written as

an indefinite Bochner integral with respect to any positive

measure X on R for which r << A. Thus, the Banach space X

has the Radon-Nikodym property if and only if every vector

measure that takes its values in X has the Radon-Nikodym

property. The R-N property of a Banach space is a global

property whereas the R-N property of a measure is a local

property.

The R-N property of a measure is important in classifying

certain tensor products of spaces of measures. In preparation

for this, we establish an important lemma.

2.1 Lemma. Suppose T:R -+ X is a vector measure of bounded

variation such that T << X << where A and v are two positive

measures on R. If r has a Radon-Nikodym derivative with respect

to v, then it has a derivative with respect to A.

Proof. By the Lebesgue Decomposition Theorem, write v = y+i

where U << and i I h. Since i I h, there exists Eo

such that i(Eo) = 0 but X(S-Eo) = 0. From M<
heL(Sn,x,) such that p(E) = IEh dX.
Thus

T(E) = IEf dv = IEf du + IEf di = JEfh dh+ jEf di







Assert that IEf di = 0 for all E E R.

Case I. E c Eo. Since i(E ) = 0, we must have i(E) = 0

and so JEf di = 0

Case II. E c S-Eo. Since h(S-Eo) = 0, we have X(E) = 0

and so JEfh dX = 0; furthermore, r < T(E) = 0.
Thus

0 = T(E) = IEfh dX+/E f di = 0+IEf di

or JEf di = 0.

Cases I and II are sufficient to conclude IEf di = 0

since E = (EnEo) u En(S-Eo) and the integral ff di is additive.

Thus T(E) = IEfh dX, that is fh = d and the lemma is
proved. O

2.2 Theorem. Let (Tk) c cabv(R;X) such that llk1k(S<+m

If 'k has the R-N property for each k E w, then so does the
measure -r = k 1 k'

Proof. We remark first that the infinite series Ek rk does

define the measure because the series Zk rk(E) converges
absolutely for each E E 51:

k 1 'k(E)I k 1 kl(:) sk iJk((S) < +" by hypothesis.

To show r has the R-N property, begin by supposing r <<.,

where X is a positive measure on R.

Note that k 1k(E) converges and consequently defines
a o-additive measure on 0 such that -n k orec
nn U.







Write v = h+k I~kl 'then v is a positive measure on
12 such that X << v; consequently, T << X < v. We intend to

show T has a Radon-RhL~odym derivative with respect to v,

and then use Lemma 2.1 to prove the theorem.

Indeed, for each n E we have also that r << v. I
n n
has by assumption the R-N property; hence In(E) = IE ndy

for some fn E X(S,R,V), where BX(S,O,V) is the space of
Bochner integrable X-valued functions.

Write Ifn/ 1 ISf du and note I'nj(S) = |fn l. In 1

is the norm of fn in BX(S,R,V).

Since nE fnl n= T (S) <+m and B (S,n,v) is a

Banach space, n~ff converges in norm to a function f t

B (S,R,V), that is, f = ~f .
But then

T(E) = fT (S) n=IE fn du = E f dv = /Ef du.

That is, f = .By Lemma 2.1 then, exists, which means,

since X was arbitrary, T has the R-N property. O


We now prove a theorem which identifies the space ca(R)BT X.
This is a generalization of a theorem of Gil de Lar~adrid (13],

where he identifies C*(H)B X, C*(H) is the dual of the Banach

space of all continuous functions on a compact Hausdorff

space H. C*(H) is of course the space of all regular Radon

measures on H. Our setting is based on an abstract measurable

space (S,R). Lamadrid's identification was that C*(H)Q X was
the class of all regular X-valued Radon measures of bounded

variation which can be represented as an absolutely series








of "step measures." Theorem 2.2 implies that such a represen-

tation does have the R-N property. Our approach is quite

different than.his and the result was independently obtained.


2.3 Theorem. Let (S,SZ) be a measurable space and X a Banach

space. Then ca(R)QnX is isometrically embedded in Ccabv(R;X),
the space of all X-valued measures with bounded variation and

relatively norm compact range.

Furthermore, ca(R)B X is the Banach space of all X-valued

measures on R with the R-N property. Symbol~ically,

ca(C)B X = RNca(52;X).

Proof. By Theorem 1.4, ca(n)0X c cabv(R;X). It is clear

that any measure 6 = .2l x.A. E ca(R)@X has relative norm

compact range since each X. does. To show the initial assertion,
if suffices to prove that on ca(R)bX, the w-topology is identical

to the bounded variation norm. Indeed, if the a-norm on

ca(R)@X is the variation norm then since Ccabv(R;X) is a

Banach space, the completion ca(R)4 X of ca(R)bX is just the

closure of ca(n)@X in Ccabv(n;X), hence ca(B)B X c Ccabv(R;X).

Take 6 = iE xig. where x. c X and 9i E ca(0). Then

n n


If we take the infimum on the right hand sice over all represen-

tations of 6 in the form Ejxjip. we obtain |6|(S) & 6 .i
n n
Suppose again 6 = .2~ x E. ca(R)@X, and put X = .E U.1,

then p. << X for each i. Write f. = dugwiheissb h

classical R~adon-Nikodym theorem. f. E Ll(h) and







n de
S= (x fi E BX(A). Note that f= dh that /9 ((S) = If 1l'

where fi 1l is the norm of fi in L1(A), and lei (S) = |$l1'
where Ifl is the norm of f in B (A).

Define B c BX(A) and M c ca(R)BX as follows
k 1 k
B {1xig.:xieX,gi eL (X), and f = .2xig.1-a.e.}
k k
M = {.E xiV.:xieX'v.Eca(D) and 6 = .E xjvi..

There exists an injection IL:B -+ M defined by
k k k
Ji( Z1x g ) = i lxi j g dX; furthermore if lii and
k g
.E xiv. are in correspondence, then
k k

We conclude that

inf. x. ~i-|g.l t inf.s Ix.|* v.(S) (b)

for we have argued that for each number from the left side,

there is a number from the right side which is at least as
small.

It is well known that BXQh = XBL1(X) isometrically

(see Treves [18]). Since f E BX(h) we have that If/1 = w'

but |fl, = inf I i ./il so that 6 ((S) = Ifll = 'l

On the other hand, |8l = inf .2 |xi|*(Vi (S).

Thus from (#), IB (S) = |fl~ r |6|T. We already have

6)I(S) < |6], so that 6 /(S) = (6 ,l which proves the first
assertion.

We now prove ca(R)B X = RNca(a;X).







If T E RNca(R;X), then r necessarily has finite total

variation; put X = TI.r Then r << X and since Ir has the

R-N property, there exists f e BX(X) such that T(E) =fEf dX
for all EC E .

Since f is Bochner integrable, we may write f in the

form f(s) = lx (E (s) X-a.e., where x E X and E e t

(the family (E ) is not in general pairwise disjoint), and

possessing the property that nE |x [A(E ) < +m. This is a
well-known result which can be derived from Theorem III.5.5

infra, or see Brooks [3].

Define In:R -4 X for each ne E by In(E) = xnX(EnEn *

Tn is easily seen to have the R-N property and In E ca(R)BX,

also n= l1 (](5 = n- |Lx *A (E < +". (1)

So we have

T(E) = j f dX = E x (= dA= x(EE)
E n1 n En n=1 Cn (nn

or T(E) = nmT (E) for Ee E (2)

As remarked above Tn E XA ca(R), hence k 1 k E XQ ca(0).

Note that (1) implies the sequence {k I k n=1 is Cauchy in

XO ca(R). From the first half of the proof, thle r-norm is

equal to the variation norm, so for n < m positive integers
m m m
k=n k n k~nklS k k=n~rlS k f

as n and m approach infinity because of (1). Regarding (2),

k1kmust converge in variation to r since it converges

to T setwise. Therefore T E X0 ca(R) since it is the sum of

a sequence ('n) in XB ca(R). Thus we have RNca(R;X) c XA ca( ).







Conversely, if T E X6 ca(R), then from the general theory

of projective products (see Treves [18], Cp. 45), there exists

x, E X and Xn E ca(n) such that n2 jz 1 ((S) < +m and such

that T (E) = n~xnh(E), where the series will converge absolutely

in X. Write n clearly has the R-N property for
each ne E also I f and 2 [T [(S) < +m. We conclude
from Theorem 2.2 that r has the R-N property and so T E RNea(R;X).

Therefore X~nca(R) c RNca(n;X), hence we have equality. O

2.4 Corollary. A measure 9:R -+ X with bounded variation has
the R-N property if and only if p is expressible as an indefinte

Bochner integral with respect to some measure X:n --* R .

Proof. If f has the R-N property, then v is expressible as

an indefinite Bochner integral with respect to any measure
with which U is absolutely continuous.

Conversely, if ~(E) = IEf dX for some positive measure

h, then there exists a sequence of simple functions (fn)
converging to f X-a.e. such that /fn-l 1 IS n-f~dk -* 0.

Write un(E) = IS n dh; consequently, Un E ca(n)@ X. We
will show that ye ca(S0)0 X so that by Theorem 2.3, U will
have the R-N property. Because ca(n)BnX is isometrically
embedded in cabv(n;X), it suffices to show that the sequence

pn E ca(R)B X converges in variation to U. This is indeed
the case because |pn-i- (S) = If -fjl and n Sm l n 1ll = 0. O

2.5 Corollary. A Banach space X is a Radon-Nikodym space if

and only if ca(S,R)q~X = cabv(S,R;X) for every measurable space





32


Proof. One always has ca(S,n)d X c cabv(S,0;X) to begin

with. If X is a Radon-Nikodym space, then that means any

X-valued measure on R has the R-N property, that is, we have

containment in the other direction, hence equality.

Conversely, if ca(S,n)0 X = bvca(S,R;X) for every measur-

able space, then, regarding Theorem 2.3, this means every

X-valued measure of bounded variation has the R-N property

regardless of the measurable space (S,R). This is the definition

of X being a Radon-Nikodym space. O


2.6 Remark. In particular ca(S,R)B X = bvca(S,0;X) if X is

a reflexive Banach space of if X is a separable dual space. O

We have shown by Theorem 2.3 that ca(R)0 X lies isometrically

isomorphically in Cbvca(R;X). The question is raised whether

this isomorphism is onto. The answer is no in general as

demonstrated by the following example.


2.7 Example. This is an example of a vector valued measure

with bounded bariation and relative norm compact range which

does not have the R-N property. This is an example due to

Yosida [19].

Let S = [0,1], R = 8_ the o-field of Baire sets of 10,1]

and X:80 -+ R+ the Lebesgue measure. Denote by m[l/3,2/3]

the Banach space of real-valued functions 5 = S(() defined on

[1/3,2/3] and normed by ([I1 = sup S(().
Define an mil/3,2/3]-valued function x(s) = S((;s) on

(0,11 by :

if o sss 6 ;
x(s)(6) = S((;s) = s-
91if 6 s s s1i.








Yosida has shown that x(s) satisfies the Lipschitz

condition: jx(s)-x(s')l & 31s-s' for all s,s' E [0,11].

Define a set function on the class of intervals of [0,1]

by x(I) = x(s)-x(s') where s is the right end point of I and

s' is the left end point. The set function x has its values

in m[l/3,2/3], and extends to the class of Baire sets on

[0,1] as a set function with values in m[1/3,2/3]. Because

of the Lipschitz condition, Ix(B)( s 3A(;B) for Be 8 B, it

follows x is o-additive, X-continuous, and of finite total

variation. Yosida has shown in [19], that x cannot be

expressed as a Bochner integral with respect to Lebesgue

measure X even though x << X. We shall show that x has

relatively norm compact range.

According to Dunford-Schwartz [12], IV.5.6, a bounded

set K in m[1/3,2/3] is relatively compact if and only if for

every E > 0, there exists a finite collection {E ,E2..,E~n~

of disjoints sets with union [1/3,2/3] and points Bk E Ek

such that sup lf(6)-f(6k)l < E, for all f EK and k = 1,2,3,...,n.
6tEk

It shall be shown that {x(I):Isl}, where I is the algebra

of unions of disjoint intervals of [0,1], is relatively norm

compact in m[1/3,2/3].

Because x is o-additive, its range is bounded.

Let E > 0 be given. Choose n so large that 1/n < E.
I k 1 k+1
For k = 0,1,2,...,9n-1, define Ek + 27'3 7) Ek

has length 1/27n. Write Bk as the midpoint of Ek.
Let 0 sks 9n-1 and T El be fixed. We claim for any

6 E Ek that Jx(I)(6)-x(I)(Bk~lCE








Suppose to begin with that 9 e Ek and 6k < 8. We may
m i41
write x(I) in the form x(I)(6) = iCl(-1) x(s )(0 where

s1 > s2 > s3 > ... > sm

Let q be the largest integer such that sq > ek and p

be the greatest integer such that sp > 9. Since Bk < 6,

we have p 4 q.

Thus,

x()() = (-)+1 ( ,s ) = (-1)i+ s i=+ (-1)i+

and

x(I)(6 ) = (1i+1 s -1 m(1)+s
k i 1 i=q+1
6 -1 6


So that

|x(I)(6) x(I)(6 )]

+ (-1)i1
i=p+1

Simplifying, we

Jx(I)(6) x(I)(6k l



9 i+1 m


Si+1 s.-1 s -1
= .l (-1) 1 -
6-1 6 -1

s s-1+ c (-1)i+
6 6 -1 =q+1

get
6 -0
1(,-1)i1 -1) ekI )(1

Bk i+1 6-s.
s. at 1 i+ 1 (-1) 1
i 6(6 -)=p 1(k-1)
k,-
s. ~a


Write Ix(I)(6) x(I) (6k l $ 1 + 2 3 f 4'

I~~- (6k-1)(6-1) (6k1(61



18

So Q .
I 3n








(2) Q2 = k *1 . ~~ (-1) i+. / Ok-6
6(6 -1 6t (6 -1)
k 54 6


(3) Q3 6 1) i=+(-1)i+1 (6-s )| 9 i= +1 (-1)i+(6-s )


Note that for p+1 < i q, we have 6k is < 6, so that

( )i+1( 5 6 6 j 1
i=p+(1) (-k l-kl 54n
Thus Q < --
3 -6n

(4) Q1 =k *C .(1i+isi < k-e 91 6 -6 5 -9
4 6*k =q+1 *k 4


4 -6n
As a result of (1)-(4),

Ix(I)(9}-x(I)( 6 ) | 5 + + + <-= < E.
k 3n 6n 6n 6n 6n n
Thus k~~ x(I)(6)-x(I) (6k)| < E where 6 r Ek. This was

shown regardless of the set l E and of k, O
In a similar manner, we can show that kg ix(I)(6),-x(I) (6k)<

where 6 eE ,k whence it follows that sup |x(I) (6)-x(I) (6k 1

for each k, o
that x(I) is a relatively norm compact subset of m[1/2,2/3].

Because Bo = o(7) and x(B ) c cl[x(I)], we conclude that x has

relatively norm compact range. O


3. The Space of Measures with Relatively Norm Compact Range.
We shall devote this section to identifying the space

ca(R)0 X. In Theorem 3.3, it is shown that this is the space
of all X-valued measures on n with relatively norm compact

range. In the next chapter, we consider an important class of

measures with relatively norm compact range, namely, the class

of indefinite integrals of "weakly" integrable functions.







Throughout this section U:n -+ X is a measure, and

h:n -+ R+ is a control measure for 19. We shall denote by

H the collection of all measurable partitions of S; that

is, r EU if and only if r = {Fl' 2'"'" n)~ where Fi E are

pairwise disjoint, X(F ) > 0 and S = iYl i. Partially order
H as follows: for sr~n' E n, write n > n' if and only if every

member of w lies in some member of n'

For each na H,~ define qn:R X by

9 (E) = FT~ A( F(E), for E E R,

and where A (E) = X(EnF). Since A is a control measure we

have that n << U and in fact hF << p; also, observe 9 I(F) = U(F)
for each Fe E .

It is clear that for each we E 9~ E ca(R)0 X and with

the partial ordering of H, (9 ) is a net (or generalized sequence)

of measures in ca(R)0 X (See Dunford and Schwartz [12], Section
I.7). We shall show that if 9 has relatively normn compact

range, then lim U, = v is semivariation.

3.1 Lemma. If the Banach space X is the scalar field, then



Proof. Because 9 << A and 9 is scalar valued, there exists

a A-integrable function f such that p(E) = /Ef dX for each

E E n, by the Radon-Nikodym Theorem.

For we H define

f (s) = E~ Ji-)I f dlu S (S), s E S;
consequently, we have

9 (E) = IE f7 du.







From [12] (IV.8.18), lim f = f is L (A). Because the

L (A)-norm of a function is the total variation of its indefi-

nite integral, we have I~m p-qS(S) 3 0. O

We use this lemma to prove the same result for the general

case of x being an arbitrary Banach space and 1J.a measure with

relatively norm compact range. We remark that Theorem 3.2

was proven by Lewis (16]; here we present a more direct proof
of the theorem.


3.2 Theorem. If u:R -* X is a vector measure with relatively

norm compact range, then I~m U-q (S)s = 0, and consequently,

y E ca(n)0 X.

Proof. Let l,H and (9 ) be as above, and define

ca(R,X) = {#Eca(R): <
Define on ca(RIA), for each we H the linear operator

Un by U 4 = 4 where E ca(RA). By Lemma 3.1, lim U 0 = O
in ca(R,X); consequently, by the Phillips' Lemma [12,IV.5.4],

I~mU # = 6 uniformly on compact subsets of ca(R,X).
For each x* E Xl* we have x*M E ca(nA) so that

I mlU (x*p)-x*M (S) = 1 mlx"~n-x*M|(S) = 0.

Since xs R1 |x*U -x*, (S) = l l-uII (S), in order to show

BmlFm ,-y (S) = 0, it suffices to show, therefore, that

lim x*M -x*M|(S) = 0 uniformly for x* E X *, that is,

lim U (x"y) = x"p uniformly for x* e X *. From the above

discussion, we need only show that r = {x*y:xfeX1*} is a

compact subset of ca(R,A).







To this end, let (x *v) C r; we shall show that there is

a subsequence which converges in variation. Since E is weakly

sequentially compact by [12,IV.10.4], there exists x0* E X1*

and a subsequence (x* )of (x *) such that x* p x0*p weakly
1 1i
in ca(R,X), that is lim x* u(E) =x0*v(E) for each E E n.
i "i
Since U has relatively norm compact range, the set R~ = {p (E):EER}

is relatively compact in X; but x* 0 onws nRipis

by the Banach-Steinhaus theorem, x* --+ x0*uiomyo

Thus, lim sup Ix* ()x*() Btte
1 Ech ni~E-O*() Btte

0 = 1m sup |x* u(E)-x0~(E l 1m |x*-x0
1 Eca. "i 1 i-X* I

(see the remarks following Proposition I.1.1).

It follows then that x* pi -+ x *u in variation, and that
n.
P is compact. Thus liml|p -u| (S) = 0.

Finally, p E ca(R)4 X since it is the limit in norm of

a net (p ) of elements from ca(R)0 X. O]

3.3 Theorem. ca(R) ~X = Cca(R;X) isometrically, where Cca(n;X)
is the Banach space of X-valued measures with relatively norm

compact range.

Proof. It is clear that any of the step measures which comprise

the space ca(R)BX have relatively norm compact range because

they are linear combinations of elements of X with bounded

scalar measures; consequently, each step measure is bounded

with range in a finite dimensional subspace of X, hence has

relatively norm compact range.

By Theorem 1.3, ca(R)0 X is isometrically embedded in

ca(R;X) and consequently in Cca(n;X). The closure of ca(R)0 X








in Cca(n;X) is ca(R)0 X and ca(R)0 X S Cca(R;X). By Theorem

3.2, we have reverse inclusion. O


3.4 Corollary. If (S,n) and (T,A) are measurable spaces with

X and Y Banach spaces, then

Cca(SxT,nB0A;X~EY) = Cca(S,R;X)0 Cca(T,A;Y).

Proof. This follows from Theorem 3.3 and the associativity

of the inductive tensor product of four Banach spaces. O


3.5 Corollary. ca(SxT,R0qA) = ca(S,0)BECa(T,A).

Proof. Any scalar measure has a bounded range, hence a

relatively norm compact range. O

4. The Space ca(R;X)fca(A;Y)

In this section (S,R) and (T,A) are measurable spaces

with X and Y Banach spaces. We shall prove that the E-norm

on ca(R;X)Bca(A;Y) is the semivariation norm, and that

ca(n;X)4 ca(A;Y) can be isometrically embedded in a certain

space of separately continuous bilinear maps.

Recall that X*0Y* is a vector subspace of (XfY)* since

each x* E X* and y* E Y* defines a linear functional x*Bfy* E

(Xi Y)* such that = x*(x)*y*(y) and Ix*By*I =

Ix* *(y* Observing the definition of the E-norm, we see that
the set

r = {x*@cy* : x*EX1* YE1*
is a norming family for X0 Y. Also, for each y e ca(R;X),

v E ca(AZ;Y), x* E X*" and y* E Y*, the linear functional x*q~y*

acting on the vector measure ye v yields scalar measure defined








by:

(G) = x*,xy*V(G),

where G E R0 A.

Thus, x*B y* can be thought of as a linear map from

ca(R;X) 0 ca(A;Y) into ca(O Ah); furthermore, x"8Ey* is
continuous when the former space has on it the E-norm and

the latter space is supplied with the total variation norm.

We prove this in the next lemma.

4.1 Lemma. Let x* E X* and y* E Y*. x* OE y* when considered

as a linear map from ca(R;X) 0E ca(A;Y) into ca(R0 A) defined

by = X*y x y*Li iS a COntinuous linear map.
Moreover, Ix*B y*| = |x* ~l*|yI

Proof. The maps = x*v and = y*v are defined

from ca(R;X) (resp. ca(A;Y)) into ca(R) (resp. ca(A)), are

both clearly linear and they are both continuous since by

Proposition I.1.1,

Ix*pl(S) 5 Ix* *(|p||(S) and ly~u (T) 5 |Y* IV (T).
From the general theory of tensor products, the map

x*8Ey* is continuous from ca(R;X) 0E ca(A;Y) into ca(Q)8 ca(A)
(see Treves [18], Theorem 43.6), and Ix"~,y*| = ix*\* y* .

The map x*0 y* can be extended to ca(R;X)@Eca(A;Y) with

values in ca(n)@Eca(A). By Corollary 3.5, ca(RDOG

caRBEca0). caA)

From this lemma, we observe the next proposition which

shall be used in this chapter and in Chapter IV.







4.2 Proposition. Let .2 9.0 v. E ca(R;X) 0 ca(A;Y). Then

the semivariation of this measure is given by
n n
.2~ ~ ~~U p.0 v.(ST u 2x*Mjxy*vil (S T),

where the supremum is taken over x* E Xl* and y* E Y1 "


Proof. The collection r = {x*B y*:x*EX1* and y*EY1*} is

a norming family for X~EY; therefore by Proposition I,1.1,
n n
fi \i (x) xs g) pl~ (SXT).

The proposition follows then from this equality and

Lemma 4.1. O3


If we now endow ca(R;X) B ca(AZ;Y) with the semivariation

it is easy to see from the above proposition that this

cross normn:


norm

is a


I@/jlEI (SxT) = sup x*pxy*\J|(SxT) = sup x*/I (S)-|y*v3 (T)


where all supremums are taken over X x 71 *

Also from Lemma 4.1, we have jI/B|(SxT) 5 le6 for any

6 ca(G;X)0 ca(A;Y); indeed, for x* e X any y* e Y *, the

function x* 0E y* is continuous and jx*0 y*l = |x* y*| = 1

so that jx*0 y*61(SxT) < 6 el. Now taking the supremum over

X X Yl* we get by Proposition 4.2, II6 ((SxT) < 6 j. We

shall see that in fact, equality reigns.


4.3 Theorem. For any 6 E ca(R;X)qEca(A;Y), we have |61 I(SxT) =

6 ,E that is,

ca(R;X)4 ca(A;Y) c ca{GO A;XB Y) isometrically.







Proof. Let p*E ca(n;X)* and v* e ca(A;Y)* and consider

U*8v* E ca(n;X)* 8 ca(A;Y)*. The norm of u* 0 v* associated
with the semivariation norm is defined by

/p*@v*l = sup l i i (1)
where the supremumn is taken over all elements p = .E p.0 v.
such that Ilp| (SxT) < 1. We claim that this norm is a crossnorm:
p*gv*J = Ip* "|v*(.
It is clear that 19*( v*I < Ip*@V*i by considering the
supremum in (1) as being over a smaller class, namely, over
all p = Fiecy such that y E ca(R;X), v E ca(A;Y) and Ilp ((SxT) s 1.
Now let p = Ci9 0 q be arbitrary with Ilp l(SxT) < 1.
Since 9* is a linear functional on ca(R;X) and Ejv*(v.)p. E
ca(R;X) we have

i i L i|1S).(2)
Choose (E ) c R, a finite collection of pairwise disjoint
sets and scalars (a ) c 4 with |aj < 1 such that

9* F*l jV*(Vi ~il (S) < + I* |p**| jaE vu*(v )9 (E )


Again choosing sets (Fk) A pairwise disjoint and scalars

(Bk) Q with |8kl 1 such that

j0* l *v* I /l |Ea p (E )vi (T)

5 + I9* v* EIk k jjii(j p i(E )@q

Combining these inequalities with (2) we get

Ei i i ka Ski~iBEi(E XFk) | 3







Now since the family {E X~k jlk is pairwise disjoint

and covers S x T, and I2 Skl < 1 for all j and k, we see that
the quantity on the right hand side of (3) is one of the numbers

overwhich the supremum is taken in the definition of the

semivariation of the measure Ly. v.
1 1 C 1.
Thus,



But now since p = Ci q Vi was arbitrary with /pl (SxT) < 1,
taking the supremum over all such p, we get by definition

Iy*@v*( < C+ ff-ju*|.

Since E > 0 was arbitrary, we get |p*@v*| < 1u* *|v*1

and the assertion that /p*8v*| = Iu* V* is proved.

Finally in order to prove le6 6 Ill(SxT) for any

6 E ca{G;X)0 ca(A;Y), if suffices to prove this for 6 of

the for .E p.V.. We have shown that |9*Bv*I = Iy*|*|v* ,
this means

Ip*@v*(8) r lu*Ov*ll l 6 SxT) = Iv*l vI*|* 6e (SxT).
So that

IBI = sup / u ( )- *(

= sup

< sup p*)*] v*l-|6 (Sxy)

= 6 (llSxy),

where the supremum is taken over |9*I = 1 and Iv*| = 1.

Thus, 9 el, 6 (llSxT). Since we have already observed
the reverse inequality, the theorem is proved.O








4.4 Corollary. ca(n;X)@cY 5 ca(n;XB Y) isometrically.

Proof. Let T = (0) and AZ = {T,#}, the power set of T, then

ca(T,A;Y) = Y isometrically. Apply Theorem 4.3. O

Let X,Y, and Z be Banach spaces. Then B(X,Y;Z) will denote

the vector space of all separately continuous bilinear maps

from Xx Y into Z. Separately continuous bilinear maps need

not be bounded; however, they are bounded whenever each factor

of the product space on which they are defined is a dual space.

For this reason, the space B(X*,Y*;Z) can be normed by

lbl = suplb(x*IU,y* where the supremum is taken over X1* x Y1
This topology on B(X*,Y*,Z) is the topology of uniform

convergence on equicontinuous (simply bounded) subsets of

X* x Y* of the form A x B. B(X*,Y*;Z) equipped with this

norm topology is denoted by B (X*,Y*;Z). It is not difficult

to see that BE(X*,Y*;Z) is a Banach space.

Analogous to the usual embedding of ca(R;X)B ca(A;Y)

into B (ca(R;X)*,ca(n;Y)*;Q), from which the definition of

the e-topology was derived to begin with, we have the following

theorem.


4.5 Theorem. There exists an isometric isomorphism from

ca(R;X)4 ca(A;Y) into B (X*,Y*;ca(nBeo '

Proof. Define 8 :ca(R;X) x ca(A;Y) -t B (X*,Y*;ca(RB0A)) by

B(plv)(x*,y*) = x*, x y*v. 8 is a bilinear map; using once

again the universal mapping property of tensor products,

there exists a unique linear map

O:ca(G;X) 0 ca(A;Y) -+ B (X*,Y*;ca(R00A)) such that

~(UBe )(x*,y*) = x*U x y*U.







To prove that 8 is a one-to-one and an isometry, it

suffices to show it is an isometry.

Let 6 = .2 p.0 9., and prove |e(6) = 8 .E

I8(6)/ = sup ~(6) (x*,y*)| = sup Jilx"pxy*v (Sxy)

= I/B( (SxT) where the supremums are taken over XI* x Y1"

By Theorem 4.3, 18 16 = I| 6 | SxT)
Thus 8(8)| = |6 ,. O

There are a few advantages as well as disadvantages to

embedding ca(R;X)0 ca(A;Y) in B (X*,Y*;ca(n00A)) rather than

B (ca(n;X)*,ca(A;Y)*;Q). Because we know very little of the
structure of the duals of ca(R;X) and ca(A;Y), it may be

advantageous to use the embedding B (X*,Y*;ca(GO A))) the
structure of the Banach spaces X* and Y* may be well-known or

more easily worked with. The range space of the bilinear

maps of B (X*,Y*;ca(R00A)) is more complicated than the scalar
bilinear maps of the other embedding, though quite a lot is

known of the structure of ca(RBcA). At any rate, both
embeddings induce the E-norm on ca(R;X) 8 ca(A;Y).













CHAPTER III
PETTIS AND LEBESGUE TYPE SPACES
AND VECTOR INTEGRATION

1. Measure Theory

Throughout this chapter, (S,R) is a measureable space,

X and Y are Banach spaces, and y:R -+ Y is a vector measure.

A set A cS is ii-null if there exists a set E R such

that A cE and Il l(E) = 0. The phrase "p-almost everywhere,"

or p-a.e., refers to y-null sets.

An X-valued R-simple function is a function of the form

f is) = Ex.((s), where x. E X, (E.) c 0 is pairwise disjoint,

and TE.(s) is the characteristic function of E The sets

E. are called the characteristic sets of f. The vector space

of all such simple functions will be denoted by S (0), and

when X = 4, by S(n). A function f:S -+ X is p-measurable if

there exists a sequence of simple functions from S (n) converging

to f pointwise p-a.e. The same function is weakly p-measurable

if for each x* E X*, the scalar function x*f is p-measurable.

Obviously, any p-measurable function is weakly p-measurable;

the two concepts coincide if X is separable, by a theorem

due to Pettis [17}. A scalar function f:R ~-> is n-measurable

provided f-l(B) n for every Borel set B. Any R-measurable

function is Fp-measurable, and any y-measurable function is

equal p-a.e. to a R-measurable function.

A sequence (fn) of Fi-measurable functions converges in

p-measure to a function f means
46








11mll~ [rl([fn-f >E]) = 0
for each E > 0. In this case, f is u-measurable and there

exists a subsequence (fni ) which converges pointwise to f
p-a.e.,this is the theorem of F. Riesz. The Riesz theorem

and the Egorov theorem are valid for vector measures because

we can choose a control measure A for p. The measures y and

A have the same null sets, and therefore the same measurable

functions; convergence in p-measurable is equivalent to

convergence in A-measure. Since these two theorems are

valid for X, they are valid for u as well. Consequently,

any sequence of functions converging p-a.e. also converges

in p-measure. The phrases "in p-measure" and "p-a.e." are

virtually interchangeable.


2. Normed Spaces of p-measurable Functions.

If f is weakly p-measurable, we can consider a number

of scalar integrals associated with f in order to define a

variety of seminorms on the space of X-valued weakly p-measurable

functions.

Define the two seminorms N and N* on the space of weakly

y-measurable X-valued functions as follows:

(1) N(f) = ysp ~jSlf d ylp ;

(2) N*(f) =x~ sug / x*fdlyl*ul.


We remark that N and N* are indeed seminorms because each

is the supremum of seminorms. Since Ixxfl < |fl pointwise

for x* E X *, we have immediately that 0 < N*(f) 6 N(f) < + m.







The N-seminorn, which is a Lebesque-Bochner type, was

introduced by Brooks and Dinculeanu [5]; this seminorm will

sometimes be referred to as the strong seminorm. The N*-

seminorm, which is aPettis type seminorm, will be called the

weak seminorm. These seminorms, of course, depend on many

parameters such as the measure y, and the Banach spaces X

and Y; it will be clear from the context which parameters

are being considered.

If f is X-valued, then Jff is scalar valued, and we shall

write N(f) = N*(lf ). Note that it is always the case that

N*(f) = sup N(jx*f ).
x*EX l

We list some properties of these seminorms

2.1 Proposition. (1) N and N* are subadditive and homogeneous;

(2) N*(f) = N(f) for f scalar valued;

(3) N*(f) = sup N*(fgA), N(f) = sup N(fgA)
AEg AER
(4) N(sup fn) = sup N(fn) whenever (fn) is increasing
and positive;

(5) N(C f ) sEC N~f ) for every sequence of positive

functions (fn)

(6) N(lim inf f ) < lim inf N(f );
(7) N(f) < +m implies f is finite p-a.e. for f R -valued.

Proof. Numbers (1), (2), and (3) are clear from the definitions.

(4): sup N~fn) = snp yU le S jdly*uli

= usup /ff dly*p|


=UE gu1S p fn d y*il = N(sgp fn "







(5): N(C f ) = N(s p Ikf ) = sup~ N( e f )

i sup n~N(f ) = 6 N(f ).

(6): From Fatou's lamma,

/S limninf Ifn d y*pl s limninffS |fn a yL*l-1 .

So for y* E Y1 '

J~lim inflff d y"~l su 41im inffS f d ly*U

= lim inf sup JS f ld y"~

=lim inf N(f ).
Finally, (6) is obtained by taking the supremum of the

left-hand inequality over Y1

(7):If fis R-valued, and N(f) < +m, then for each y* E Y ,
f is finite ly*p -a.e., from the classical theory. By Theorem
1.1.2, U-null sets are determined by only a countable family

of { y*p|}, that is, there exists (yn C 1* such that a subset

A cS is p-null if and only if A is lyn~u -null for each
ne E As a result, f is finite p-a.e. O

The set FX(S,R,I-;Y) of functions f:S -t X which are
p-measurable and satisfy N(f) < += is a vector space with

seminorm N. When no confusion will arise, we write FX~u

for FX(S,R,p;Y). The set WX(S,Q,p;Y), or simply WX(p), is
the set of all functions f which are X-val~ued and p-measurable

that satisfy N*(f) < +=. WX(u) is also a vector space with
seminorn N*. It is clear that FX( ~ 'X(p) and the topology
induced on FX(u) by the seminorm N* is weaker than the N-norm

topology of FX(y) since N*(f) < N(f).







Brooks and Dinculeanu [5] have shown, and it follows

from (5) in Proposition 2.1, that the system (FX(u),N) is
a Banach space if functions equal p-a.e. are identified.

(WX(y),N*) need not be a Banach space however, since it may
not be complete, even if functions equal p-a.e. are identified.

We can make WX(y) into a complete metric space by con-
sidering the metric:

d(f,g) = N*(f-g) + inf~Ca+ p1 ([lf-g >a])}, f1g E W,(p).
Recall that the second term in the definition is itself

a metric equivalent to convergence in p-measure (see Dunford

and Schwartz [12], p. 1021.

2.2 Proposition. The semimetric space (WX(u),d) is complete.

Proof. Suppose (f ) W (p) is d-Cauchy, then (f ) is Cauchy
in p-measure; consequently, there exists a function f from s

into X which is p-measurable and to which (fn) converges in

p-measure, that is, lim Ilil(l[lf -f >E]) = 0, for each E > 0.
To show 1 m d~fn f) = it suffices to show 11m N*(fn-f) = 0.

Let x* E X be fixed. Since Ix*f -x*f| s If -fl

pointwise, we must have x*fn -+ x*f in Ii-measure too. Now
for each y* E Y1* ISIx*fn-x*fmldly*pl N*(f -fm), so (x*f )

is Cauchy in L (y*y), the classical Lebesgue space. But

x*f, -+ x*f in y-measure implies x*f, x*f in y*y-measure,
so therefore x*f, x*f in L (y*M).
Let E > 0 be given, choose K E w Such that whenever

m,n K, N*(f -f ) < E.

But J jx*f -x*f id(y*yj s N*(f -f ) < for every

[x*,y*) E X1*xY1* and m,n r K.







Because x*fn- x*f in L (y*u) for each (x*,y*) E X <'~f

we have 11m fS x*fn-xffmlddYy~l = IS X*fn-x*fldjy*p|.
Therefore, for n 2 K,

ISIx*fn-x*f d y*li =11m JSIx*fn-X*ffmidly*ul s E.

Taking the supremum over X1*xyl*, we get N*(f -f) s E for all
n 2K. O

This semimetric topology of WX(v) is the topology where

a sequence (fn) 'X(y) converges to a function f in WX(~

if and only if N*(fn-f) -* 0 and fn --* f in p-measure. It

is possible, though we shall not do so here, to consider a

slightly more general spa-e, the space of weakly p-measurable
functions with finite N*-seminorm.

We next prove that N* is a norm on WX(p), if we agree
to identify two functions which disagree only on a u-null set.

2.3 Proposition. If f EWX(p) and f = g 11-a.e. for some X-valued

function g on S, then g E WX(u) and N*(f-g) = 0, in particular,

N*(f) = N*(g).

Conversely, if N*(f) = 0, then f = 0 u-a.e.

Proof. It is clear that g is y-measurable since it is equal

p-a.e. to a p-measurable function.

Now for each x* E X1* and y* E Y *, x*f = x*g y-a.e.,

and therefore ly*p|-a.e. since |y*ul II1-|| by Proposition

I.1.1. This being the case, from the Lebesgue theory of

integration we have ISlx*f-x*g dly*yl = 0. N*(f-g) = 0 is

obtained by taking the supremum over X1*XYl '
Conversely, N*(f) = 0 implies xsu J p ~ l~(=0

for each y* E Y1*. This supremum is the Pettis norm of f







with respect to the measure ly*p ; it follows then from
Pettis [17] that f = 0 ly*F|-a~e. By Theorem I.2.1, we have

f = 0 -a.e. OI


Thus the space (WX(Fr),d) is complete metric space; in
fact, it is a Frech~t space. To see this, it suffices to

show that lim d(af,0) = 0, where as i and fe W w(9). This
fact was proven in Dunford and Schwartz [12], p. 329.

Notice that SX(8) is a vector subspace of both FX(~

and WX(p). We shall denote by B (9), the closure of SX(~

in FX(u) and remark that BX(y) is a Banach space with norm

N. PX(p) will denote the closure of SX("?) in the metric
topology of WX() PX(") is a Frechit space.
As a result we have

(1) f i BX(p) if there exists a sequence (f ) SX )!

converging y-a.e. to f such that Iim N(f -f) = 0.

(2) f eP (V) if Ilim N*(f -f) = 0 for some sequence

(fn) ES (0)n converging p-a.e. to f.

For a simple function g we have N*(g) C N(g), therefore,

BX( ~ ~ X(u). If X = 4, we write BX(y) = B(p) and P (u) = P().

2.4 Proposition. B(p) = P(p) and N(f) = N* (f) for all f i B(p).
Proof. By Proposition 2.1 (2), N(f) = N*(f) whenever f is
scalar valued and u-measurable. Thus N N* on S(0) so

B(p) = P(y). O

2.5 Proposition. Any bounded, u-measurable, X-valued function

on S is in BX(~







Proof. Let g:S -+ X be bounded and p-measurable and write

K=su /g(s) There exists a sequence (gn) X S(n) converging
in p-measurable to g and uniformly bounded by 2K.

For given E > 0, there exists M E Such that n 'M

implies I/j j([ g-gn >c]) When n r M,

N(9-n) ysu1* /S n -9 d y*p


5~F Iu Jn nj9,dly*ll +y~~P u S-E n49ndly*p|I

I 3K* (IIEn) + E*(191 (S- En

5 3KE + E (llS) = E (3K+ p (lS) ).

TEherefore, limn N(g-g ) = 0 and g e BX(u). O

Because BX( ~ 'X(p), it follows that WX(u) contains
the bounded, p-measurable functions too.

2.6 Proposition. A function f E WX(p) is in PX(y) if and only
if 1 m N*(fzA ) for every sequence (A ) C R with n lAn =~

Proof. (--4 Suppose f EP (u), then letting E -> 0 be arbitrary,
choose a simple function g such that N*(f-g) <2

Now if (An) t and n lAn and K = su lg(s)|, then

there exits M F such that n 'M implies II1 I(An, 2K
For n r M,

N*(ftA ) *fg *gA ) < E+ K.II In 19 E(A
n n

(+-) Assume N*(flA -+ 0 whenever n lAn p.

Put Bn = [ f En] and An = S-B ; obviously n 1A = ~. fS

is bounded, so it is in PX(y) by Proposition 2.5. Since

11m N*(f-ftgn) = 1 m N*(flAn) = 0 by assumption, f E PX(p). O








2.7 Proposition. For f E WX ~) f E PX(y) if and only if


Proof. The condition N*(f (.) <

0,

there exists 6 > 0 such that if IIF1||(A) < 6,r then N*(f5A) (i) Let E > 0 be given, choose a simple function g so that

N(-)< T. Put K=su ((s) , then for ilpl (A) < ZXwe
have

N*(f ;A) 5 N*(f-g) + N*(g5A)

< + K*N* (A) = + K. (A)
E C
<2 +K2K
Thus for (lilA) < 2 ehveN(g

(4-) Let n 1An = Q. Then HI(IAn) -+ 0 so by assumption
11m N*(fg~n = This implies by Proposition 2.6 that

f E PX(11). O
2.8 Remark. In Propositions 2.6 and 2.7, the properties

of N* that distinguish it from N where not used; consequently,

2.6 and 2.7 remain valid when WX() PX(u) and N* are replaced

with F (V), BX(U) and N, respectively.

3. Convergence Theorems.

We now consider convergence properties of the N*-norm

in order to obtain criteria for a function to be in PX(~

given that it is the pointwise limit of a sequence of functions

in. PX(~

3.1Theo~rem. (Vitali Convergence Theorem) Let (fn) 'X(1

and f:S --* X. Suppose (1) fn -+ f in p-measure;

(2) N*(fn (*)) << 9 uniformly in n.
Then f eP (9) and lim N*(f-f ) = 0.







Conversely, if fn -+ f in PX(y), that is, in the d-metric,
then (1) and (2) hold.

Proof. We first show that (f ) is d-Cauchy. Since by

(1), (fn) is Cauchy in p-measure, it suffices to show this

sequence is N*-Cauchy.

For E > 0 given, there exists, by assumption (2), a

6 > 0 such that II|9| (A) < 6 implies N*(fnSA) < for all
ne W.

f_ f in Ip-measure implies the existence of M e (

such that I|9) ( [ ,fnml FE ]-iii~;i ) < 6, when n,m r M.

Fix m,n 2 M1 and write B = [ fnfml 3 Tp (S) *]

N*(f -f ) < N*(f -fS ) + N*(f -f S _)

SN*(fn B) + N*(fm B) + N*(fn fm S-B)



S(f)is N*-Cauchy as well as in U-measure, thus

(fn) is d-Cauchy. Since PX(u) is complete and fn -t f in
y-measure, we see that l~im N*(f -f) = 0 and f eP (9).

Conversely, suppose d(f ,f) -+ 0; then fn -t f in

pt-measure and N*(fn-f) -+ 0.

Let 8 > 0 be given, there exists M e ( such that n >_ M

implies N* (fA) < N*(f5A) + for all As E Ti olw
from the inequality

N*(f CA) N*(fSA) N*(fn-f

Since fcP (l9), we have N*(fz; d) << Thus there exists
a 6 0 o tat N(fg)
N*(fn A) < N*fA + < = c whenever 1I (IA) < 6.







Because fl f2' M f X 9), it must be true that N*(fkS (-) '<~
for k = 1,2,...,M. There are only finitely many, so we may

find a single 61 > 0 such that ful l(A) < 61 implies N*(fk5) A '

1 i k M. Finally, putting 62 = min {6,61), we see that

N*(fn A) < E for all new whenever (1 I(A) < 62. This is
condition (2); condition (1) follows from the assumption

that d(fn,f) -* 0. O

3.2 Theorem. (Lebesque Dominated Convergence) Let (fn) P (9).
Assume g E P(G) and f:S -+ X such that

(1) fn -+, f in p-measure;

(2) |fn] 191 pointwise y-a.e. for every ne E .
Then f E PX(pJ) and N*(fn-f) - 0.

Proof. Note that |x*fnl n fj jg pointwise p-a.e. for every
ne W and x* E Xl*. Let As E .

N*(fnSA A x5gu N(x*fn A) N(ggA)

g E P(Fl), and P(p) = B(y), so N(gg .) <<9
by Proposition 2.7. By the above inequality we then have

N*(fn (.)) << 9 uniformly in ne E .
By the Vitali Theorem 3.1, f E PX(y) and N*(fn-f) -m 0- O

3.3 Corollary. (Bounded Convergence) If [fn| M pointwise
p-a.e., for every n, where M is some positive constant, then

f E PX(FI) and N*(f -f) -* 0.

Proof. Put g(s) = M for s E S. g is a constant function so

g E P(p) by Proposition 2.5. Apply Theorem 3.2.








3.4 Remark. No crucial role was played by the N*-norm;

consequently, the Vitali and Lebesgue Dominated Convergence

theorems are valid when N* and PX(p) is replaced by N and

BX(p), with only minor changes in the proof.

Brooks and Dinculeanu [5] have studied the space BX(~

in more detail and generality. Under suitable conditions,

the space B (9) is weakly sequentially complete, a workable

dual space has been identified, and sufficient conditions

have been given for subsets of BX(y) to be weakly compact.

The space PX(u) is more difficult to work with because it

is not a Banach space but a Frechit space.


4. Integration.

Let p:n -+ Y be countably additive and X a Banach space.

In this section we develop an integration theory for functions

in PX(u) and B (9).
Suppose f(s) = 1xE(swhrx xad:)

forms a measurable partition of 5, define for any Ee E ,


ffa Edu =~~i .2x.(EnE.).

We note that the value of the integral of a simple function

lies in the space X OE Y

4.1 Proposition. (1) The integral of a simple function is
well defined.

(2) The integral is linear, homogeneous,

and countably additive.
n n
Proof. Suppose f = .Z1 x.:E and g = j4 X.( j. Then.
n m
f-9 C i C 1 (x -x )5EA and








n m
JE fEdu lEg 9E du = i1x 0C p(EnE )- xO(n)
n m m n
= .1l x.0(.E y(EnEinA))) .1l X.0(.E p(EnEjnA,))

nm m n
I.Ex.@u(EnEinA.) .E .E X.(EnEjnA.)
i=1 ]= ] 1] = 11]
n m
=~ ~~j .E .E(.X. (EnEinA.) = I f-g 0 do.

This proves the linearity and uniqueness; indeed, if

f = g u-a.e. then whenever Il|9 (E nA ) i 0, we have xi xj = 0.
It is always true then that I1ul (EnE nA ) = or xi Rj = 0;
therefore,
n m
lff 0 dp Iff 8 du =~ .E .E(x.-X) Op(EnEjnA.) = 0.

This proves the integral is well defined.

In order to show the countable additivity of the indefinite

integral of a simple function, it suffices to consider a simple

function of the form f(s) = x5E(s), for some x E X and E E *.

indeed, if (An) o is disjoint and A = n lAn, then since

y is >-additive, u(EnA) = nE y(EnA ) converges unconditionally
in Y.

It is clear that nE x~p{EnA ) converges unconditionally
in X8 Y and

nE xBp(EnA ) = x0 E p(EnA ) = x~p(EnA).
But this is equivalent to

IfA edy 1 A fEdl. C

4.2 Propositon. Let f E SX(RJ). Then for every (x*,y*) E X* x Y*,

(x*0Ey") IE Edy = TEx~fd(y"p).







Furthermore,

IfE 6dyl < N*(f5E '

Proof. The first assertion follows immediately upon writing

the integral as a finite sum determined by the simple function

f. The cannonical image of Xl* x Y1* in (X 0E Y)* is norming
for X 0 Y so that

|IfE edyl = sup (~x*By*)JE BEdy
= sup IISx*(fgE)d(y"p)

SN*(f A '
where the supremum is over (x*,y*) E X1* x Y1*

4.3 Corollary. Let f E SX(0) and T(E) = /E 8Edy, for EE G .
Then r:R -+ X 0 Y is a vector measure and << pi.

Proof. The fact that -r is a vector measure follows from

Proposition 4.1. Observing Proposition 4.2, (1 (8)I < N*(fSE "

N*(f5( ) << p by Proposition 2.6, so r << p. O

5. The Weak and Strong Integrals.

Consider a function f E PX(y). There exists a sequence

(fn) SX(R) which converges to f in y-measure and 1 m N*(fn-f)=0.
The sequence (fn) is said to determine f. Now, for each E E n,

the sequence {fE n Edp} is Cauchy in the norm of X 8E Y.
Indeed, by Proposition 4.2,

llff 0du /ff 8 dpE N*(f -f ).

Define IE edy = 11im IE n edy, E E n.
This integral takes its values in X 0e Y and is called the

weak integral of f over E with respect to p.

The weak integral in unambigously defined for if (fn)

and (g )both determined f, then lim N*(f-f ) = 0 and
n n








l~im N*(f-g ) = 0.

|IfE nEdy /E n Edp|~ N*(fn n,

< N*(f -f) + N*(f-gn

From this we see that

lim /ff 0 du = lim I g 0 do.

Let now f E BX(p) and (fn) 'X(R) a determining sequence

for f in BX(p), that is, 1 m N(f-fn) = 0.
Because

llff -0du Iff 0 dul N(f -f ),
we see that the sequence {/ff 0 dp) is Cauchy in the norm

of X 0~ Y; hence, it tends to a limit in that space.
Define the strong integral of f by

JE 6tdy = Ifm JE n Edy, for E E 51.

Since B (u) PX(y) and the norm N < N*, it follows

if f E BX(U), then any sequence in S (R) that determines f

in BX(y) also determines f in PX(u); consequently, a determining
sequence for f in BX(y) defines the same value in XB Y for
both the weak and strong integrals. In this way we see that

the strong integral is well defined and can be unambiguously

denoted in the same way as the weak integral.

We next consider the countable additivity of these

integrals as well as a decomposition theorem for weakly

integrable functions.

5.1 Propositon. The indefinite integral of a weakly (respn.

strongly) integrable function is countably additive.







Proof. The countable additivity of the strong integral will

follow from the countable additivity of the weak. To that

end, let fe PX(p) and write r(E) = IE Edy, where E E R.
There exists a sequence (fn) of simple functions that determines

f in P (i-).

If I (E) = Iff 0 dp, then by definition lim r (E) = T (E)
for every E E n.

By Proposition 4.1, each rn is countably additive, and
by Corollary 4.3, rn << p for each ne E ; therefore, by the
vector form of the Nikodym theorem ([12], IV.10.6), r is

countably additive. O


5.2 Proposition. If f E PX(y), then for each x* E X1* and
y* E YI*, we have x*f E L (y*y), the classical Lebesgue space.

Furthermore, IEx*f d(yfy) = (x*Bfy*)JE Edy.

Proof. The first assertion follows from the inequality

ISlx*fld y"pl N*(f) < +m. The left-side of this inequality
is the L1-norm of x*finLyp)thts xf|<+.Te

second assertion follows by considering a determining sequence

for f, Proposition 4.2, and the continuity of x*0 y*. O


5.3 Proposition. If f E PX(y), then JIE Edyl N*(f5E '

If f E BX(y), then IIE Edyl < NECE '

Proof. For f eP (u),

IIE EdpQE = sup I(x*0 y*)/E E~dyl


= sup JISx*fgfE) d(y*u)l







< sup IS x*(f5E)|dlyy~l

= N*(fSE '
The second assertion follows because B (p) c PX(u) and
N*(f) s N(f). O


5.4 Theorem. If h = .L1 x. EP (1J), where x. e X and the
family (E )c R is pariwise disjoint, then for each E E n
we have

I he du = .~ x.0pl(EnE.), and the series converges
unconditionally in X 0~ Y.

Proof. Define T(E) = / h~edp.
Since T is o-additive, Proposition 5.1, and S = i 1E ,

we hae T(E = T(EnE ) = i IT(EnE ) where the last series
converges unconditionally in X B_ Y.

Now for each i, T(EnE ) = IEnE~h@ dy, and for (x*,y*) E

X x y1*, we have

x*B y*T(EnE ) = JEnE.x*h d(y*p) = x*(x )y*y(EnE )

= x*B y*(x.0p(EnE.)).

But this implies T(EnE ) = x yu(EnE ) because {x"E*I(*E "

X xY1*} is forming for X Y.
Thus IEh~cdy = T(E) = T~l(EnE ) = i EE) ovre

unconditionally in X 9s Y. O

5.5 Theorem. (Decomposition Theorem) Suppose f E PX()
then f can be written in the form f = g + h p-a.e., where

(1) g is bounded (hence g E B (9));

(2) h = .2 x.CE where x. E X and E. E 0 are disjoint.







Furthermore,

(#)/ f@ dp = I go du i 1x.0p(L~EnE. where the last series
converges unconditionally for each E E a.

Proof. Since f is >-measurable, it has an almost separable

range, so that we assume from the beginning that the range

of f is separable.

Let a 40O be summable. Define, for each n, S(n,f(s))

to be the sphere of radius an about f~s). For each n, f(S) c

skSS(n,f(s)). The range of f is separable metric space --
hence It is Lindelif; consequently, there exists a sequence

(s ") in S such that f(S) f uSnm ~ )
i c= in fsin)

Pettis has proved in [17] that the function s ifs)-f~sin)
is p-measurable, hence An=ffsn l(0a)eis mauab.
iI i nl1-

+c n n n =Ofri~j
Note that for each n, S = iu E. an E.nE o jj

Obviously, for any n, and s E S, we must have s EnEi for some

i; but this implies that [f(s)-f(s n)l < an and, therefore,

If(s) f (s) < a This means fn -+ f uniformly on S.

Write g(s) = E (f ~(s)-f(s)s)) then g is measurable,

and bounded since Ig(s) sn-el !n+1(s)-fn(s)| < 2n l
1 1
Finally, define E. = E. x. f(si ) and h(s) = f (s)=
i~ 1 i 1 =X'
1 1
We clearly have f(s) = g(s) + h(s), for all s E S, because

f(s) = n ~mn(s) = flts) + 1 (f~~s-n+(s)-f~)) h(s) + g~s).








g is bounded so that g E PX(y) (Proposition 2.5); consequently

h E PX(y), since h = f g and f,g E PX ')
(#) follows from Theorem 5.4. O


The Decomposition theorem is similar to the one published

by Brooks in [3].

We now turn to a deeper study of the weak and strong

integrals by comparison with well known,more familiar integrals.


6. The Weak Integral and its Relationships to Other Integrals.

The purpose of this section is to explore the various

relationships between the weak integral, the general bilinear

integral of Bartle [l], and a Pettis-type integral which will

be introduced below.

Let us first consider the Bartle general bilinear integral.

Bartle considers u-measurable functions f:S -- X and a measure

p:R -+~Y with a bilinear map b from X x Y into a third space

Z. In our context, b:XxY -+ X8 Y is the canonical bilinear

map defined by b(x,y) = x~y. Note that Ib(x~y)le = |x y .

Bartle requires the "control" set function for the measure

CI to be Ill9 : |x R+ the semivariation of v with respect to

X (and E), defined in Chapter I. From Lemma I.2.2, we have

that |jp l(A) = 11 1(A) for every AE t COnsequently, Bartle's

control set function turns out to be the "usual" one. It is

important to note, that the measure then has the *-property

(see Bartle [1]).

In order for a function f from S into X to be integrable

in the sense of Bartle, there must exist a sequence (fn) ESX~





65

converging p-a.e. such that the sequence {(f fdy} is Cauchy

is the norm of X 0E Y for each E E 0, where the integral of
a simple function is defined in the usual fashion. In this

case, one defines

(B)/Efdp=lqmfE ndy.

We say the sequence (f ) determines the Bartle integral of f.
We. now consider the relationship between the weak integral,

defined for functions in PX(y), and the Bartle integral.

6.1 Theorem. A function f:S -+ X is Bartle integrable if and

only if f E PX(p). Moreover,

(1) a sequence (fn) SX(R) determines the Bartle integral

of f if and only if lim~ N*(f -f) = 0;

(2) JEf E dp = (B)IEf dp for every E E R.

Proof. If f EP ,(9), then there exist a determining sequence

(fn) 'S SX(0 for f in PX(U), that is, 1 m N*(f-fn) = 0. From
our observations preceding the definition of the weak integral

in section 5, the sequence {IfEn 8~ du) is Cauchy in X 8E Y
for every Ee E Since the Bartle and weak integrals of simple

functions obviously coincide, (f ) determines the Bartle
integral of f, f is Bartle integrable, and

lJE 8 du = (B)JEf dp.
Conversely, suppose f:S -+ X is Bartle integrable, that

is, there exists a sequence (fn) C SX(n) converging to f

p-a.e. such that lim E fn du exists in X 0E Y for every

E E n, the limit being (B)/E n, dp. We wish to show f EP (9) ,
to do so, it suffices to show that lim N*(f-f ) = 0, this will
also prove (1).







Write

T (E) = (B)/E n du and TO(E) = (B)IEf du, E E R.

By the definition of the Bartle integral, lim T (E) = T (E)
for each E E 51. The measures rn are integrals of simple

functions so they are a-additive and ?n << 9 for each n E w

consequently, by the vector form of the Vitali-Hahn-Saks theorem

(see [l2,III.7.2]),we have ?n << 0 uniformly for n E w.

Because |x*0 y*Tn IInl for- x* E X1* and y* E Y1*

we have IxXE*T yl << p uniformly for ne E x* E Xl*an

y* E Y *. -NOte that |x*0 y*Tn (E) = IE x*fn d y*@I for E E 5?.

Taking the supremum over X1* x Yl*, we have N*(f C d)) <

uniformly in ne E But fn -t f U-a.e., and N*(fn ())

uniformly implies by the Vitali Theorem 3.1, that f c PX(i

an~d lim N*(f -f) = 0. The validity of (2) follows because

(fn) determines the weak and Bartle integrals of f. O

In [12], Dunford and Schwartz developed a theory of

integration of scalar valued functions with respect to a

vector valued measure. This theory is that of Bartle's for

X = 4, in this case we shall say a scalar function is Bartle-

Dunford-Schwartz integrable, or B-D-S integrable. We have

the following corollary to Theorem 6.1.


6.2 Corollary. A scalar valued, p-mearurable function f is

B-D-S integrable if and only if f E P(p).


6.3 Remark. By Proposition 2.4, P(u) = B(u); we shall denote

this space by D(p). From Corollary 6.2, D(y) is the Banach

space of all scalar functions which are B-D-S integrable with

respect to p.








7. A Pettis-type Integral.

In this section, we will introduce an integral which

is more general than the weak integral in the sense that

more functions are integrable. The definition of this integral

is reminiscent of B. J. Pettis' integral introduced in [17],

and it will be shown that for strongly measurable functions,

PX(p) is exactly the class of all functions integrable in
the new sense. Again, p:n -+ Y is o-additive.

A function f:S -+ X is Xd Y-integrable, or Pettis-integrable,

on a set E E n, if there exists an element BE e X0 Y such that

for all x* e X* and y* E Y* we have x*OE ( E) = Ex*fd(y*p).

We shall denote the element BE by (P)IEf dy. A function of

this type is X0 Y-integrable if it is X@,Y-integrable over every

set E E R. Any XO Y-integrable function is weakly p-measurable.

Because X* 0 Y* is a subspace of (Xi Y)* which is norming

for X0 Y, the Pettis integral as defined above is single
valued, linear, and finitely additive. Note that if Y is the

scalar field, then this integral is Pettis' "weak" integral.


7.1 Theorem. If f is XO Y-integrable, then the range of the
indefinite integral of f is bounded.


Proof. Put T(E) = (P)/E f dy and consider the family

K = {x*B y*T:x~EX and y*EYI '.
We have K c ca(S,R).

For Ee 0 r x*O y*T(E)I C |X* y*|* T 05)| = T 05) .
This shows the set K to be pointwise bounded; by a result

of Nikodym ([12], IV.9.8), the set K is uniformly bounded.








that is, there exists a number M such that Xx*0 y*T(E) < M

for all x* E Xl* and y* E Y *. Taking the supremum over

Xl x 1* we get /T(E)IE M for all Ee E that is, T is
bounded. O


7.2 Corollary. If f is an X 0E Y-integrable, then N*(f) < +m.

Thus if f is p-measurable, f E WX M '


Proof. By Theorem 7.1, sun|(P)J fdd sMc
for some number M.

Consequently, for x* E X and y* E Y1* we have

jS x'fld y*pl _c 4 su jlEX~dY

=4 su lx*0 y*(P)/Ef dpl

S4 ~Eu (P) Ef dul
5 4M < +m.

If f is u-measurable and N*(f) < +m, then by definition,

f E Wx(u). O

We now prove the countable additivity of the indefinite

integral of a Pettis-integrable function. Pettis proved that
the weak integral of [17] was countably additive by showing

weak countable additivity implies countable additivily; in

our context, it is not clear that we have weak countable

additivity.


7.3 Remark. The dual of X 8E Y is J(X,Y), the space of integral
forms on X x Y. For any functional z E (XO Y)*, there exists
closed and bounded subsets P c X* and Q c Y* and a positive







Radon measure v on the w*-compact set P x Q with total variation

5 1, such that for all 6 E X$ Y

z{6) = I xp(x*,y*) dv(x*,y*).

Here we consider 6 as a bounded bilinear map on X* x Y* restricted

to P x r ; the integral is the ordinary Lebesque integral.


7.4 Theorem. Let f be X0 Y-integrable. Then the indefinite

integral of f is countably additive.


Proof. Suppose (E ) n is a disjoint family and EO= i 1E .

Write T(A) = (P)I f dp for As G W att hw.T(
A 1= 1i
converges unconditionally in X 0E Y and converges to T(Eo "
Because of the Pettis lemma ([12]IV.10.1), it suffices

to prove that I is weakly countably additive, that is,

= i3 for each z E (X@ Y)* with |zl 51.
Let z be fixed.

Indeed, z is an integral form on X x Y, regarding Remark

7.3, there exists closed and bounded subsets P c X* and Q c Y*,

and a Radon measure v on P x Q such that

z(6) = I xp(x*,y*) dv(x*,y*), for 6 E X8 Y.

Define T (x*,y*) = IE.x*f d(y*y) for i = o,1,2,....

Note that T (x*,y*) = .2 T.(x*,y*) for x* E P and y* E Q,

and that T. E C(PxQ) for i = 0,1,2,....
Write K = sup { x*(*|y*I : x*EP and y"EQ} < +m, and

since T is bounded by Theorem 7.1,

M = supT(A) < +m.

Finally, define Fn = in E., then for x* i P and y* EQa we have








l.IT.~(x*,y*) I = ljIf xf d(y'p)1

= fF x*f d(y*M)|

= )

< Ix*I y* *((P)IF f dyle

5 KM < +m.
n m
Thus the sequence {.E T.}n~ is pointwise dominated by
K'M on P x Q. By the Lebesgue Dominated Convergence Theorem

Ipx To(x*,y*)dv(x*,y*) = il x C*y*d1*y)
But this says

dV(x*,y*)


= IpxQ (~E f do, (x*,y*)>dV(x*,y*)

= IJ I~ x*f d(y*M) dv~x*,y*)

= IpxQ o(x*,y*) dv(x*,y*)

= f QT.(x*,y*) dv(x*,y*)

= .f fxQIi*f d(yip) dV(x*,y*)

= f <(P)/i f~ dy(x*,y*)>dv(x*,y*)

=i l.

= l, and the theorem is


That is,

proved. O


7.5 Theorem. If f is Xd Y-integrable, then the indefinite
Pettis integral of f is absolutely continuous with respect
to U.








Proof. Write T(A) = (P)j f dp; is an XB Y-valued measure
on R by Theorem 7.4; consider K c ca(S,Q) defined by

K = {x*@Ey*T:x*EX and y*EY *.
It is clear that K << p, that is, x*0 y*Tr << for each

x*0 y*T E K. Furthermore, the family K is uniformly strongly
additive; this means that for any disjoint sequence (E ) R

we have 1 m x*0 y*T{E ) =- 0 uniformly for x*BEY*T E K. To
see this, by Theorem 7.4, the series T'(E.) converges
unconditionally in X @E Y, from the Orlicz-Pettis lemma [17],

we have lIm T(E )/ E 0 Since Ix"Bqy*T(E ) 5 (T(E )( for

x* E X and y* E Y *, we must have limlx*0 y*r(Ei )
uniformly for x* E X1* and y* E Y1

In (2], Brooks has shown that K < pi and K uniformly
strongly additive together imply K << uniformly, that is,

x*@ y*T << 0 uniformly for x* E X1* and y* E Y1 '
Given E > 0, there exists 6 > 0 such that when Ee E

and liu ((E) < 6, then Ix*B y*T(E)) < E. Taking the supremum
over Xl* x Yl*, we have T~(E)I < E whenever li l(E) < 6,
that is, r << p. O


7.6 Corollary. If f is X@ EY-integrable, then N*(f6( ) << U.

Proof. Let T(A) = (P)j f dy. By Theorem 7.4, T << v which

implies x*0 y*T << U uniformly for x* e XI* and y* e YI*. But
this means for every E > 0, there exists a 6 > 0 such that

whenever E t and IIu||CE) < 6, then

)x*B y*T|(E) = lE x*f djy*pl < E







uniformly for x* E Xl* and y* E Y1*. Taking supremum over

XI* x Y we get N*(fSE) s E whenever |19110E) < 6. O

We now prove that for y-measurable functions, to be in

PX(u) and to be XB Y-integrable are equivalent notions and
the weak and Pettis integrals coincide.


7.7 Theorem. A Ii-measurable function f is X~i Y-integrable if

and only if f EP (#). In this case,

IE ~EdY = (P)IEf du for every E E 9.

Proof. Suppose f is u-measurable, by Theorem 7.1, N*(f) < +m,

hence f E WX(11); from Corollary 7.6, N*(f5( ) < is sufficient, by Proposition 2.7, to imply that i s PX '~

Conversely, if f E PX(u), then by Proposition 5.2, we
have (x*0 y*)JE ~Ed = IExff d(y~y) for each x* E Xl" '" E
and E E n. Thus it is true that for each Ee E there exists

a vector aE E XO Y, namely BE = E Edy, such that x*B y*(oE)

IEx*fd(y*Y); this is the definition of the Pettis integral.
Therefore, f is XB Y-integrable and (P)IEf du = IE ~Edy. O

7.8 Remark. Using a Pettis definition, we can define an integral

for functions in WX(u) in such a way that this integral, for

functions in PX(u), is the weak integral defined in section

5. For f E WX(u), define a linear map on (XBEY)* as follows:

Z -+ lAxB Ex*f d(y*M) dv(x*,y*),

where z E (X~EY)* and jAxB(*)dv(x*,y*) is the representation
for z as an integral form -- see Remark 7.3. We designate







this linear map by (P)IEf dy and is defined on (X@/Y)

Since N*(f) < +o=, it is easy to see that (P)IEf du is a
continuous linear functional on (XqEY)*, hence (P)JEfdpE(XO Y)**.
Indeed, we can choose, for each lzl r 1, positive Radon measures

vz on X x Y such that z represented as an integral form is

0 -+ IX1*xY1*8 (x*,y*)dvz (x*,y*),6 s XBEY

For lz| < 1, we have


s X1'xyl* EIx*f Idly+ldvz Ix*,y*)
& N*(f5E) < +m.

Thus (~P)JEfdp| < N*(f5E), that is, (P)IEfdy E (X8 Y)**.
Theorem 7.7 states that for f E PX(p) we have jE ~Edy =

(P)JEfdy E Xq~Y.

7.9 Remarks. We have defined a general integral which takes
its values in the inductive tensor product of two Banach

spaces. In the most general case, the integral is that of
Bartle's bilinear integral with the improvement that convergence
of integrals is characterized by norm convergence. Bartle's

integration theory yielded a Vitali convergence theorem; however,
in our context, Lebesgue's Dominated convergence theorem is
obtained as well.

A generalization of the Pettis integral is given. If
the range space of the measure is the scalar field, our

definition is exactly the Pettis definition and N*(-) = ((*))1'
It was shown that the weak and Pettis integral are equivalent

for strongly measurable functions.







If the range space of the functions is the scalar field,

then the definition of the weak integral is equivalent to the

Bartle-Dunford-Schwartz integral.

Finally, if X = Y = 4, then the weak integral is the

Lebesque integral for scalar functions.


8. The Strong Integral.

The strong integral was defined for the functions in

BX(y), and had its values in X 6E Y. We now show that the
strong integral is included in the integral of Brooks and

Dinculeanu introduced in [5}. They defined a Lebesque

space of integrable functions with respect to an operator

valued measure. We shall outline very briefly the basic

development of the theory inl [5].

Let E and F be Banach spaces and consider an operator

valued measure m:n -t L(E,F), where L(EF) is the space of

bounded linear operators from E into F. For each z E F*,

define mz:SZ E* by mz(A)x = ; mz is countably
additive and of finite total variation. For an m-measurable

function f:S --+ E, define Nl zs p,E (l~z. etn

FE(N ) be the collection of all m-measurable functions f with

NI(f) < +", it follows that FE(N ) is a Banach space which
contains SE(0). Finally, LE(N1) is the closure in FE N1) of

SE ')
An integral can be defined for functions f E FE(N1)

as follows: Igfdm is the vector in F** defined by

= Igfdmz, where z E F*.
If f E LE(N1), then IS f dm E F; indeed, the integral of







simple functions is F-val~ued, the mapping f --* ISf dm from

LE(N1) to F** is continuous, and the simple functions SE(R
are dense in LE(N ).
The following theorem shows that the strong integral

is included in the integral of Brooks and Dinculeanu.


8.1 Theorem. Let p:R -+L Y and define m:R -+ LI

m(A)x = x~p (A). Then

(1) f is ct-measurable if and only if f is

(2) N(f) = NI(f) for all p-measurable f:S

(3) LX N 1) = BX()

(4) /Sf dm = ISf E dp, f E BX(~

The proof of this theorem is contained in

sequence of lemmas.


(X,XG Y) by


m-measurable;







the following


8.2 Lemma. f is p-measurable if and only if f is m-measurable.


Proof. It suffices to proved and m have the same null sets.

For E E R, each of the following are pairwise equivalent:

E is p-null; sup p(EnA) O;

sup sup xpEAj=0


supl(n) sq 0|(n)X =0
AEU X


E is m-null. C]


8.3 Lemm~a. Let r = {%*0 y*:x*EX *,y*EY *}. Then

N (f) = su JS f djm |, feS(0).







Proof. Write f(s) = ~i E(s) where (a ) 4 and (E ) R

are disjoint. Put J(f) =s Slfid mm then J(f) < NlQf

since r c (X0 Y)1 '

Let E > 0 be given, there exists z E (XO Y)1* such that

Nl~f x Eftd/mzI = C+ i~l \) mz (E ).
There exists an integer p e and set Ai E 1 < j <

and I < j s n, such that

Im (E ) E Z Im (A j)/.


Substituting this into the above inequality:
n P
1(jf) 5 C Ei1 aii .T at j 11 mZ(Ai )

=2E+E.E a. Im (A..)l.

Choose for each pair (i,j), R.. E X with |x.. I '1 and

Im (A..) P&ii + Im (A j)X..).

Then
C -)
NI(f) 2E + Z .E .a {Plli +jm (A )'X }

= 3E + Z .2 .|3. <7. Op(A..), z> .

Choose complex numbers 6.. with (6. I 1 and
1] ij
= 6..
1] ij 13 13 17


Substituting once again,

N1(f) i j+E a(4 x0( ,z







= 3E + ~~~ 3E 1 Iijaliij(A~l] i


r is norming for X 0E Y, there exists x*B y* E r

if W = x*qEy*, then

S3E + E +|~~lE l. a.|6..
S4E + C. a.|- m (A )7..j
< 4E + Ci~ljm lij 1

= 4E + ci ail .m (A ))


= 4E + Is f dim |

5 4E + J(f).

il(f) 5 4E + J(f) which is sufficient to conclude
i), and in turn; NI(f) = J(f). O


Since
such that i

N (f)


Thus N\

N (f) & J(f


8.4 Lemma. For all f:S -+ X p-measurable we have

N1(f) = J(f) where J~f) =sup I)fldlm ).

Proof. Case I: N (f) = +m

Let n E be arbitrary, there exists z E (X8 Y)*1 such
that n < IS f d mzI. We can choose a simple function g e S(R)
with O 6 g(s) < jf(s) for all s ES and

n < ISg dimz "
By Lemma 8.3, N1(g) = J(g), since gs J(g) r J(f); finally,

n < ISg d mZI '1(9 1 '~) This implies that NI(f) = +" so that for this case Nl~f ~)







Case II: N1(f) +m.
Letting E > 0,w a hoeze(q)* such that

Nlf ~ Slf d mzI

Choose g > 0 simple such that gs < fl and

JSlf dlmz/ < Sgf1 d mz '

Thus,

Nl"f) < E + /Sgd mzI N1(
= E + J(g)

< E + J(f).

We deduce NI(f) s E+J(f), that N1(f) J(f) and Nl~f
J(f). O


8.5 Lemma. Given w = x*0EY y rr Im ((E) Iy*/ (E) for
every E E n. Conversely, for each y* E Y *, there exists

x* E Xl* such that |y*p (E)< m l(E), where w = x*BEy* and



Proof. Given w = x*BEy* e T, E ESZ and E > 0, we can choose

(A ) C a partition of E and vectors (x ) C X with Ixi 1
so that

Im (E) < E+Z. m (A.)x.l
= C+E.X*(xi) y*M (A.)

< EC+E y*p (A ) s E+ y*ul3).
In this way we obtain the inequality |mw (E) & Iy*U (E).

Let (A i) E G be an arbitrary partition of Ee E Choose
x E X with Ix| < 1 and x* E XI* such that x*(x) = 1. Then if

we put w = x*BEy*, we have







iy*plA )l = Ci x*(xi)|* y*p(A ) = m l~(A )xj




From this we obtain ly*pf(E) < Im (E). O

8.6 Lemma. N(F) = N (f) for all p-measurable f:S -+ X.


Proof.

N(f) =


By Lemma 8.4, it suffices to show N(f) = J(f), where

5up le Sf d y*9 and J(f) = suI IS fldim .


The equality follows from 8.5. Since ly*p (-) < Imaj() for
some as 7 given y* E Y ehv

ISf dly*pl /Slfdlmwj
and therefore N(f) s J~f).

Also, given me 1 r mw (') Iy*p (*) for some y* E Y *,
which implies in much the same way as above that J(f) I N(f).

Putting these two together: N(f) = J~f). O


8.7 Lemmna. L (N ) = B (p) and Iff dm = I fO dy, f E B 9).

Proof. By Lemma 8.6, N(f) = N (f) for all p-measurable f,

and since the spaces LX(N1) and BX(u) are the closure of S (R)
with respect to norms Nl and N, respectively, we must have

LX(N ) = BX(U
To prove the second assertion, it suffices to prove it

for simple functions; this is because SX(R) is dense in EX(~

( = LX(N )) and the linear maps

f -+ jS edy and f -t /Sf dm
are continuous from B (u) into X &E Y. To show ISf dm = /S 8Edy
for f E SX ~)' we need only show Iff amz CS ~Edy,z>, where








z c (Xe Y) *. Writing f(s) = Ejx.G ((s) where x.E X and

(E ) c R are disjoint, we have
IOf dm = E m (E.)X. = E 0 Z 1 Z 1 1 1

Cix Op (E )'Z> = . O

This ends the proof of Theorem 8.1. O


8.8 Remark. For the elementary cases, the strong integral

reduces to some well-known integrals. For X = 4, we have

the Bartle-Dunford-Schwartz integral as proven in section 6.

If Y = 4, that is, if p is a scalar measure, the strong

integral takes its values in X, the range space of the functions

integrated; in this case, we clearly have the Bochner integral

with N the Bochner norm.


9. The Spagces PX (lJ) and BX ('

In preparation to proving some theorems concerning the

topological properties of the weak and strong integrals, as

well as the classification of certain natural linear operators,

it is necessary to make a few remarks on essentially bounded

measurable functions. As throughout this chapter iy:R -t Y

is a vector measure.

Let f:S -+ X be weakly y-measurable. Define the following

functions:

(1) N (f) = p-ess sup f(s) =inf sup |f(s) ,
a seS H saS-H

where the infimumr is taken over all p-null sets H 5 S.

(2) N* (f) = sup (u-ess sup x~f(sl) )
m x*EX1* sSE







It is clear that N* (f) = sup N (X*f) and that



The magnitude of Nk,(f) and Nm(f) depends on the collec-

tion of U-null sets. Sometimes this dependence will be

denoted by Nim(f;lp) and N,(f;p). For example, if v:R -+ R+

is a positive measure such that v << 9, then the collection

of all v-null sets contains the U-null sets; consequently,

any y-measurable function is also v-measurable. It is always

true, therefore, that N* (f;V) & N*m[f;u) and No[;)
for any y-measurable function f, because the infim~um in N*m~f~

and N,(f;V) is taken a larger collection of null sets.

A weakly p-measurable function f is weakly U-essentially

bounded if N* (f) < +m, and is (strongly) p-essentially

bounded if N (f) < +m; obviously, if it is >-essentially

bounded, then it is weakly iJ-essentially bounded.

Define the following spaces:

(1) PX (p) is the space of all weakly p-measurable functions
f:S -+, X which are weakly y-essentially bounded.

(2) BXm(p) is the space of all p-measurable, y-essentially
bounded functions f:S -+ X.

We have BXm ( ~ X (y), and the inclusion is in general

strict. If X is separable, then BXm ( ~ X (p) (see Pettis

[17]). The space B m(I)) with the norm N (-) is a Banach space
if we identify functions equal Fu-a.e., P ~(U) is a normed

linear space with norm N",('). For the case X = 4, the spaces

BXm( ~ =Xm(p) and we shall denote this space by Lm()
the classic Lebesgue space of p-essentially bounded scalar







functions with norm

(Ila l m = p esse up |4 I(s ) ,l 4 E L

9.1 Proposition. If f E BX (Fc), then f E BX(p) and

(#} N(fcE) Nm (f) pII j(E)
for E E n.


Proof. ~Since f E B m(p), f is p-measurable. For each y* E Y *,
we have y*u <<9 and consequently

Na(f;y*p) 5 Nm(f;p) < +ao.
Thus f t L (y*p) c L (y*p) and

/E fldly*pis Ng(f,y*y) ly*Ul (E)



Taking the supremum over Xl* of the left hand side,

N(ftE) s N (f;y)* p (IE) < +m
This proves part of the assertion and shows f E FX(u) since

N(f) < +m. Also, N(f (~) << p by Proposition 2.7 and

Remark 2.8, f e BX(u). O

9.2 Proposition. If fe B X (9) and $ E L (u), then f a BX 1

and N(f SE) 5 Nmof /lm* III (E) for E E s?.

Proof. Obviously Noo~~ < m*l~l
The result then follows from Proposition 9.1:



9.3 Proposition. If fOE PX(p) and 4 E Le~(p), then f a PX(~
and


N*(f )< I/ mN*(f).







Proof. Let x* E X1* and Y* e Y1* be arbitrary, then x~f E

L1(y"p) by Proposition 5.2 and $ E L (y"p). From H61lder's

inequality, we have for E E n

/El~x*fldly*1pl IIII 4 E x~f d y*u i

aI~PI N*(fsE)

Taking the supremum over Xl* x Y1* of the left-hand side:

N* (f kE) < IIPI N* (f5E) #

f E PX(11) implies N*(f ~~) << p by Proposition 2.7; this
fact and (#) combine together to imply that N*(f i~) << Il,

but then fQ E PX(y) by Propositon 2.7 again. The second
assertion is (#) for E = S. O

10. Compact Operators.

Let X and Y be Banach spaces and V a bounded linear map

from X to Y. V is said to be a compact operator if V maps

bounded sets in X onto relatively compact subsets of Y.

Compact operators (and weakly compact operators) have been

studied by many people in connection with integral representa-

tions of operators on spaces of continuous functions; see

Dunford and Schwartz [12] for a discussion of the known results

in compact operators.

In this section, we classify a certain natural linear

operation as being compact provided the measure p:R -+ Y has
a relatively norm compact range.


10.1 Lemma. Let f E S (n). Define a map V:LO~U) -+ X0 Y by







V is linear and continuous. Furthermore, V is a compact

operation if p has a relatively norm compact range.


Proof. Let f a SX(n) and 4 E L (y). Then by Proposition

9.2, fQ E BX (p) and therefore f$ E BX(Fu) by Proposition 9.1.
The map V(O) = j@f$OcdV is then well-defined since the integral
exists, and maps Lm(y) into X 0 Y. V is clearly linear; it
is also bounded:

IV = ulp_ J~llSf$0Ed d



= N,(f)- pl (S)S < +m.
Suppose v has relatively norm compact range; by Theorem

II.3.2, y e ca(R)B Y; consequently, there exists a sequence

(1-k) 5 ca(R)0Y of step measures such that [1u-pk U~S) 0.
We may assume, according to Theorem II.3.2, that Uk~~

for each ka E ; in fact, we may take nk i=1iiwhere
k k k
yi E Y, v. ca(n) and vi < p.
For each k E W, define Vk( ~ IS Edyk. Then

Vk:L (p) -+ X8SY, this is because fO E BX (p) and pk
implies fO E BX (k) BX k) which makes Vk well-defined.

Vk is bounded:

IVk ( INf;k] Il;klm I* pl (S)


We have used here the fact that uk << p implies Nmo; k k

Noo(f;u) and llQ;lull i ll;ll (see the remarks in section
9). We have thus shown that IVI N,(f;y)- jp U(S) < + .








Assert that the operators Vk are compact for each k E w.

Proof of assertion. Let k em b e fixed and write uk = .E yiv.
where y. E Y and miE ca(n) with v. << U. We again deduce that
& L (V ), fe B X (V ) and so ft E BX (i) for i = 1,2,...,n.
Since vi << p we have L (U) E L (V ) for each i. Define
V,:Lm(V.) -+ X by V, (3) = j f$i dv.. This is well-defined
since fil E BX Ui) for each J E L (vi) by Proposition 9.2.
Since f is a simple function and v. is a scalar measure, we
can apply a lemma of Pettis [17], Lemma 6.11, to conclude

Vk is a compact operator on Lm(v ). Because v << 9, we
always have 11;Vil 5 m;ul for all Ji E Llmvi), thus any
set in Lm(yl) which is bounded in the ||*FM m,-nnorm is bounded
in the |*;q||m-O-nom and so Vk will map this set into a
relatively compact subset of X -- this means then that Vk
restricted to Le~(v) is a compact operator.

It is easy to see that Vk(9 i IVk )~i, for O E L (II);
since each Vk is compact, and Vk is a finite sum of compact

operators, Vk is compact too. This proves the assertion.

Now for E L (F) with 'I1, =I

(V-Vk) IS ~Ed( p-pk lE





= N ~)l~- ki (S) .
Thus IV-Vk/ s N,() ~ Fk (S). But since ll-u]k (S)-+ 0
we also have IV-Vkl -+ ; hence Vk -+ V in the uniform operator







topology of L(Le(p),XO Y). By Lemma VI.5.3 of [12], V is
a compact operator since it i~s the limit in the uniform

operator topology of compact operators. O?

102 Theorem. Suppose u:R -+ Y has relatively norm compact

range. Then for each f e PX(v), the map

V{4) = IS ~Edy,
is a compact operator from L (y) into X 8E Y.

Proof. By Proposition 9.3, fO E P (9) so V is well-defined.

Since f E PX(y), there exists a sequence (fn) S (R) which

determines f in PX(p), that is, fn -+ f in p-measure and
N* (f-fn -+t 0.

Define V (4) = IS f 40 dp. By Lemm~a 10. 2, V :LI(1) -
XB Y is a compact operator. Let 4 E Lm(y) with I1)(. = 1.

((V-vn) ~ S Il(- n Edpd ~


I(IN*({f-fn)

N*(ff-f )


Thus |V-VnI n "ff); since N*(f-fn) -* 0 we have
V -+ V in the uniform operator topology so that V is compact

also. O

U). 3 Corollary. If the range of y is relatively norm compact,

then the indefinite integral of any function in PX(u) has
a relatively norm compact range too.


Proof. Let B = {CE:Est). Then B c L (y) is bounded. For

any f EP (9), the map V(4) = JS ~Edy is a compact operator,







therefore, sends the set B onto a relatively norm compact

subset of X 0E Y. But V(B) = {V(SE):EER} = {If~E dp:EcG},

that is,V(B) is the range of the indefinite integral IffedFy.

V(B) is relatively norm compact in X 8E Y. O

1Q.4 Corollary. The indefinite integral of functions in BX(I
has a relatively norm compact range if u does.


Proof. Recall B (u) C P (9) and apply Corollary 10.3. O

105 Corollay. Let 9 have relative norm compact range. Then

PX(p) ca(n)B XB Y
isometrically.


Proof. The space ca(a)B XB Y is the space of all XB Y-valued
measures on n with relatively norm compact range by Theoren
II.3.3.

Define T:PX(pI) -* ca(n)b XB Y by

T(f) = / fO dpl, FEPX(I
By Theorem 10.2, the indefinite integral T(f) has relatively

norm compact range so that T(f) E ca(R)B XB Y. T is clearly
linear, it suffices to show N*(f) = IIT(f) /(S) for T to be

an isometry since the norm on ca(R)B X EY is the semivariation
norm by Theorem II.3.3. But this is obvious, from Proposition
I.1.1 we have

T~f (S =(x*,ys pX1 xYli Ix*0 y*T(f) (S)
So that

lTcf) l (S) = sup xf|dyp
(x*,y*)EX1sxylisxl YV
IT( f) (lS) = N*(f ). O














CHAPTER IV
THE FUBINI THEOREM

1. Preliminaries.

Throughout this chapter, (S,R) and (T,A) are measurable

spaces; X and Y are Banach spaces; p:R --+ X and v:A -+ Y are

vector measures. The symbol RBA denotes the algebra of

rectangles of R and A while R8 A is the o-algebra generated

by GAe.

We shall consider three Fubini type theorems for integrals

of scalar functions with respect to the inductive product

measure yq^v. In section 2, we prove the multiplicative

property of product integration; it is obtained in the most

general form possible. The classic Fubini theorem is proven

in section 3 with only minimal restrictions placed in the

hypothesis. The existing vector valued Fubini theorems place

severe restrictions on the measures by requiring both measures

to have finite total variation (see [10] and [14]); we require

that only one of the measures, M or v, have the Beppo Levi

Property. Finally, in section 4, we derive a Fubini theorem

for continuous function.

We use the integration theory developed in Chapter 2

throughout this chapter. Recall that for a vector valued

measure X:n -+ X, the spaces P (A) and B (X) coincide (Proposition

III.2.4), and they define the Banach space of all scalar functions

integrable in the Dunford-Schwartz sense with respect to X







(Corollary III. 6.2). As in Remark IrT.6.3, we use the notation

D(S,n,X;X) or simply D(A) for this space. Consequently,

we shall write D(u) for D(S,O,U;X), D(v) for D(T,A,V;Y), and

D(p@ v) for D(SxT,00 A,90 v;X@ Y).

In this chapter, the variables of integration will some-

times be written in for clarity; for example:

(1) ISf du will be written /Sf(s) du(s), f E D(y);

(2) ISxyh d(uq~v) will be written IsxTh(s,t)d(pqv))(s,t),
for h E D(p@ 9).

2. The Product Theorem.

In this section, all functions are measurable; I and g,

with or without subscripts, will always denote functions defined

on S and T, respectively.


2.1 Proposition. Suppose f and g are scalar simple functions.

The function (fg)(s,t) = f(s)g(t) is a scalar simple function
on S xT and for each Ec E and F EA

IEx~fg d[8e v) = JEf duIfpg dv.

n m
Proof. Suppose f(s) = Cla GE.(s) and g(t) = b (F

n m
Egxpfg d(p8a) = ExF i 1 j 1 ib (EixPjd(pBEv
n m n m
i 1 j la b (ueav) (EE xPF)) = i ~~aib M(EE )Bv(FF )
n m
(ila U(EE ) 8 ( b v(FF ))


= IEf du 0 IFg dv. O







2.2 Lemma. Let f and g be scalar simple functions,

Define T(E) = IEf dp, for E E n,

and p(F) = jFg dv, for Fe A.~

Then rO p(G) = /fG d"Ev) for each G E n00 '

Proof. Both I and p are o-additive taking their values in

X and Y, respectively. The inductive product measure always

exists and agrees with the indefinite integral jfg d(p~e ) on
the algebra of rectangles:

TO p(ExF) = r(E)Bp(E)

= J~f dv8/Fp dv

= /ExFfg d(8e v),

by Proposition 2.1.

TO p agrees with ffg d(p~E ) on 00A, so they must agree
on n8 A because both measures have unique extensions from the

algebra to the o-algebra. O

We now prove the main theorem of this section.


2.3 Theorem. Suppose f e D(Fl) and g e D(v). Then fg E D(p@ V)

and IEx~fg d(Be 0) = JEf dv B IFg dv,
for each Ee E and Fe A.n

Proof. Let X and Q be control measures for M and v, respectively.

Let (fn) and (g ) be two sequences of simple functions which
determines the integrals of f and g, respectively.

Define r (E) = JE ndv and p (F) = /F ndy for n =- 1,2,3,...,
and Ee E Fe A By the Vitali-Hahn-Saks Theorem, In~

pn c<< uniformly in n.







Now by Lemma I.2.3, we have T 8 p << Xx# uniformly

on R 0 A. Write yn = T 0 p .

Note that Yn(G) = /f~G nd(X0 v), for Ge 9 A8~, by Lermma
2.2.

Assert that (Yn) converges on COA. To see this, let
A = .uEixFi be a disjoint union, E. E SZ, F. E A.

For n,m r w, we have
IY,(A)-Y (A)( = E InEnEx m E m(E xF )l


il n(E )@pn(F ) T (E )Bp (F )



+ i 1 nm(E ~'PF) ImE)*pn Fi)


The sequences {Tn(E)} and {pn(F)} are Cauchy in X and Y,
respectively since

lim In(E) = Ef dp and lim pn(F) = jFf dv.

Therefore, limln(E ) Im(E )l = 0, 1 s i < k,

and li~nF)-p( )=0, 1 < i k.

From this and the above inequalities we have that

~Slimy (A) Y (A)] = 0;

that is, {yn(A)} is Cauchy in X6 Y and therefore converges in
X4 Y.

The measures (Yn) converging on RSA and yn << Xx 4
uniformly on R 0 A are sufficient to imply that the sequence

(Yn) converges on R00A ([4], Corollary 4).








Thus, lim y (G) = lim Iff g d(Be v) exists for each
Ge E OCA, and fng n fg pointwise pa v-a.e.; this implies

by Theorem III.6.1, fg E D(p0 v) and the sequence (ffn)! of
simple functions determines the integral of fg.
Consequently, since the theorem is true for simple functions,
we have,

jgxpfg d(Oe V) = 11m JExF fn n d(p~eV

= lim JE fn du 8 /F gn du


= JE f du 8 I, g du.O

2.4 Remark. The crucial point in the proof of Theorem 2.3

was invoking Lemma I.2.3 to conclude r 8 p << Xx $ uniformly

on n 8 A; this is because for the inductive product In~I~

II I(-) so that condition (3) of that lemnma is fulfilled

(Tn X uniformly if and only if |lrn [ << A uniformly). For
the projective product measure, ye v, the proof of Theorem 2.3
will not work since n << A uniformly and p << 4 uniformly

need not imply T 0 P << Ax4 uniformly on RBA; consequently,
further hypothesis may be required for a result analogous to

Theorem 2.3 for ye v. If y and v both have finite variation

the Theorem 2.3 is true for u0 v. O

2.5 Corollary. Let i s D(p) and g E D(v}. Then for each
Ee E and F E A,

(1) the function s -t /Ff(s)g(t)dv(t) is a member of B (pl);

(2) the function t -i /Ef(s)g(t)du(s) is a member of BXv)
(3) IExFf(s)g(t)d(p~eV )(s,t) = JE IFf(s)g(t)dv(t)@cdp(s)

=, IF f(s)g(t)duds)sdv(t).







Proof. Fix E E and F EA and write

x = /E f du and y = I, g dv

wh re x E X and y E Y.

Obviously, y-f(*) E B (p) and x~g{*) E BX(v); indeed,

if (f C S(R) determines f in D(u), that is, N(f-fn) -+ 0,
then the sequence (yf ) S (R) determines y~f:

N(yin-Yf) = xsuR, l*S ~n-yf d x*UI

= Iylxsu 1* S n-f dlxx*M

= Iyi'N~fn-)

Thus lim N(yf -yf) = |y lim N(f -f) = 0, and yf E B (u).
Similarly, xg E BX~v

Because /E n BEdy = (IfE du)0y, we must have

(E eBdy = (/Ef du)0y too.

Finally,

Ipf(s)g(t)dv(t) = f(s)/Fg(t)dv = yf(s)
so that (1) is just the function s -+ yf(s) which we have

shown to be in B (p), and using Theorem 2.3 we have

Ixf~s)g(t)d(peEV)(S~t) = Egf(s)du(s) 0 Fpg(t)dv(t)

= jgf(s)du(s) 8 y

= jEyf(s) 8E dp(s)

= JE Fg(t)dv(t)f(s) 0E dils)

= lE Ff(s)g(t)dv(t) 0E dU(s).
Condition (2) and the second equality of condition (3)
are proved similarly. O







3. The Classic Fubini Theorem.

In this section, we shall use the following notation

for the norms of the spaces D(u0 v), D(u), and BX(v) which
was introduced in Chapter III.

(1) For h E D(p@ v),
Nth) =Px (*sul*xyl SxTh(slt)|Id x*,xy~vl(s,t),

(see Remark 3.4 infra);

(2) for f E D(N)

N1(f) =sup fs)dx s)

(3) for g E BX V'

N2(g y 1~LL j |g(t) d y*v( (t).

Let As GO AB. For s ES and t E T, the s-section and
the t-section of A are, respectively,

As LEtT:(S,t)eA},

and A" = (seS:(s,t)eA}.

From the classical theory of Lebesgue integration, we know

that As E A and At E R.

3.1 Theorem. Let Ae C A8_ then

(1) the map t -* 1 (At) from T into X is in BX()

(2) the map s -+ V(As) from S into Y is in B (u);

(3) ~e V(A) = j (A )@ dV(t) =J(s0d~)

Proof. Let Hi be the class of all sets Ae G AB~ for which

the conclusions (1), (2), and (3) hold. We shall show that

H is a monotone class containing the algebra of rectangles.








H contains the class of rectangles. If A = .uEx
1= 1 1~i
where (E ) are disjoint, then
v(As) = vF)E()


which is a Y-valued simple function, clearly in By (u).

Also,

jlSy(As d(s ) I~lS (i~i E.Ed(s)0us

k k
= .2U{E.)@V(F.) = .E pe v(EixF.)
1= 1 1 l= E 1


This proves (2) and half of (3).

We can write A = .0 E xF~ where now the sets (F ) are
pairwise disjoint and undergo a similar analysis to obtain

(1) and (3).

Thus if Ae E RA, then A satisfies (1), (2) and (3) and

therefore, A E H. Finally we conclude R0A c H.

We now demonstrate that H is a monotone class.

Suppose (An) C H is a monot-one sequence and A = 11m An,
the pev(A = im O vA )andp(A) = lim U(A ) and
th n E~VA i Bv(n) an n(t
V(As) = lim V(A ) for each s ES and t E T.
n n
The functions u(A ) t BX(V) advA)eB()sneA
and (1) and (2) hold. Now because vector measures are bounded

we see that there exists constants P and Q such that Ipi(An) 5 P

and /V(As)I < Q for all ne 0 Is E S, and t E T. By the

Bounded Convergence Theorem (Corollary III.3.3), p(A ) BX "'

v(A) B(9, (A) -+yA) in BX(V) and v(An -+(A)i




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