~ (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