Topics in Banach space integration

Series in Real Analysis - Volume 10

TOPICS IN BANACH SPACE INTEGRATION

SERIES IN REAL ANALYSIS VOl. 1:

Lectures on the Theory of Integration R Henstock

Vol. 2:

Lanzhou Lectures on Henstock Integration Lee Peng Yee

VOl. 3:

The Theory of the Denjoy Integral & Some Applications V G Celidze & A G Dzvarseisvili translated by P S Bullen

VOl. 4:

Linear Functional Analysis w Orlicz

VOl. 5:

Generalized ODE S Schwabik

Vol. 6: Uniqueness & NonuniquenessCriteria in ODE R P Agarwal & V Lakshrnikantharn VOl. 7:

Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces Jaroslav Kurzweil

VOl. 8:

Integration between the Lebesgue Integral and the Henstock-Kurzweil Integral: Its Relation to Local Convex Vector Spaces Jaroslav Kurzweil Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane Douglas S Kurtz & Charles W Swartz

VOl. 9

Series in Real Analysis - Volume 10

TOPICS IN BANACH SPACE INTEGRATION

Stefan Schwabik Czech Academy of Sciences, Czech Republic

Ye Guoju Hohai University, China

N E W JERSEY

1: World -Scientific -

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SHANGHAI

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TOPICS IN BANACH SPACE INTEGRATION Series in Real Analysis Vol. 10

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Preface

A general integration theory based on the concept of Riemanntype integral sums was initiated around 1960 by Jaroslav Kurzweil and independently by Ralph Henstock. Much of this theory can be presented at the level of undergraduate courses and this fact is reflected in the growing number of university textbooks or elementary expositions which include at least elementary facts about the Henstock-Kurzweil theory. This concerns especially the last two decades and the publications [BaOl], [BSOO],[DPS87], [G94], [H88], [H91], [KS04], [K80], [LM95], [LPY89], [LVOO], [M97], [McL80], [McS83], [Pf93], [S99],

[sol].

The main virtue of the presentation of the Henstock-Kurzweil integral of real-valued functions is that no measure theory is required and that even sophisticated convergence results can be derived using merely elementary tools from the calculus without advanced topology. The relatively new concepts of the Henstock-Kurzweil and McShane integrals based on Riemann type sums are an interesting challenge also in the study of integration of Banach spacevalued functions. The advantage of a relatively transparent and easy definition is undoubtedly an invitation to do so. The investigations started around 1990 by the work of R. A. Gordon and since then attention has been paid to this field. One of the crucial problems is the comparison of the new concepts with the classical ones of the Bochner and Pettis integral. It V

vi

Banach Space Integration

should be mentioned at this point that some results concerning the basic facts of integration of Banach space-valued functions using Riemann type sums are also included in the early book [K80] of Jaroslav Kurzweil. This text presents an overview of the concepts and results achieved during the past 15 years. The Henstock-Kurzweil and McShane integrals play the central role in this text. In Chapter 1 elementary facts concerning the definition and properties of the Bochner integral are presented, Chapter 2 is devoted to the Dunford and Pettis integrals. In Chapter 3 we present the McShane and Henstock-Kurzweil integrals and Chapter 4 gives an overview of some special properties of the McShane integral. In this parts of the book special attention is paid to convergence theorems and the results are compared with the general Vitali Convergence Theorem. Chapter 5 is devoted to the interrelations of the Bochner and McShane integrals while in Chapter 6 the more delicate problem of the relation between McShane and Pettis integrability is studied. Properties of the indefinite integrals (primitives) for integration theories introduced based on Riemann-type integral sums are investigated in Chapter 7, some other convergence results (controlled convergence) are also presented. In the final Chapter 8 Denjoy and Henstock-Kurzweil extensions of the classical Bochner, Dunford and Pettis integrals are presented and a short overview of known results is given. An appendix at the end of the book collects basic facts from functional analysis, function spaces, etc. for the reader’s convenience with references to the respective sources. We would like to express our sincere thanks to our friends and colleagues who supported us by valuable advice, constructive criticism and patience especially in the Prague seminar on integration theory and also in China. Our work was supported by the grant No. 201/04/0690 of the Grant Agency of the Czech Republic and by the Academy of Sciences of the Czech Republic, Institutional Research Plan

Preface

vii

No. AVOZ10190503 in the case of the first author and partially supported by the Science Foundation of Hohai University, Foundation of mathematical key program of Hohai University and the Foundation of postdoctoral fellows of Lanzhou University in the case of the second author. May 2005

The authors

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Notation

The following basic notation will be used in this book: N = {1,2,. . . } is the set of positive integers, R stands for the reals, R", m 2 1 is the rn-dimensional space under the usual Euclidean norm, I = [ a l ,bl] x . . . x [a,, b,] c R" will be a compact interval endowed with the (outer) Lebesgue measure p. Given a set E c I , we denote by X E the characteristic function of the set E . X will be a Banach space with the norm 11 - 1 1 ~ . By X*we denote the dual to X and

a@>= {x E x;J J Z J J5X 1) is the unit ball in the Banach space X. Given a functional x* E X * its value on the element x E will be denoted by z*(z).

ix

X

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Contents

Preface

V

ix

Notation 1. Bochner Integral 1.1 Simple functions, measurability . . . . . 1.2 The integral of simple functions . . . . . 1.3 Bochner integral . . . . . . . . . . . . . 1.4 Properties of Bochner integrable functions the Bochner integral . . . . . . . . . . .

1 . . . . . 1 . . . . . 9 . . . . . 11 and of . . . . . 22

2 . Dunford and Pettis Integrals 27 2.1 Dunford integral . . . . . . . . . . . . . . . . . . 27 2.2 Pettis integral . . . . . . . . . . . . . . . . . . . 33 2.3 Some properties of the Pettis integral . . . . . . 36 3. McShane and Henstock-Kurzweil Integrals 3.1 Systems. partitions and gauges . . . . . . 3.2 Definition of the McShane and HenstockKurzweil integrals . . . . . . . . . . . . . . 3.3 Elementary properties of the McShane and Henstock-Kurzweil integrals . . . . . . . . 3.4 The Saks-Henstock lemma . . . . . . . . . 3.5 A convergence theorem . . . . . . . . . . . xi

45 . . . . 45

. . . . 46

. . . . 48 . . . . 55 . . . . 64

xii

Banach Space Integration

3.6 The strong versions of the McShane and Henstock-Kurzweil integrals . . . . . . . . . . . . 70 3.7 Integration over unbounded intervals and some remarks . . . . . , , , . . . . . . . . . . . . . . . 85 4. More on the McShane Integral 87 4.1 Special properties . . . . . . . . . . . . . . . . . . 87 4.2 An equivalent definition of the McShane integral . 113 4.3 Another convergence theorem . . . . . . . . . . . 119

5. Comparison of the Bochner and McShane Integrals 5.1 Strong McShane integrability and the Bochner integral . . . . . . . . . . . . . . . . . . . . . 5.2 The finite dimensional case . . . . . . . . . . 5.3 The infinite dimensional case . . . . . . . . . 5.4 An example . . . . . . . . . . . . . . . . . . .

133

. . . .

. . . .

133 150 153 159

6. Comparison of the Pettis and McShane Integrals 171 6.1 McShane integrable functions are Pettis integrable . . . . . . . . . . . . . . . . . . . . . . 171 6.2 The problem of P c M . . . . . . . . . . . . . . 173 6.2.1 Functions weakly equivalent to measurable ones . . . . . . . . . . . . . . . . . . . . . 183 6.2.2 P C M does not hold in general . . . . . . 188 7. Primitive of the McShane and HenstockKurzweil Integrals 7.1 Absolutely continuous functions and functions of bounded variation . . . . . . . . . . . . . . . . 7.2 Generalized absolute continuity and functions of generalized bounded variation . . . . . . . . . . 7.3 Differentiability . . . . . . . . . . , . . . . . . . 7.4 Primitives . . . . . . . . . . . . . . . . . . . . . 7.4.1 The strong Henstock-Kurzweil integral . 7.4.2 The McShane and the strong McShane integral . . . . . . . . . . . . . . . . . . .

191 . 192

. 200 . 202 . 211 . 212 . 2 18

Contents

xiii

7.4.3 The Henstock-Kurzweil integral . . . . . . 223 7.5 Variational measures and primitives for S M and S7-K . . . . . . . . . . . . . . . . . . . . . . 226 7.6 Controlled convergence . . . . . . . . . . . . . . . 231 8. Generalizations of Some Integrals 8.1 Bochner integral . . . . . . . . . . . 8.2 Dunford and Pettis integral . . . . 8.2.1 Denjoy approach . . . . . . . 8.2.2 Henstock-Kurzweil approach 8.2.3 Some examples . . . . . . . 8.3 Concluding remarks . . . . . . . . .

251

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . 251 . . . . 254 . . . . 255 . . . . 269 . . . . 272 . . . . 273

Appendix A Classical Banach Spaces A . l Spaces of sequences . . . . . . . . . . . . . . . A.2 Spaces of functions . . . . . . . . . . . . . . . A.2.1 The spaces C ( I ) and L p ( I ) ,1 < p < 00 A.2.2 The spaces L1 and L , . . . . . . . . .

. . .

.

277 . 277 . 279 . 279 . 279

Appendix B Series in Banach Spaces

283

Bibliography

29 1

Index

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Chapter 1

Bochner Integral

1.1

Simple functions, measurability

Definition 1.1.1. A function f : I + X is called simple if there is a finite sequence Em c I , m = 1 , .. . , p of measurable sets such that

Em n El

=0

for m # 1

and P

I=

U Em m=l

where

f ( t ) = ym E X

for t E Em, m

=

1,.. . , p ,

i.e. f is constant on the measurable set Em.

Denote by J ( p , X ) = 3 the set of all simple functions defined on I . Clearly 3 is a linear space and if f is a simple function then also l l f l l : I -+ R is a simple function.

Definition 1.1.2. A function f : I -+ X is called measurable if there exists a sequence (fn), fn E J ,n E N with lim IIfn(t) - f(t)Ilx = 0

n4cc

for almost all t E I . 1

2

Banach Space Integration

Clearly, if f E 3 then f is measurable.

Proposition 1.1.3. Iff : I + X is measurable then the real function 11 f JIx: I 3 R is measurable.

Proof. Let fn E 3,n E N be the sequence corresponding to f . Then llfnllx are simple real functions for all n E N and because

for t E I we conclude that lim Ilfn(t)llx = Ilf(t)Ilx a.e. in I n-+m

and therefore (1 f I\x is measurable. 0

Remark. It has to be mentioned that (in the case X = R) a function f : I --+ R is measurable in the sense of Definition 1.1.2 if and only if for every finite a E R the set { t E I ; f ( t ) > a } (or equivalently {t E I ; f ( t ) 2 a } , {t E I ; f ( t ) < a ) , { t E I ; f ( t ) 5 a } ) is measurable. For details see e.g. [WZ77], Theorem (4.13). Definition 1.1.4. A function f : I --+ X is called weakly measurable if for each x* E X* the real function z*(f): I + R is measurable. The concepts of measurability and weak measurability are closely related. The relation is given by the well known theorem of Pettis presented below. First we give the following lemma.

Lemma 1.1.5. Assume that X is a separable Banach space. Then there is a sequence {x; E B ( X * ) ;m E N} such that for every x* E B ( X * ) there exists a subsequence {x;; k E N} of {x; c B(X); m E N} such that lim zi(x) = x*(x)

k4oo

for every x E X

Proof. Assume that {xn E X; n E the separable space X .

N} is a dense sequence in

Bochner Integral

Consider for n E

3

N the mapping

x* E B ( X * )+ (pn(x*)= {x*(q),. . . , x*(xn)} E R". The space R" with the Euclidean norm is separable and therefore for any fixed n E N there is a sequence { x : , ~E B ( X * ) ;k E N} such that the set {'P"(x;,k); k E

N

is dense in the image (p,(B(X*)) c R" of the unit ball B ( X * ) . This means that for every x* E B ( X * )there exists a subsequence (x&J of {xi,k E B ( X * ) ;k E N} such that

Ix;,&i) for i

=

- x*(xz)l

1


1 , 2 , . . . , n. Therefore we get lim x;,,, (xi)= x*(xi) "+03

for every i E N. Since the sequence { Ilx;ll~*; n E N} is bounded and lim x;,",(zi) = x*(xi)

n-+m

for the dense sequence (xi)in X we obtain lim ~ ; , ~ , (= x )x*(x)

n+cc

for all x E X and the lemma is proved.

0

Remark. The final part of the proof is based on the following theorem: If X is a Banach space then a sequence xi E X* weak*converges to some xT, € X * if and only if the sequence (11xi11;n E N} is bounded and lim xi(.) = z,(x) on a dense 71-03

subset in X . (See [Y65],p. 188.) Let us also mention that the conclusion of Lemma 1.1.5 is in fact the weak* separability of the ball B ( X * ) in X * . This

4

Banach Space Integration

means that if the Banach space X is separable then B ( X * ) is weak*-separable. Theorem 1.1.6. (Pettis) A function f : I t X is measurable if and only i f f is weakly measurable and almost everywhere separable valued, i.e. there is a set N C I , p ( N ) = 0 such that the set

{f ( t ) ;t E I \ N ) c x is separable.

Proof. Let f : I + X be measurable. Then there is a sequence f n E 3,n E N such that (1.1.1)

lim Ilfn(t) - f (t)Ilx= 0

n+cc

for almost all t E I . If x* E X*then (1.1.1)implies

a.e. in I . Since f n E 3, n E N we obtain that x * ( f n ) : I -+ R is a simple real function for every x* E X*. Therefore x*(f ) is measurable for each x* E X* and f is weakly measurable by definition. We will use the following result (see [DS] Theorem 111.6.12 and its corollaries). Theorem 1.1.7. (Egoroff’s theorem) Let f n : I be a sequence of measurable functions such that

+

almost everywhere in I . Then for every 7 > 0 there is a measurable set H that p ( I \ H ) < 7 and

uniformly on H.

X,n E N

5I

such

Bochner Integral

5

Thus for each n E N there is a measurable set En c I such that p(E,) < and lim IIf,(t) - f(t)IIx = 0 uniformly on I \ En. Since f, E J , the range f n ( l ) c X of f , is finite for every n E N and it follows that UnENfn(I) is countable. Hence for every n E N the set f ( I \ En) is separable and

f( U(1\ ~

n )=)

is separable as well. Using the fact that k(E,)

E,

=

< $, n E N we have

I\

nEN

and

,(n

E,)

nEM

(1.1.2)

nEN

nEN

n

U f ( r \ En)

U(I\En) nEN

= PV

\

U( I \ En)) = 0. nEM

Putting N = n,,,E, we get by (1.1.2) the separability of the set { f ( t ) ; t E I\ N } . Let us prove the converse. Without loss of generality assume that the full range f ( I ) of the function f is separable. Therefore the space X can be also assumed to be separable (X can be taken Assume as the smallest closed linear subspace containing f(1)). that x, E X , n E N is dense in X. First we show that the function Ilf(t)llx is measurable. For a 2 0 and x* E X*consider the sets

A

=

{t E I ; Ilf(t)))xI a}

and

Ax* = {t E I ; I~*(f(t))l 5 a}. We have

6

Banach Space Integration

and since by the Hahn-Banach Theorem for every fixed t E I there exists xT, E X* with I I x T , ~ ~ X *= 1 such that zG(f(t))= Ilf(t)Ilx we have also

n

Ax* C A

x* E B ( X * )

and consequently

n

A=

A~..

x*EB(X*)

According to Lemma 1.1.5 we have 00

n=l

X*EB(X*)

where x;T,is given by this lemma and therefore 00

n=l

The sets A,,, n E N are measurable by the weak measurability of f (cf. Remark after Proposition 1.1.3) and henceforth the set A c I is measurable. This yields the measurability of the function Ilf(t)llx on I (see the same Remark). Let { y n E f ( I ) ; n E N} be a dense set in f ( I ) . Similarly as we have shown the measurability of Ilf(t)llx above we can show that the functions gn : t E I +. Ilf(t) ynllx E R, n E N are measurable. Taking a fixed k E N put

The measurability of gn : I 4 R yields that E," c I are measurable sets and because for every t E I there is an n E N such that Ilf(t)- gnllx < $ we get U,"==lEk = I .

Bochner Integral

Define

u

B,"= E," \

7

Ej", n e N,k

E

N.

j
I?; c I are measurable sets, B;f l I?; = 8 for n # m,

u Bk = I oc)

n=l

and oc)

n= 1

Hence for every k E

N there exists n k

E N such that

00

1

Define h k ( t ) = Yn

if t E B,"

and hk(t) = 0 otherwise. For t E I we have Ilf(t) - h,+(t)llx< and of course

uniformly in I . The range of Let us set

hk

is countable by the definition.

gk(t) =

h,(t) for t E

u

nk

l3: and g k ( t ) = 0 n=l Ilf(t) - g k ( t ) l l x = 0 a.e. in

wise. Then g k E 3 and lim k-+m this gives the measurability of f.

other-

I and 0

Looking at the proof of the Pettis theorem 1.1.6 we can use the countable valued functions h k to present the following. Corollary 1.1.8. A f u n c t i o n f : I + X is measurable i f and only if

lim hn(t)= f(t)

n+m

Banach Space Integration

8

uniformly for almost all t E I where (h,) is a sequence of countable valued measurable functions.

Remark. The Pettis measurability theorem can be found in classical books on vector integration (e.g. [DU77]) sometimes with a shorter proof. We use essentially the approach given in the book [Y65]because Lemma 1.1.5 will play a certain role later in our considerations. Proposition 1.1.9. I f f : I + X is measurable then there is a bounded measurable g : I X and a measurable h : I --+ X with --f

03

h ( t )=

ZnXE,(t),

Zn

E

x,

E

N,t E 1,

n= 1

where En c I , n E that f = g + h.

N are pairwise disjoint measurable sets, such

Proof. Using the Pettis theorem 1.1.6 we can suppose that the range f ( I ) of f is a separable subset in X with {zn,n E N} being an at most countable dense subset in f ( I ) . Define

Then I

c U:=, En, En n Em = 0 for rn,n E N,rn # n. Put

for t E I . h : I

+X

is measurable and if t E En n I then

Bochner Integral

9

Proposition 1.1.10. If X is a separable Banach space then f : I -+ X is measurable if and only i f f is weakly measurable. Proof. For a separable space X the range { f ( t ) ;t E I } c X off is automatically separable and the statement follows immediately from the Pettis measurability theorem 1.1.6. 0

1.2

The integral of simple functions

It is a simple task to define the integral of a simple function. Assume that f : I t X is a simple function given by Definition 1.1.1. Define the integral off : I t X as

If A

c I is measurable and f

E

J then define

f A ( t ) = f ( t )if

tEA

and fA(t) =0

if t $ A,

i.e. f A = XA. It is easy to see that the function set f

a

f A

is again simple and we

The integral of simple functions f E 3 defined in this way is evidently a linear mapping J : J -+ X. If A, B are disjoint measurable sets in I then from the linearity of the integral and from the obvious identity f A U B = f A f B we have

+

LUBfs +/ = A

f

B

f.

(1.2.2)

10

Banach Space Integration

Remark. In the special case when X = R and f 5 g where f ,g E 3 we have

L f sLg*

(1.2.3)

I f f 2 0 and A c B , then (1.2.4) For the integral of a function f E 3 of the form presented in Definition 1.1.1and a measurable A c I we have (1.2.5)

nEm) = A. and umxl(A For a given f E 3 define

Ilf 111 = The mapping

/ Ilf I

IIX.

(1.2.6)

(1 - ((1: 3 -+ R has the following properties:

11

Bochner Integral

By 11 - 111 from (1.2.6) a seminorm on 3 is given; the implication llflll = 0 + f ( t ) = 0 for all t E I does not hold. To see this it suffices to take A c I such that p ( A ) = 0 and a function f for which f ( t ) = 0 provided t $ A. The triangle inequality (1.2.9) can be shown using a decomposition of the interval I into measurable sets with respect to which each of the functions f and g is simple, i.e. f and g have constant values at each measurable component of the decomposition; the inequality (1.2.9) then results from the triangle inequality valid in the Banach space X. The seminorm 11 - /I1 defined above for elements of 3 is called the L-seminorm.

1.3

Bochner integral

Let us now consider sequences of simple functions fn E 3,n E N with the seminorm 11 . 111 given in the previous section.

Definition 1.3.1. A sequence called L-zero if

( f q ) , fq

E

3, q

=

1 , 2, . . . is

Two sequences ( f q ) , ( g q ) f q , gq E 3, q = 1 , 2 , . . . are called equivalent if their difference ( f q - gq) is L-zero. A sequence ( f q ) = (fq)gl,fq E 3, q = 1 , 2 , . . . is called LCauchy if for every E > O there is an N = N, E N such that

We will consider the completion of the linear space 3 of simple functions on I with respect t o the L-seminorm 11 111. The completion of 3 is given as the space of equivalence classes of L-Cauchy sequences of functions from 3. For details concerning the concept of the completion of a seminormed space see e.g. [L93].

12

Banach Space Integration

The set of L-Cauchy sequences of simple functions has the structure of a linear space, i.e., if (f,) and ( g q ) are L-Cauchy sequences of simple functions and a E IR then both (f, g,) and ( a f q )are L-Cauchy sequences of simple functions. The following statement is fundamental for the construction given below.

+

Lemma 1.3.2. Let (f,) be an L-Cauchy sequence of simple functions defined on I . Then there is a subsequence ( g k ) of (fq), which converges pointwise almost everywhere to some function f : I -+ X and for every E > 0 there is a measurable E C I with p ( E ) < E such that this subsequence converges uniformly on I \ E . Proof. Since the sequence (f,) is L-Cauchy, for every k E there is Nk E N such that if q, r 2 Nk, then

Without loss of generality it can be assumed that We set

ivk

N

< Nk+1.

then

for m 2 n. Next we define for t E I the series

k=l

and show that it converges absolutely for almost all t E I to an element in X and that this convergence is uniform except for a set with arbitrarily small measure.

Bochner Integral

13

For n E N denote

Then

I

1

1 Ilgn+dt>- gn(t>Ilx= IlSn+l - Snlll < 22"

and this yields

1 P(Mn) < 5' Let us define

2, = Mn u Mn+lu . . . *

Then Zn+l c

For t

Z,,n E N and

# 2, and k 2 n we have

and therefore the series ~ ~ , ( g ~ + l (g kt( )t ) ) converges absolutely and uniformly for t # 2,. Assume that E > 0 is given. Putting N = Zk,we have for sufficiently large k P")

=P(Zd

1

< 2"-1 < E

and this leads t o the assertion that the series CEn(gk+l(t) gk(t)) converges absolutely and uniformly on I \ N . If we take M = nz,, then evidently p ( M ) = 0 and if t $! M , then t # Zn for some n. Therefore the series

Banach Space Integration

14

+

g1 ( t ) CE,(gk+l ( t )- g/c ( t ) )converges for t 4 M and this means that lim gk(t) = lim f N k ( t ) exists for almost all t E I and the k-cc

k-cc

sequence g k ( t )= f N k ( t )converges uniformly on I

\ N. 0

Lemma 1.3.3. a) If ( f q ) is an L-Cauchy sequence of simple functions then the limit lim JI f q exists. q'm

b) If (f q ) and ( g q ) are equivalent L-Cauchy sequences of simple functions then lim

q-00

l f q ;iElgq. =

(1.3.1)

c) If (f q ) and ( g q ) are L-Cauchy sequences of simple functions which converge almost everywhere to a function f : I -+ X then (f q ) and (gq) are equivalent and (1.3.1) holds.

Proof. The existence of the limit in a) is easy to show. Indeed, for simple functions f q we have (see (1.2.5))

This means that the sequence of integrals JI f q E X ,q E N is a Cauchy sequence and therefore it is convergent, i.e. the limit lim S, f q exists. q+m

For proving b) let E > 0 be given. Then by a) and the equivalence of the L-Cauchy sequences (f q ) and ( g q ) there is an N E N such that for r > N we have llfT

- 9'7'111 =

1

llfT

I

- gTIIx

< &,

Bochner Integral

15

and

and b) is proved. For the proof of c) let us set h, = f, - g, and assume that E > 0 is given. It is clear that lim hq(t)= 0 for almost all t E I q-00

and that the sequence h, is L-Cauchy, i.e. there is an N E such that for T , q 2 N we have Ilhq - hrlll

N

< &*

This implies by a) that the sequences of integrals S,h, and

J, JJhqJJx are convergent. It remains t o show that

Define

M = {t E I ; h N ( t ) # 0) For q 2 N we have

c I.

16

Banach Space Integration

because h ~ ( t=) 0 for t E I a subset 2 c M with

\ M.

By Lemma 1.3.2 there exists

and a subsequence h,, which converges to zero uniformly on the set M \ 2. Hence there is an SO E N,so 2 N such that for s 2 so and for t E M \ 2 we have

Therefore

provided s

2 so.

For s

2 so we also have

Hence rb

< E + E + 2~ = 4~ and because

17

Bochner Integral

we

obtain

lim JI Ilhqs(t)llX =

0 and therefore also

S'OO

J' Ilhq(t)llx= 0, and c) is proved.

q-+m lim

0

Definition 1.3.4. Denote by B the set of all functions f : I --+ X for which there is an L-Cauchy sequence f q , q E N of simple functions which converges to f almost everywhere in I , i.e.

for almost all t E I . We say in this case that the L-Cauchy sequence q E N determines the function f E B.

fq

E

3,

By a) from Lemma 1.3.3 it is easy to see that to every LCauchy sequence ( f q ) of simple functions a value x ( f q E ) X can be assigned by the relation X(f4) = qlim ~ O

LO f q .

Using b) from Lemma 1.3.3 we can see that the same value x ( f 4 )E X belongs to all L-Cauchy sequences which are equivalent t o the sequence ( f q ) . This allows us now to present the following concept.

Definition 1.3.5. For f E B define

lf

=;i&$-q

(1.3.2)

where ( f , ) is an arbitrary sequence of simple functions which determines f E B. The value f given by (1.3.2) is called the Bochner integral of the function f. If necessary the more detailed notation (a)JI f will be used for this concept of integral. The set of functions B is called the set of Bochner integrable functions.

s,

It is easy to see that the set B is linear.

18

Banach Space Integration

By (1.2.1) the integral was defined in a very natural way for simple functions while by the relation (1.3.2) this integral is extended to functions f E B. The correctness of this definition is clear by Lemma 1.3.3. In our presentation we follow the lines given in [L93] by S. Lang but the reader can find the Bochner integral in many books, e.g. [M78] or in general books on functional analysis, e.g. [DS]. Lemma 1.3.6. I f f E B and (f,) is an L-Cauchg sequence of simple functions which determines f , then 11 f J J Xis integrable and the sequence (Ilf,llx) determines the real function llfllx in the sense of the set B. I n this case we have (1.3.3) Moreover, (1.3.4)

Proof. Since

we get

and this means that the sequence Ilf411x of real-valued simple functions is L- Cauchy. Moreover,

19

Bochner Integral

for almost all t E I and consequently llfllx : I + IR is integrable by Definition 1.3.5 and Ilfqllx, q E N determines llfllx where (1.3.3) holds. Since by (1.2.5) for f q E J’ we have

(1.3.2) and (1.3.3) can be used for obtaining (1.3.4) by passing to the limits with q --+ 00 on both sides of this inequality. 0

By Lemma 1.3.3 we know that lim

Ilfqlll

does not depend on

q-00

the choice of the L-Cauchy sequence ( f q ) which determines the same f ; therefore the seminorm 11. [I1 defined for simple functions f E J’ can be extended to functions f E l3 by the relation (1.3.5) In this way

11 . /Il

:B

t

llflll

R is defined and the following holds:

L 0 for every f

E

a,

(1.3.6)

llaf 111 = lalllfll1 for every f E l3 and a E R,

(1.3.7)

Ilf + 9111 Illflll + 11g111 for every fd E B.

(1.3.8)

These relations are immediate consequences of the analogous relations (1.2.7) - (1.2.9) for II.111 given on J’,showing that II.111 is a seminorm on B. Lemma 1.3.7. I f f E B and ( f n ) is an L-Cauchy sequence of simple functions determining f , then

Proof. Since ( f q ) is an L-Cauchy sequence of elements f q E J’ which converges almost everywhere to f , for every E > 0 there

Banach Space Integration

20

is N, E

N such that

llfr - f q l l l < E

(1.3.9)

provided T , q > N,. Let us fix r > N, and put gq = f r - f q E 3 for q E N. Then lim g q ( t ) = f r ( t )- f ( t ) E B for almost all t E I and q-00

because 1191 - g k I ( 1 = 11 fi - f k l l l the sequence (gq) is L-Cauchy and determines fr - f E B. Hence

and this implies lim 11 fr r-+m

-

fill = 0. 0

Corollary 1.3.8. Iff E B then for every simple function gE E J’ such that

Ilf - 9 E l I 1 < &,

E

> 0 there is a (1.3.10)

i.e. the set J’ of simple functions is dense in B with respect to the seminorm 11 . [I1. Lemma 1.3.9. The space B equipped with the seminorm is complete.

II.111

Proof. Assume that gq E B , q E N is a Cauchy sequence with respect to the seminorm 11.111. By Corollary 1.3.8for every q E N there exists a simple function fq E J’ such that 1194

-fqlh

1

< -. Q

Hence

and therefore the sequence (fq)is L-Cauchy. By Lemma 1.3.2 the sequence (fq) contains a subsequence (f q s ) which converges almost everywhere in I to a certain function f : I + X and this

Bochner Integral

21

subsequence is L-Cauchy. Hence f E B. For this subsequence (f,,)we have II9% - f 111 5 119% - f*sII1+ Ilf% - f Ill

and the subsequence ( g q s ) of ( g q ) converges in the seminorm )I 111 to f by Lemma 1.3.7. This implies that also the original sequence ( g q ) converges in this seminorm to f E B and henceforth B is complete. 0

-

Using Lemma 1.3.9 we can see easily that the following holds.

Corollary 1.3.10. A function f : I --+ X belongs t o B zf and only if there is a sequence f n E 3, n E N such that

for almost all t E I and

By this corollary we get that f E B is necessarily measurable. On the other hand, this corollary in fact gives another definition of Bochner integrability which is equivalent to Definition 1.3.4 (and Definition 1.3.5 can be used for defining the integral).

Definition 1.3.11. f : I -+ X is Bochner integrable if there is a sequence of simple functions fn : I --+ X , n E N such that lim f n ( t )= f ( t ) a.e. in I and

n+03

Let us note that the following holds.

Theorem 1.3.12. I f f : I 4X is such that f ( t ) almost all t E I then f E B and f = 0.

s,

=

0 for

Proof. The L-Cauchy sequence of simple functions from Definition 1.3.4 can be chosen as functions which are identically zero.

I7

22

Banach Space Integration

Corollary 1.3.13. Iff : I t X is Bochner integrable and g : I t X i s such that f ( t )= g ( t ) f o r almost all t E I t h e n g i s Bochner integrable and JI f = 9.

lI

+

Proof. Since g = g - f f and g - f is Bochner integrable by Theorem 1.3.12, we obtain the statement immediately. 0

Corollary 1.3.13 makes it possible to identify functions which are equal almost everywhere as is usual in the Lebesgue theory.

Remark 1.3.14. For the case X = R, i.e. for f : I + R, the definition of Bochner integrability and the Bochner integral (Definition 1.3.5 or Definition 1.3.11) give an alternative approach to Lebesgue integrability and the Lebesgue integral. This means that f : I --+ JR is Bochner integrable in the sense of Definition 1.3.5 if and only if f is Lebesgue integrable and the two integrals of f have the same value.

1.4

Properties of Bochner integrable functions and of the Bochner integral

Fkom the definition of the class B it is clear that every f E B is measurable in the sense of Definition 1.1.2. By the Pettis measurability theorem 1.1.6, if f E B then f is also weakly measurable and almost everywhere separable valued. For a given measurable set E C I and f E B we define

where f n E 3,n E N determines f. This definition makes sense because x ~f n ,. n E N is evidently a sequence of simpIe functions which determines x E . f . Let f : I -+ X be a countable valued measurable function of

Bochner Integral

23

the form

f ( t >=

C

YmXE,(t),

t E I,

(1.4.1)

m=l

c I, m X ,m E N.

where Em Ym

E

E

N is measurable, Emn El

=

8 for m # I ,

Lemma 1.4.1. A countable valued measurable function f I -+ X of the f o r m (1.4.1) is Bochner integrable if

:

00

m=l

Proof. Define for 1 E

N functions

J for every 1 E N and lim f l ( t ) = f ( t ) for t l-tW For t E I and k < 1 we have by definition

Then

fi E

E I.

1

IIfi

- fkII1 =

C

IIYrnIIxpL(En>.

m=k+l

Now we can see that the sequence if the series 00

m=l

(fi)

is L-Cauchy if and only

24

Banach Space Integration 00

converges. In this case the series

C ymxEn

converges in X to

m=l

f and by definition we have f E B and 00

n

and also

J1

m=l

Corollary 1.4.2. A countable valued measurable function f : I --+ X for which Ilf(t)lIx 5 g ( t ) a.e. in I with g E B is Bochner int egrab1e .

Proof. Using the sequence (fi) from the proof of Lemma 1.4.1 we can see that 11 fi 11 1 5 JI g < 03 for every 1 E N and therefore the condition given in Lemma 1.4.1 is satisfied. 0

Theorem 1.4.3. A measurable function f : I -+ X is Bochner integrable if and only if 11 f (Ix : I -+ R is Bochner integrable.

Proof. If f E B then Lemma 1.3.6 implies the integrability of

IlfI I X .

Assume that llfllx is Bochner integrable. Since f is measurable (see Corollary 1.3.10), by Corollary 1.1.8 for every k E N there is a countable valued measurable function fk of the form 00

(1.4.2) m= 1

where Ek,m C I , m E €V is measurable, f l Ek,J= 8 for m # 1, E X , m E N and fk has the following property:

Bochner Integral

there exists N have

c I , p ( N ) = 0 such that for

25

every k E N we (1.4.3)

for t E I \ N . Hence

a.e. in I and, since p(1) < 00,Corollary 1.4.2 implies that Bochner integrable and

N such that

Ilf

- fkllx is measurable and (1.4.3) holds the function - fkllx is integrable and

Since

[If

is

00

F

Choose an rk E

fk

00

We have also

and therefore f E B. 0

Banach Space Integration

26

Corollary 1.4.4. I f f : I -+ X is measurable and bounded by an integrable function g : I -+ R, i.e. 11 f (t)Ilx5 g ( t ) for almost all t E I , then f is Bochner integrable. Proposition 1.4.5. Let f : I

+X

be measurable of the form

n= 1

where g : I + X is measurable and bounded, En are pairwise disjoint measurable subsets of I , xn E X , n E N (see Proposition 1.1.9).

T h e n f is Bochner integrable if and only if x, and En, n E N can be chosen such that the series C,"==, x, p(En) converges absolutely in X , and in this case we have +

for every measurable E

cI

Proof. Assume that f E B is of the form (1.4.4). Since g is bounded we have g E B (see Corollary 1.4.4) and also f - g = C,"=1x n X E , E 13. By Theorem 1.4.3 we have JI 11 C,"=, X n * x E n J l x< 00 but this means, because of En n Em = 8, m # n, that *

2, . p ( E n ) is absolutely convergent in X . and C,"==, Conversely, if g is bounded and the series C,"=, x , . p(E,) converges absolutely, then g E B by Corollary 1.4.4 and X n - p ( E n ) E B by Lemma 1.4.1. Hence f = g h E B.

c,"==l

+

Chapter 2

Dunford and Pettis Integrals

If f : I X is a function that is only weakly measurable an approach similar to the Bochner theory presented in the previous Chapter 1 does not apply even if the real function x*(f) : I t R belongs to L1 for every x* E X*. It can be used for an individual x* E X*only. Nevertheless, methods of elementary functional analysis allow us to define a reasonable concept of integral if for a weakly measurable f we have x*(f ) E L1 for every x* E X*. The resulting integrals have a relatively rich structure and we will present the fundamental results in this direction in the present chapter. We are presenting only elementary and basic facts concerning the Pettis integral in this chapter. The reader interested in more detail should consult e.g. [DU77], [M91], [M02], [T84]. Especially the excellent survey [M02] gives a contemporary overview of the state of art in this interesting field of functional analysis. --+

2.1

Dunford integral

Lemma 2.1.1. (Dunford) Assume that f : I -+ X is weakly measurable and that for each x* E X*the function x*(f ) : I --+ R is Lebesgue integrable (x*(f) E L1). Then for each measurable E C I there exists a unique x z E X** such that (2.1.1)

Banach Space Integration

28

for every x* E X*. Proof. For a given measurable E JI x*(f - X E ) and we can define TE(X*) =

c I we have

J E x * ( f )=

- XE).

.*(f

is a linear map of X* into the space L1 of real Lebesgue integrable functions on I and TE

is a linear functional on X*. Assume that xi -+ x* in X * and n + 00, i.e.

T'(xi)

-+

g in L1 for

Then xi(f X E ) converges for n -+ 00 in measure to g and by the Riesz theorem there is a subsequence x i k , k E N of xi such that

for almost all t E I . Since x i ( f ( t )- X E ( ~ ) ) x * ( f ( t ) x E ( t ) ) , n -+ 00 for all t E I it follows that g ( t ) = x*(f(t) . x ~ ( t )for ) almost all t E I and x*(f . XE) E L1. This means that the graph of the linear map TE : X * L1 is closed and by the Banach closed graph theorem the operator T E is bounded. Hence -+

-+

-

Dunford and Pettis Integrals

29

and it follows that

Therefore JE x*( f ) is a continuous linear functional on X*defining an element xg E X**for which (2.1.1) holds.

Remark. It is worth mentioning that the Lebesgue integrability of the real function x*(f) required in Lemma 2.1.1 can be replaced by the equivalent Bochner integrability of x*(f)as was presented in Remark 1.3.14. The previous Dunford lemma 2.1.1 makes it possible to introduce the following definition. Definition 2.1.2. If f : I -+ X is weakly measurable and such R is Lebesgue integrable for each that the function x*(f) : I x* E X*then f is called Dunford integrable. The Dunford integral (27)JE f of f over a measurable set E C I is defined by the element xg E X** given in Lemma 2.1.1, i.e. --f

n

( D )J f

E x**,

= x;

E

where x g ( x * )= JE x*(f) for all z* E X * . Denote by 23 the set of all Dunford integrable functions. For f : I --+ X with f E D we have x*(f) E

x* E x*.

L1

for all

Let us define

T ( z * )= z*(f), x* E

x*.

(2.1.2)

T : X*-+ L1 is a linear operator which is bounded according to the Banach closed graph theorem (cf. the proof of Lemma 2.1.1 for the case E = I ) . Let T* : LT = L , + X** be the adjoint of the operator T defined by

T*(g)(x*)= l g T ( x * )= *

l

g

*

x*(f) E R,

g E L; = L,.

Banach Space Integration

30

T*(g)is a linear functional on X * for any g E L,(= because

(L1)*)

and it is also bounded because the boundedness of the operator

T gives

5 11911Lm IlTll *

*

Hence T*(g)E X** for every g E L,. Assuming g = X E E L, where E

IIx*IIx*.

cI

is measurable, we have

Then T * ( x E E) X** for every measurable E

cI

and

v ( E ) = T * ( X E ) = (D)

(2.1.3)

The function v ( E ) = (D) JE f defined for all measurable E C I is called the indefinite Dunford integral o f f . Proposition 2.1.3. A s s u m e that f : I -+ X i s Dunford integrable. T h e n the following assertions are equivalent. a) T h e operator T : X * -+ L1 given in (21.2) i s weakly compact. b) T h e adjoint operator T* : L , + X** t o T is weakly compact. c) T h e set {x*(f ) ;x* E B ( X * ) }c L1 i s uniformly integrable, i.e. r p t i ~ JE o

x*(f ) = O uniformly for x* E B ( X * ) .

d) T h e indefinite Dunford integral v ( E ) given by (2.1.3) i s countably additive, i.e., if En C I , n E N are pairwise disjoint

Dunford and Pettis Integrals

31

measurable sets then n=l ' ( E n ) is n o r m convergent in X**).

n=l

in X** (the series C,"=l

Note that a weakly compact operator takes bounded sequences into sequences having weakly convergent subsequences or, equivalently, maps bounded sets into weakly compact sets.

Proof. Let us mention that by Gantmacher's theorem ([DS], VI. 4. 8. Theorem) an operator T is weakly compact if and only if its adjoint T* is weakly compact and therefore a) and b) are equivalent. Let us consider the set

T ( B ( X * )= ) {x*(f);x* E B ( X * ) }c L1. We have

for x* E B ( X * ) because the operator T is bounded. Hence the set T ( B ( X * )is) bounded. ) L1 is By Theorem A.12 in Appendix A the set T ( B ( X * ) c weakly compact if and only if we have lim,(E),o JE x*(f) uniThis means that c) is equivalent to formly for x* E B(X*). a>.

Assume that c) holds. It is easy to see that

Then for every q > 0 there is an

IT*(XE)(Z*)I for every x* E

=

E

> 0 such that

llx*(f)l <7

B(X*) if p ( E ) < E and therefore IIT*(XE>II L rl.

Banach Space Integration

32

If En c I , n E N are pairwise disjoint measurable sets dew note E = Un=l En. Then limN,, p(E \ U,"=, En) = 0 and consequently

we have by finite additivity

This means that

and lim

N+,

Hence v ( E ) is countably additive. Assume now that c) does not hold. Then there is a K > 0 and a sequence E, C I , n E N of measurable sets with p ( E n ) -+ 0, n + 00 and r

for some XI*, E B(X*). Since the measures of En tend to zero, it is possible to take a subsequence of En assuming that for m < n we have

Dunford and Pettis Integrals

Take A, = En,\ Urn<,Em;A, m # 1 and

33

c I are measurable, A, n Al

=

0,

6

Iz;(f>l

Hence there exist B, disjoint) such that

> 5'

c A,, B, measurable (B, are pairwise

and therefore IIT*(XB,>II> f for every n. Therefore the series co T*(xB,)= v(B,) cannot converge and d) is not satisfied. This gives the equivalence of c) and d).

crxl

0

2.2

Pettis integral

The Dunford integral (D) JE f from the previous section is an element of the second dual X** of the Banach space X. This situation is not very pleasant, one would expect that the values of an integral of an X-valued function belong to the same space

X. To define another integral having this property let us recall that the space X itself is in a natural way embedded into X**. If it happens that (D) JE f E X c X**the following definition can be presented. Definition 2.2.1. If f : I -+ X is Dunford integrable where ( D )S, f E X (or more precisely ( D )S, f E e(X) c X**,where e is the canonical embedding of X into X**)for every measurable E c I , then f is called Pettis integrable and

is called the Pettis integral of f over the set E .

Banach Space Integration

34

We denote by P the set of all Pettis integrable functions

f :I-+X. It can be seen immediately that the Pettis integrability of f : I -+ X can be defined equivalently as follows.

Definition 2.2.2. A weakly measurable f : I 4 X with z*(f) Lebesgue integrable for every x* E X* is Pettis integrable if for every measurable E C I there is an element X E E X that satisfies

for every x* E

X*.

It is immediate that if X is a reflexive space (X** = X) then the Dunford and Pettis integrals coincide. If X is not reflexive then they can be different as is shown by the following classical example.

Example 2.2.3. (A function that is Dunford but not Pettis integrable.) Assume that co is the Banach space of real sequences

for which

zn = 0, with the norm

Let us define

f ( t ) = (X(O,lI(t),2 X ( o , ; ] ( t ) ) , . . , n X ( o , $ ] ( t ) , * ) * *

for t E [O,13. Evidently f(0) = 0 = (0, . . . , 0, . . . ) and if t E (0,1] then +] and f(t) = there is an n* E N such that t E ( 1 , 2 , . . . , n*,0, . . . ) for this t E (0,13. The values of f belong t o co.

(A,

Dunford and Pettis Integrals

35

If x* E (co)* then there is a sequence a = ( a n ) E

Z1,

00

n=l

such that

n=l

Then 00

n=l

is a measurable real function, i.e., f is weakly measurable and

n= 1

JU

n=l

Hence f is Dunford integrable while rl

r1

00

and we see (by Lemma 2.1.1 and by Definition 2.1.2) that the Dunford integral is

1'

(27)

f ( t ) d t = (1,1, . . . ) E 1,

= (Zl)* = (co)**.

0

On the other hand, (23)S,'x*(f(t))dtf co and therefore the function f : [0,1] t co is not Pettis integrable.

Example 2.2.4. (The indefinite Dunford integral is not countably additive in general.)

Banach Space Integration

36

Using the function f given in Example 2.2.3 above it is easy to see that

and

Jkf ( t > d t

(27)

=

( 1p p2 . .. , * ,kl , . . . )

€1,

0

for every k E N. Because of this the Dunford integral cannot be countably additive.

2.3

Some properties of the Pettis integral

First of all let us relate the Pettis integral to the Bochner integral described in Chapter 1.

Proposition 2.3.1. I f f : I f i s Pettis integrable and

J

(P> f E

f o r every measurable E

t

=

X i s Bochner integrable t h e n

(a>JE f

(2.3.1)

cI.

Proof. Since f E B let ( f q ) be an L-Cauchy sequence of simple functions determining f (see Definition 1.3.4). Then

and for x* E X * we have

Dunford and Pettis Integrals

37

because

and limq+co(B)J' II(fq - f)lIx = 0 by Corollary 1.3.10. Hence f E P by Definition 2.2.2 and (2.3.1) holds.

0

Theorem 2.3.2. I f f : I -+ X is Pettis integrable define for a measurable set E c I the function

L

v(E7)= ( P )

f

EX

(the indefinite Pettis integral). The function v is countably additive.

Proof. Assume that En En n Em = 8, n # m.

c I, n

E

N

are measurable sets,

Then

n=l

for every x* E X * . This means that v is weakly countably additive, i.e. the series of real numbers Cr=,x*(v(E,))is convergent for every x* E X * . Hence it is also unconditionally convergent (see Proposition B.6) and by Theorem B.5 this means that it is also weakly subseries convergent. The Orlicz-Pettis theorem B.16 yields that the series C,"==, v ( E n ) is unconditionally convergent and henceforth convergent in norm while

n= 1

n=l

Banach Space Integration

38

The theorem is proved.

0

Proposition 2.3.3. Let f : I

-+

X be measurable of the form

(2.3.2) n=l where g : I + X is measurable and bounded, En are paimuise disjoint measurable subsets of I , x n E X , n E N (see Proposition 1.1.9). Then f is Pettis integrable if and only if x, and En, n E N can be chosen such that the series C,"=lX n - p ( E n ) converges unconditionally in X , and in this case we have CO

(P)] f = ( P ) ] E

E

for every measurable E

c

g+Cxn.p(EnEn) n=l I.

(2.3.3)

Proof. Assume that f E P is of the form (2.3.2). Since g is bounded we have g E B c P by Proposition 2.3.1 and therefore also h = f - g = C,"=l xn X E , E P . If E C I is measurable then, because the indefinite Pettis integral is countably additive by Theorem 2.3.2, we have 0

n

0

c 00

n

h=

h=):

x , - p ( E n En).

n= 1

Taking any rearrangement of the series C,"=l the same function h, i.e. 00

h=

00

~ X n ' X E = ,

n=l for any one-to-one map

x , XE,,we obtain

x r ( n ) * XEn(n)

n= 1 7r

of N onto N and of course

co

00

h=

C xn(n) P(E n Er(n)) n=l *

E X-

Hence the series C,"=lx , - p(En) is unconditionally convergent.

Dunford and Pettis Integrals

39

To show the converse let us mention that the function g : I + X being bounded (11g(t)llx5 K for almost all t E I ) is Bochner integrable (see Corollary 1.4.4) and therefore g E P by Proposition 2.3.1. Now it suffices to show that h = - X E , is Pettis xn . p(E,) is unconditionally integrable provided the series convergent in X. Without loss of generality it can be assumed for simplicity that p(En)> 0, n E N. Assume that E c I is measurable. Then the series

xrZ1

E,"==,xn

is unconditionally convergent in X because p ( E n ) 5 1 for all n E N (see Theorem B.5) in Appendix B. If x* E X * then x*(xn) p ( E n En) converges unconditionally in R and therefore by Proposition B.6 from Appendix B we have

C,"=l

+

This means that z*(h)is integrable (see Theorem 1.4.3). Hence

s,

03

C

~ * ( h=)

~t-*(lt.,)

-p

( n~E,) = z*

n=l

and h E P while

This yields f

=g

+ h E P and also the equality (2.3.3).

0

Theorem 2.3.4. Suppose that X does n o t contain subspaces isomorphic to co and let f I 4 X be Dunford integrable. Iff is measurable, then f i s Pettis integrable o n I .

40

Banach Space Integration

Proof. Since f is measurable, we have by Proposition 1.1.9 the relation OD

f =g +x

x n

*

XE,

n=l

where g : I -+X is measurable and bounded, E, are pairwise disjoint measurable subsets of I , xn E X , n E N. Since the interval I is compact, the function g : I + X is Bochner integrable and by Proposition 2.3.1 also Pettis integrable. The Dunford integrability of C,"==,x,x E , yields the Lebesgue integrability of x*(C,"==, x, - xE,) for every x* E X * and we have also z*( x, X E , ) = 00 x*(x, ~ x E , ) because the sets En are pairwise disjoint. Therefore we have

c,"=l

OD

for every x* E X * . This implies that the series C,"=l x, . p(E,) weakly absolutely converges (see Definition B.18 in Appendix

B).

Since X does not contain subspaces isomorphic to C O , by the Bessaga-Pelczyriski Theorem B.22 presented in Appendix B, the series C,"=lxn p(E,) converges unconditionally in X. Hence C,"=lx, - X E , is Pettis integrable by Proposition 2.3.3 and we have f E P . 0 Theorem 2.3.4 shows that the situation presented in Example 2.2.3 is typical. The role of the space co is essential also in another respect.

Theorem 2.3.5. Suppose that X does n o t contain subspaces isomorphic t o co and let f : I + X be Dunford integrable. If (D)JJ f E X f o r every interval J C I t h e n f i s Pettis integrable on I .

Proof. First of all we have the following statement:

Dunford and Pettis Integrals

If J k C I , k E N then (Dl J"Jk f E

iS

a sequence

of

41

non-overlapping intervals

x*

Indeed, we have

with ( D )JJk f E X for all k and

for every x* E X*. Hence the series C k ( D )JJk f weakly absolutely converges. Since X does not contain subspaces isomorphic to co the Bessaga-Pelczyriski Theorem B.22 from Appendix B implies that the series z k ( D )JJk f ) unconditionally converges to a certain element xu Jk E X and (D) Ju Jk f = xu Jk E X. By Theorem (1.11)in [WZ77] every open set in R", m 2 1 can be written as a countable union of non-overlapping (closed) intervals an therefore by the statement above we obtain that (D) S, f E X for every open set G c I . If F c I is closed then I \ F is open (in I ) and

Note that if 2 C I is such that p ( 2 ) = 0 then ( D ) J z f = 0E Let now E c I be an arbitrary measurable set. Then by Theorem (3.28) in [WZ77] we have E = H U 2 where p ( Z ) = 0 and H is of type F,, i.e. H = Hk where Hk C I , k E N are closed. Define L, = Uizl Hk. The sets L, C I are closed and L, C Ln+1, n E N. Set Lo = 0, K, = L, \ L,-l, n E N. Then

x.

uk

Banach Space Integration

42

Kn n Kl

=8

for n # 1 and H

=

U,"==,.Note that

Further

and

for every x* E X*. Similarly as above the Bessaga-Pelczynski Theorem B.22 from Appendix B implies that the series C n ( D )JKn f unconditionally converges to a certain element X H E X and (D) JH f = X H E X. Hence (D) JE f = (D) JH f (D) Jz f E X.

+

0

Using the fact that there are Banach spaces in which an unconditionally convergent series need not be absolutely convergent (see Example B.9 or Corollary B.12 in Appendix B) it is possible to find a series of the form xn - p(En) which converges unconditionally but not absolutely (En and xn for n E N are the same as in Proposition 2.3.3). Hence by Proposition 2.3.3 the function h = C,"=lx, xE, is Pettis integrable while by Proposition 1.4.5 this function cannot be Bochner integrable. In this way, in addition to Proposition 2.3.1 we obtain the following result.

x,"=l

Theorem 2.3.6. W e have 13 C P and the inclusion is proper for general Banach spaces X ,i.e., there exist Banach spaces X

Dunford and Pettis Integrals

and functions f : I Bochner integrable.

--+

43

X which are Pettis integrable but not

Since by Definition 2.2.1 a Pettis integrable function is Dunford integrable, Theorem 2.3.2 shows that d) from Proposition 2.1.3 holds. In this way we arrive at the following result.

Theorem 2.3.7. I f f : I ---f X is Pettis integrable then the following equivalent statements hold: a) The operator T : X* -+ L1 defined by T ( x * )= x * ( f )f o r x* E X* is weakly compact. b) The adjoint operator T* : L , X** to T is weakly compact. c) The set {x*(f ) ; x* E B ( X * ) }c L1 is uniformly integrable, 2. e., --f

d) T h e indefinite Pettis integral u ( E ) given by v ( E ) = f o r E c I measurable, is countably additive, i.e., if En c I , n E N are pairwise disjoint measurable sets then

( P )J'f

in X (the series

C,"=lv ( E n )is norm convergent in X ) .

Note that c) from Theorem 2.3.7 guarantees that for every E > 0 there is an 7 > 0 such that if E c I is measurable with P ( E ) < rl then Ilv(E)llx I E . In connection with Theorem 2.3.7 the following natural problem arises: Is it true that if f : I t X is Dunford integrable and one of the equivalent statements a) - d) of Proposition 2.1.3 is fulfilled then f is Pettis integrable? The answer to this is negative. It was proved by R. Huff in [H86], Proposition 3, that if a Dunford integrable function is given, then it is Pettis integrable if and only if the operator T :

44

X*

Banach Space Integration

t L1 is weakly compact and another condition is satisfied which is not satisfied automatically.

Chapter 3

McShane and Henstock-Kurzweil Integrals

3.1

Systems, partitions and gauges

Let a compact interval I c R", m >_ 1 be given. A pair (7, J ) of a point T E R" and a compact interval J c R" is called a tagged interval, T is the tag of J . Two compact intervals J , L c R" are called non-overlapping if int J n int L = 8 (int J , int L denote the interiors of J , L , respectively). A finite collection { ( T ~I ,j ) , j = 1,. . . ,p} of pairwise nonoverlapping tagged intervals is called an M-system in I if Ij C I for j = 1,.. . , p . An M-system {(~j, I j ) , j = 1,.. . ,p ) in I for which ~j E I j , j = 1,. . . ,p is called a K-system in I . An M-system { ( T ~I,j ) ,j = 1,.. . , p } in I is called an Mpartition of the interval I if P

j=1

Similarly a K-system { ( T ~I,j ) , j K-partition of the interval I if

u

=

1,. . . ,p} in I is called a

P

Ij

= I.

Clearly, every K-system in I is also an M-system in I and 45

Banach Space Integration

46

similarly every K-partition of I is also an M-partition of I . Given a positive function 6 : I + (0, +m) called a gauge on I , a tagged interval (7, J ) is said to be 6-fine if

where B(7,6 ( ~ ) is ) the ball in R" centered at

7

with the radius

6(7)* M- or K-systems or partitions are called 6-fine if all the , j = 1,.. . ,p are &fine with respect to tagged intervals ( ~ jIj), the gauge 6. The following is well-known.

Lemma 3.1.1. (Cousin) Given a gauge 6 : I exists a 6-fine K-partition of I .

3

(0, +oo) there

Lemma 3.1.1 was discovered by P. Cousin and published in 1895 in Acta Maternatica for the case of a twodimensional interval in R2; it was many times rediscovered and can be found in all books concerning Henstock-Kurzweil integration, e.g. [G94], [H88], [H91], [K80], [M97], etc. Since every K-partition is evidently also an M-partition, Lemma 3.1.1 ensures that the set of 6-fine M-partitions of I is also nonempty. This property gives the background for the definitions in the next section.

Remark.

3.2

Definition of the McShane and Henstock-Kurzweil integrals

Assume that a function f : I

--+

X is given.

Definition 3.2.1. f is McShane integrable and J E X is its McShane integral if for every E > 0 there exists a gauge 6 : I + (0, +m) such that for every 6-fine M-partition {(ti,I i ) , i =

McShane and Henstock-Kurzweil Integrals

47

1,. , . ,p } of I the inequality

sIf

holds. We denote J = ( M ) McShane integrable functions.

and M denotes the set of all

Definition 3.2.2. f is Henstock-Kurzweil integrable and J E X is its Henstock-Kurzweil integral if for every E > 0 there exists a gauge 6 : I t (0, +m) such that for every 6-fine K-partition (ti,Ii),i = 1,. . . ,p of I the inequality

sI

holds. We denote J = ('FIX) f and 'FIX denotes the set of all Henstock-Kurzweil integrable functions. If a subset E C I is given, a function f : I + X is called integrable over the set E if the function f X E : I 4 X is integrable. This concept concerns both the Henstock-Kurzweil and the McShane integral. Since every K-partition of I is an M-partition, we have directly from Definitions 3.2.1 and 3.2.2 the following result.

Theorem 3.2.3. If a function f : I + X is McShane integrable then it is also Henstock-Kurzweil integrable and

Remark. In comparison with an M-partition of I , a Kpartition of I imposes a greater restriction by assuming the tag to be in the corresponding interval of the partition. The number of 6-fine partitions is decreased in this way and leads to smaller restrictions on the corresponding class of sum integrable functions. In the case of real functions the Henstock-Kurzweil integral is equivalent to the non-absolutely convergent Perron integral (the

Banach Space Integration

48

narrow Denjoy integral), while the McShane integral coincides with the strictly less general absolutely convergent Lebesgue integral, see e.g. Gordon’s book [G94]. The inclusion M c 7-K given by Theorem 3.2.3 is therefore strict even for the case of real functions.

3.3

Elementary properties of the McShane and Henstock-Kurzweil integrals

Let us start with the following result.

Theorem 3.3.1. Let f : I -+X . I f f = 0 almost everywhere in I then f as McShane integrable on I and ( M ) f = 0.

s,

Proof. Assume that E > 0 is given. Let N = { t E I ; f ( t ) # 0 } and for each n E

Nn

= {t E

N,let

N ; n - 1 5 Ilf(t)llx< n}.

Since p ( N ) = 0, we have also p ( N n ) = 0 for n E N and therefore & there are open sets Gn such that Nn c Gn and p(Gn) < -. n2n Define a gauge S : I -+ (0, +a) in such a way that d ( t ) = 1 for t E I \ N and B ( t ,S ( t ) ) c Gn if t E Nn. Suppose that (ti,Ii),i = 1,. . . , p is a &-fineAd-partition of I. Then

n=l

n=1

McShane and Henstock-Kurzweil Integrals

Hence f : I

t

X is McShane integrable and ( M )JI f

49

= 0.

0

Using Theorem 3.2.3 we can see immediately that the following holds.

Corollary 3.3.2. Let f : I t X . I f f = 0 almost everywhere in I then f is Henstock-Kurzweil integrable o n I and (fix) f = 0. The next set of results will be formulated for the case of the Henstock-Kurzweil integral but all of them hold for the McShane integral as well, it suffices to check their proofs with the necessary replacement of K-partitions by M-partitions, etc.

s,

Theorem 3.3.3. A function f : I t X is Henstock-Kurzweil integrable if and only if for every E > 0 there exists a gauge 6 : I --+ (0, +m) such that for every 6-fine K-partitions { ( t i ,A ) , i = 1,.. . ,p } and { ( s j , J j ) , j = 1,. . . ,r } of I the inequality

holds.

Proof. It is clear that if f E 7-K then for every E > 0 there exists a gauge 6 : I 3 (O,+m) such that for every &fine K partition { ( t i ,I i ) , i = 1,. . . , p } of I we have

Hence

50

Banach Space Integration

and (3.3.1) holds for any 6-fine K-partitions { ( t i , I i ) , i = 1,.. . , p } and { ( s j , J j ) , j = 1,.. . , r } of I . Given E > 0 assume that (3.3.1) holds for any b-fine Kpartitions {(ti,Ii), i = 1,.. . , p } and { ( s j ,J j ) , j = 1,.. . , T } of I . Denote

c k

S(4=

Mf,0) =

f(tz)P(Ji);

i=l

D

=

{(ti,Ji),i = 1,.. . , k } }

cX

where D is an arbitrary 6-fine K-partition of I . The set S ( E )c X is nonempty because by Cousin's lemma 3.1.1 there exists a &-fine K-partition {(ti,Ji), i = 1,.. . , k } of I . Since by (3.3.1) we have

for all 6-fine K-partitions {(ti,J i ) , i { ( s j , L j ) , j = 1,. . . , I} of I , we have also

=

1,. . . , k } and

diam S(E)< E (by diam S(E)the diameter of the set S ( E )is denoted). Further, evidently S(E1)

provided ~1 sponding to set

<

E:!

~ 1 ~2 ,

c S(E2)

because we can choose gauges 61, 6 2 corresuch that 6,(t) 5 &(t)for t E I . Hence the

n E>O

=

sf x

McShane and Henstock-Kurzweil Integrals

51

consists of a single point because the space X is complete (by clS(&)the closure of the set S ( E )in X is denoted). For the integral sum S(f,D ) we get

whenever D = {(ti,Ji), i = 1 , . . . , k } is an arbitrary 6-fine Kpartition of I , and this means that f E 'FIK. 0

Theorem 3.3.4. Assume that f : I t X is HenstockKurzweil integrable and let J c I be a compact interval. Then f is Henstock-Kurzweil integrable over the interval J .

Proof. By Theorem 3.3.3 for any given E > 0 there exists a gauge 6 : I --+ (0, +m) such that for every &fine K-partitions {(ti,Ii), i = 1 , . . . , p } and { ( s j , J j ) , j = 1,.. . , r } of I the inequality (3.3.1) is satisfied. Let { ( ~ i , Ki), i = 1 , . . . , q } and { ( a j L , j ) , j = 1,.. . , s} be arbitrary &fine K-partitions of the interval J . The complement I \ J can be expressed as a finite union of intervals contained in I . Taking an arbitrary 6-fine Kpartition of each of those intervals we obtain a finite collection { (pl, Ml), 1 = 1,.. . , t } of tagged intervals which together with { ( T ~Ki), , i = 1 , . . . , q } or { ( a j L, j ) , j = 1,.. . , s} form two 6-fine K-partitions of the interval I . Taking the difference of the integral sums corresponding t o these two &fine K-partitions of I we can see that its value is 4

S

i= 1

j= 1

because the remaining C:=,f ( p l ) p ( M l )is the same for both of them. Therefore by (3.3.1) we have

52

Banach Space Integration

and this inequality shows by Theorem 3.3.3 the HenstockKurzweil integrability of f on J . Theorem 3.3.5. Assume that J, K c R" are compact intervals such that J U K is again a n interval in R". Assume that f ; J U K -+ X is Henstock-Kurzweil integrable o n each of the intervals J and K. Then f is Henstock-Kurzweil integrable o n the interval J U K. Moreover,

provided the intervals J and K are non-overlapping.

Proof. Let us consider the case when the intervals J and K are non-overlapping. In this case F = J n K is the common face of both intervals J and K in R". By hypothesis, there is a gauge 61 on J and a gauge 62 on K such that for every &-fine K-partition {(ti,J i ) , i = 1,.. . , p ) of J we have

and for every &-fine K-partition { ( s j ,Kj), j = 1,. . . , q } of K we have

For t E J \ F define & ( t )> 0 so that 6,(t)) < dist (t,F ) and similarly for t E k \ F we choose 6,(t) > 0 so that 6,(t)) < dist ( t ,F ) . Define 6 on J U K by min(&(t>,6,(t)) min(6l(t), 6,(t)) min(&(t), 6,(t))

if t E J \ F, if t E F, if t E K \ F.

McShane and Henstock-Kzlrzweil Integrals

53

Let (ti,Ii),i = 1,.. . , T be a 6-fine K-partition of J U K . Consider the tagged intervals (ti,I i ) ,i = 1,.. . , T for which ti E F . Then (ti,Ii n J ) is &-fine and (ti,lin K) is &-fine while for the corresponding term in the integral sum we have

f ( t i ) P ( I i )= f ( t i ) P ( I i n J )

+ f ( t i ) P ( An K).

The system of tagged intervals (ti,I i ) , ti E J , i = 1,.. . , T , ( t i , I i n J ) , ti E F,i = 1 , .. . , T is a h1-fine K-partition of J and the system of tagged intervals (ti,Ii),ti E K, i = 1,.. . , T , (ti,Iin K ) , ti E F, i = 1,.. . , T is a &-fine K-partition of K. Now we have

Banach Space Integration

54

Hence f is Henstock-Kurzweil integrable on J U K and

( H I c )JJ", f

=

( H I c ) JJ f

+ ( H I c ) JK f ' 0

The case when the intervals J and K overlap ( p ( J f? K ) > 0) can be treated in a similar way using Theorem 3.3.3 to obtain the existence of the integral ('FIX) J,, f.

Theorem 3.3.6. Let f , g : I + X be Henstock-Kurzweil integrable o n I , c E R. Then (a) c . f is Henstock-Kurzweil integrable and

(b) f

+g

is Henstock-Kurzweil integrable and

Proof. The statements of the theorem follow easily from the fact that the integral sums for c - f equal c times the integral sums for f and the integral sums for f $ 9 are the sum of integral sums for f and for 9. 0

X be Henstock-Kurzweil inteTheorem 3.3.7. Let f : I grable on I and g : I + X . Iff = g almost everywhere in I , then g : I t X is Henstock-Kurzweil integrable o n I and --$

55

McShane and Henstock-Kurzweil Integrals

Proof. By Theorem 3.3.1 the function g

f is McShane and therefore also Henstock-Kurzweil integrable on I and ('HK)J f ( g - f) = 0, see Corollary 3.3.2. Theorem 3.3.6 (b) yields that g = ( g - f ) f is HenstockKurzweil integrable on I and ('HK)Jf g = (EX) J I ( g - f) -

+

s,

('FIK) f

=

+

(W s, f *

0

3.4

The Saks-Henstock lemma

The following results are crucial for any advanced theory of integration based on Riemann-type integral sums. It is used for proving many results as will be clear in the sequel.

Lemma 3.4.1. (Saks-Henstock) Assume that f : I -+ X is Henstock-Kurzweil integrable. Given E > 0 assume that a gauge 6 on I is such that

every 6-fine K-partition { ( t i J, i ) , i = 1 , . . . , k } of I . Then if { ( r j ,K j ) , j = 1 , . . . , m ) is an arbitrary &-fineKsystem we have

for

Proof. Since { ( r j ,K j ) , j= 1, . . . , m } is a 6-fine K-system the m

complement I

\ U int Kj j=1

consists of a finite system

Ml,

I

=

1, . . . ,p of non-overlapping intervals in I . The function f belongs to 'HK and therefore the integrals ('HK)JMl f exist by Theorem 3.3.3 and, by definition, for any 7 > 0 there is a gauge 61 on Mm with & ( t )< 6(t) for t E Ml such that for every 1 = 1 , . . . , p we

56

Banach Space Integration

have

provided { (sf, J,), i = 1,. . . , kl} is a b,-fine interval Ml. The sum

K-partition of the

represents and integral sum which corresponds to a certain -fine

K-partition of I and consequently by the assumption we have

2f j=1

ccf P

(Tj)P(Kj)

+

kl

/f

( s : ) P ( J ; )- (‘FIX)

1=1 i=l

< E.

I

Hence

< X P

Since this inequality holds for every ately the statement of the lemma.

> 0 we obtain immedi0

If we replace I(-partitions in the proof of Lemma 3.4.1 by M partitions, we obtain the following result for the McShane integral.

McShane and Henstock-Kurzweil Integrals

57

Lemma 3.4.2. (Saks-Henstock) Assume that f : I + X is McShane integrable. Given E > 0 assume that a gauge 6 on I is such that

for every 6-fine M-partition {(ti,Ji), i = 1,.. . , k } of I . Then if { ( r j ,K j ) , j = 1,.. . , m } is an arbitrary 6-fine M system we have

Corollary 3.4.3. I f f : I + X,f E N K , the Banach space X is finite-dimensional and if for a given E > 0 a gauge 6 on I is such that

for every 6-fine K-partition {(ti,J i ) ,i = 1 , .. . , k } of I , then we

have

for an arbitrary 6-fine K-system { ( r j ,K j ) ,j = 1,. . . , m ) . C is

a constant which depends on the dimension of the Banach space X only. The same holds if N K is replaced by M and M-partitions and M-systems are used instead of K-partitions and K-systems. Proof. It is easy to see that there is no restriction in assuming dim X = 1. The more-dimensional case can be treated componentwise.

58

Banach Space Integration

Assume therefore that f : I indices j = 1,. . . , m for which

t

R. Define M+ as the set of

f ( T j ) P ( K j ) - ('FtK) and M- as the set of indices j

=

6,f

20

1,. . . , m for which

Then by the Saks-Henstock lemma 3.4.1 we have

and

Hence

McShane and Henstock-Kurzweil Integrals

59

The constant C for the general case comes from the relation on X and the norm given for between the given norm 1) . example as the sum of absolute values of the coordinates of a point in X. 0

Remark. It should be mentioned at this point that the SaksHenstock lemma 3.4.1 for the Henstock-Kurzweil integral and its analogue 3.4.2 for the McShane integral are presented in a different form from that used sometimes in literature, see e.g. the paper [C92] of S. S. Cao. For real functions (or functions with values in a finite dimensional Banach space) the Saks-Henstock lemma is given usually in the form of our Corollary 3.4.3 and in this form it is presented also for the case of general Banach spaces by many authors. We use the present form because it works well and we present the special form (with the norm inside of the sum) for special purposes below in Section 3.6. Let us show a consequence of the Saks-Henstock Lemma 3.4.1 in the case of a function defined on a one-dimensional interval. Proposition 3.4.4. A s s u m e that f : J --$ X i s a function defined o n a bounded interval J c R of any kind (open, half-open, closed). A s s u m e that f o r any subinterval K c J the integral (‘FIXS, ) f exists. T h e n f o r every E > 0 there exists a function A : J + (0, +GO) such that i f { ( r j ,K j ) , j = 1, . . . , m ) i s a A - f i n e K -s ystern with K j c J for j = 1,. . . , m, then the inequality

holds.

Banach Space Integration

60

Proof. Let

E

> 0 be given. Assume that

Upzl

is a sequence of closed intervals for which we have Jp = J . If t E J and t is not an endpoint of the interval J , then there is an index k ( t ) such that t belongs to the interior of the interval J k ( t ) . If t E J and t is an endpoint of J , pick k ( t ) E N such that t E J k ( t ) ; such a k ( t ) exists because we have U,”=,J p = J . Since by the assumption for every p = 1,2, . . . the integral JJp f exists, by definition for any p = 1 , 2 , .. . there is a gauge Ap : Jp t (0, +m) on J p such that for every A,-fine K-partition (ti,Ii) of the interval J p we have

For t E J choose A ( t )> 0 such that

and

W)) n J c Jk(t).

B(t,

Then for a A-fine K-system { ( r j ,K j ) , j = 1,. . . , m} we have

52

00

&

p= 1

P=l

- & --<&,

2

McShane and Henstock-Kurzweil Integrals

because if Ic(rj) = p , then K j inequality

c B ( r j , A ( r j ) ) C J p and

61

the

immediately follows from the Saks-Henstock Lemma 3.4.1 used for the interval Jp.

Remark. Looking at the proof of Proposition 3.4.4 it is easy b) ---t X is a function defined on to recognize that if e.g. f : [a, a half-open interval [a,b) and if for any c, d E [a,b ) the integral (‘FIX) s,” f exists, then for every E > 0 there exists a function A : [a,b) t ( 0 ,+GO) for which Proposition 3.4.4 holds. Theorem 3.4.5. (Hake) Let [a,b] c R, a < b be a compact interval and let a function f : [a,b] + X be given for which the s,” f exists for every a 5 c < b. Assume that the integral (7-K) limit lim (‘FIX)J, f

c+b-

exists. Then the integral

=A E

X

(‘FIX) Jab f exists and the equality (‘FIX)/

b

f

=A

a

holds.

Proof. Let E > 0 be given. By the definition of the limit there exists a B E ( a , b ) such that

b). for every c E [B, The function f satisfies on the interval [a,b) the assumptions of Proposition 3.4.4 and therefore there exists a A,-, : [a,b) --f

Banach Space Integration

62

(0, +GO) such that if ~1

F r1 F 211 F ~2

5 7-2 < - 02 < - . * . 5 um 5 T m 5

V,

are elements of the interval [a,b) for which

b j , 4c B ( T j , Ao(Tj)) for j

=

1 , .. . ,772, then

Put &

and

for t E [a,b ) . Looking at the choice of the gauge A it is easy to see that for any A-fine K-partition {(ti,[ai-l,ail), i = 1,.. . , k } with a0

< a1 <

* * '

< ak-1 < a k

we have tk = b

and also

ak-1

> B.'

Using the facts presented above we have

..

lNote that this does not hold for the case of A-fine M-partitions.

McShane and Henstock-Kurzweil Integrals

63

for any A-fine K-partition { ( t i , [ai-l,ail), i = 1,. . . , k } of the interval J . This gives by Definition 3.2.2 the existence of the integral ('FIX) s," f as well as the fact that its value is A E X . 0

Remark. An analogous theorem holds also for the case of the b ] ,i.e. limit from the right at the left endpoint of the interval (a, we have the following statement. Let [a,b] c If%, a < b be a compact interval and let a function f : [a,b] -+ X be given for which the integral ('FIX) f exists f o r every a < c 5 b. Assume that the limit

scb

exists. Then the integral ('FIX) f exists and the equality

s,"

( ' F I X )a[ f = A holds. The proof of this theorem can proceed analogously to the case of Theorem 3.4.5. Hake's Theorem 3.4.5 does not hold for the McShane integral. For the case X = JR functions which are Henstock-Kurzweil integrable but not Lebesgue (=McShane) integrable are examples for this. Theorem 3.4.5 is one of the most important differences between the Henstock-Kurzweil and McShane integrals.

64

Banach Space Integration

Similar results are possible also for the case of moredimensional intervals I c R", m > 1. They are rather technical and we are not presenting them here.

3.5

A convergence theorem

This section is devoted to a convergence result for our sum integrals defined in Definitions 3.2.1 and 3.2.2. It is based on the observation that defining an integral is in fact a certain limiting procedure. Convergence theorems for the integral concern the possibility of interchanging the limit and the integral. From classical calculus it is known that this can be done if one of the limiting processes is uniform with respect to the other one. The point is to consider pointwise convergent sequences fk(t) t f ( t ) , k t 00, t E I and to fix the meaning of uniformness of the integration process with respect to k . In a printed book form first results of this type had been presented in [K80] and in the evidently independent book [McL80]. Let us start with the following.

Definition 3.5.1. A collection M of functions f : I t X is called 'FIX-equi-integrable (M-equi-integrable) if every f E A4 is Henstock-Kurzweil integrable (McShane integrable) and for any E > 0 there is a gauge A such that for any f E M the inequality

holds provided {(ti,Ii), i (M-partition) of I .

=

1 , . . . , p } is a A-fine K-partition

Theorem 3.5.2. A s s u m e that M = {fk : I t

X ;k

E

N} is

McShane and Henstock-Kurzweil Integrals

65

an XK-equi-integrable sequence such that lim

fk(t) =

k+cc

Then the function f and

;

I

---f

f(t),t E I .

X is Henstock-Kurzweil integrable

holds.

Proof. If A is the gauge from the definition of equiintegrability of the sequence (fk) corresponding to the value E > 0 then for any k E N we have

for every A-fine K-partition {(ti,I i ) , i = 1,.. . , p } of I . If the partition {(ti,Ii), i = 1,. . . ,p } is fixed then the pointwise convergence f k + f yields P

P

Choose ko E N such that for k > ko the inequality

i=l

holds. Then we have

IIX

66

Banach Space Integration

for k > ko. This gives for k , 1 > ko the inequality

which shows that the sequence ('FIX)JI X is Cauchy and therefore

fk,

k E

N of elements of

r

fk =

J E X exists.

(3.5.2)

Let E > 0. By hypothesis there is a gauge A such that (3.5.1) holds for all k G N whenever {(ti, Ii),i = 1,. . . , p ) is a A-fine K partition of I. By (3.5.2) choose N E N such that II('FIX) JI fk Jllx < E for all k 2 N . Suppose that {(ti,Ii),i = 1,.. . , p } is a A-fine K-partition of I. Since fk converges to f pointwise there is a kl 2 N such that

/I i=l

i=l

IIX

Therefore

i=l

11

i= 1

IIX

P

and it follows that f is Henstock-Kurzweil integrable on I and 0 limk,,('FtK) J, fk = J = ('FIX) J, f. The following McShane variant of Theorem 3.5.2 follows analogously.

McShane and Henstock-Kurzweil Integrals

Theorem 3.5.3. Assume that Ad = {fk : I an M -equi-integrable sequence such that lim

k+cc

Then the function f : I

fk(t) =

-+

67

X ; k E N} is

-+

f ( t ) ,t E I .

X is McShane integrable and

holds. Theorem 3.5.2 and its McShane variant 3.5.3 represent convergence results the power of which is not very clear at this moment. Nevertheless, it should be emphasized that the proof of these theorems is rather elementary. For the case of the McShane integral we will come back to this topic in Section 4.3.

Proposition 3.5.4. A function f : I + X is HenstockKurzweil integrable (McShane integrable) if and only if the set {x*(f); x* E B ( X * ) } is 'FIX-equi-integrable (M-equiint egra b 1e) .

Proof. I f f is Henstock-Kurzweil integrable then for every E > 0 there is a gauge 6 : I t (0, +m) on I such that

for every b-fine K-partition {(ti,I i ) } of I . For an arbitrary x* E X * we have

and therefore {x*(f); x* E B ( X * ) }is 'FIX-equi-integrable.

68

Banach Space Integration

If on the other hand {z*(f);x* E B(X*)}is 'FIX-equiintegrable then for every E > 0 there is a gauge 6 : I -+(0, +GO) on 1 such that

for every 6-fine K-partition {(ti,I i ) } of I and x* E B ( X * ) . Hence if { ( t i , I i ) } ,{(sjlJ j ) } are 6-fine K-partitions of I we get

I

for every z* E B ( X * ) .Hence

and by Theorem 3.3.3 the function f is Henstock-Kurzweil integrable. The McShane variant of the proposition can be proved analogously. 0 Concerning the concept of an equi-integrable collection given by Definition 3.5.1 let us note that we have the following result which represents a certain Bolzano-Cauchy condition for equiintegrability of an equi-integrable collection A4 of functions f : I --+ X. The similarity to Theorem 3.3.3 is evident.

Theorem 3.5.5. A collection M of functions f : I + X is ('FIK- or M - ) equi-integrable if and on13 iffor every E > 0 there

McShane and Henstock-Kurzweil Integrals

exists a gauge A : I

-+

69

(0, +GO) such that

for every A - f i n e (K- or M - ) partitions { ( t i ,I i ) , i = 1 , . . . , p ) and { ( s j , J j ) , j = 1 , . . . , r } of I and every f E M .

Proof. If M is equi-integrable then the condition evidently holds for the gauge A which corresponds to > 0 in Definition 3.5.1 of equi-integrability. If the condition of the theorem is satisfied then every individual function f E M is (RKor M-)integrable with the same gauge A for a given E > 0 independently of the choice of f E M (cf. Theorem 3.3.3 and its proof) and this proves the theorem. 0

Let us close this section by an analogue of the Saks-Henstock Lemma 3.4.1 or 3.4.2 which holds for equi-integrable collections 0ff:I-X.

Lemma 3.5.6. (Saks-Henstock) A s s u m e that a n ‘FIX-equiintegrable (M-equi-integrable) collection M of functions f : I --+ X is given. Given E > 0 assume that the gauge A o n I is such that

for every A - f i n e K-partition (121-partition) { ( t i , J i ) , i

1, . . . , k ) o f I .

=

Banach Space Integration

70

Then if { ( r j 7K j ) , j = 1,.. . , m ) is an arbitrary 6-fine K system (M-system) we have

for any f E M ,

For the proof of this statement the proof of Lemma 3.4.1 can be repeated word for word.

3.6

The strong versions of the McShane and Henst ock-Kurzweil integrals

Let Z denote the family of all compact subintervals J function F : Z--+ X is said to be additive if

c I. A

F ( J u L ) = F ( J )+ F ( L ) for any non-overlapping J , L E Zsuch that J U L E Z.

Definition 3.6.1. A function f : I --+ X is said to be strongly Henstock-Kurzweil integrable on I if there is an additive function F : Z --+ X such that for every E > 0 there exists a gauge 6 on I such that k i=l

for every &fine K-partition {(ti,J i ) , i = 1,. . . , Ic} of I . Denote by S H X the set of functions f : I + X which are strongly Henstock-Kurzweil integrable on I .

71

McShane and Henstock-Kurzweil Integrals

Definition 3.6.2. A function f : I + X is said to be strongly McShane integrable on I if there is an additive function F : Z-+ X such that for every E > 0 there exists a gauge 6 on I such that k

i=l

for every 6-fine M-partition { ( t i , J i ) , i = 1 , .. . , k } of I . Denote by S M the set of functions f : I + X which are strongly McShane integrable on I . Let us note that in [SS98] V. Skvortsov and A. Solodov define the Henstock and McShane variational integrability of functions f : I + X and this notion coincides with our strong Henstock-Kurzweil and McShane integrability from the previous Definitions 3.6.1 and 3.6.2. This concerns also many other papers devoted to this topic. From the Definitions 3.6.1 and 3.6.2 and from the fact that every K-partition is also an Ad-partition we get immediately the following.

Remark.

Proposition 3.6.3. If f : I -+ X is strongly McShane integrable then it is strongly Henstock-Kurzweil integrable, i.e S M c SIFIK. Later we will show that in general the inclusion S M C SNK: is proper, i.e. for any infinite dimensional Banach space X there is a function f : I -+ X for which f E SIFIK but f $! S M (see the examples in 5.4). In addition to Theorem 3.3.1 let us present the following.

X we have f = 0 almost Theorem 3.6.4. If for f : I everywhere in I then f is strongly McShane integrable and consequently also strongly Henstock-Kurxweil integrable. Proof. By Theorem 3.3.1 the real function llfllx : I McShane integrable and ( M )J, llfllx = 0.

+

IW is

Banach Space Integration

72

This means that for every such that

E

> 0 there exists a gauge 6 on I k

k.

i=l

i=l

for every 6-fine M-partition {(ti,Ji),i = 1,. . . , k } of I and therefore f is strongly McShane integrable with the additive function F ( J ) = 0 for every J E 1. 0

Theorem 3.6.5. Iff : I -+ X is strongly Henstock-Kurzweil (McShane) integrable on I then it is Henstock-Kurzweil (McShane) integrable on I and ('HK)j, f = F ( I ) ( ( M ) f = F ( 1 ) ) where F is the additive function from the definition of strong integrability.

sI

Proof. The statement follows easily from the inequality

i=l

which holds evidently for every K-partition (M-partition) {(ti, Ji)} of I . n U

Remark. It is easy to see that a Henstock-Kurzweil (McShane) integrable function is strongly Henstock-Kurzweil (McShane) integrable if for every E > 0 there is a gauge 6 on I such that

McShane and Henstock-Kurzweil Integrals

73

for every &-fineK-partition (Ad-partition) {(ti,J i ) } of I . In other words F ( J ) = ('FtlC) JJ f ( E ( J )= ( M )JJ f) is the additive interval function for defining the respective strong int egrability. Using this simple observation the following result is easy to prove.

Proposition 3.6.6. If the Banach space X is finitedimensional then a function f : I + X is Henstock-Kurxweil (McShane) integrable if and only i f it is strongly HenstockKurzweil (McShane) integrable. Proof. The fact that the strong version of integrability implies integrability is stated in Theorem 3.6.5. Since X is finite-dimensional and f is assumed to be e.g. Henstock-Kurzweil (McShane) integrable then Corollary 3.4.3 implies that for every E > 0 there is a gauge 6 on I such that

for every 6-fine K-partition (Ad-partition) {(ti,Ji)} of I and the proposition is proved.

Definition 3.6.7. A function f : I + X has the property S*M (S*'FtlC) if for every E > 0 there is a gauge 6 on I such that k

1

for any &fine Ad-partitions (K-partitions) { ( t i ,Ji),i = 1,. . . , k} and { ( s j , L j ) , j = 1,.. . , Z} of I . Similarly as in the case of Proposition 3.6.3 we obtain

Proposition 3.6.8. I f f : I it has the property S * X K .

+X

has the property S*M then

74

Banach Space Integration

Theorem 3.6.9. If a function f : I -+ X has the property S*M (S*'FIIc) then f is McShane (Henstock-Kurxweil) integrable.

Proof. If {(ti,Ji), i = 1,.. . , Ic} and { ( s j , L j ) , j are &fine M-partitions (K-partitions) of I we have

=

1 , .. . ,I}

1

P ( J d = CP(JZ n Lj) j=1

and k

P P j ) = CPVi f-l 4) i=l

Hence 1

1

k

and by Definition 3.6.7 and Theorem 3.3.3 this yields the state0 ment .

Corollary 3.6.10. If a function f : I + X has the property S*M then the real function 1) fllx : I + R is McShane integra bl e .

McShane and Henstock-Kurzweil Integrals

If a function f : I

75

has the property S*'FIK:then the real is Henstock-Kurzweil and therefore also

+X

function (1 f I(x : I + Iw McShane integrable.

Proof. Since

IIlf(ti>Ilx - Ilf(sj>llxl5 Ilf(ti>- f(%) IIX for any choice of t i , s j E I we can see that the function llfllx : I R has the property S*M (S*'FIIc)and Theorem 3.6.9 implies its McShane (Henstock-Kurzweil) integrability. The second part of the corollary follows from the well-known fact that every nonnegative Henstock-Kurzweil integrable function is also McShane integrable. (See also the remark after Theorem 3.6.13.) -+

Remark. It is worth to mention that the condition S*'FIK: implies Henstock-Kurzweil integrability of f (Theorem 3.6.9) as well as of 11 f IIx (Corollary 3.6.10) an it represents a certain "absolute integrability" condition for f : I + X. Using Theorem 1.4.3 and the McShane integrability of 11 f I ( x provided f has the property S*M or S*'FIK:we know that for a measurable f we have f E B in this cases. This shows how far the conditions S*M and S*?-IK go what it concerns the above mentioned absolute integrability. Lemma 3.6.11. I f f : I + X has the property S*M (S*'FIK) then it is strongly McShane (Henstock-Kurzweil) integrable on T 1.

Proof. We prove the lemma for the McShane case only. It can be checked easily that the proof of the Henstock-Kurzweil case remains the same with appropriate changes (K-partitions instead of M-partitions, 'FIX instead of M ) . I f f : I -+ X has the property S*M then, by Definition 3.6.7, for every E > 0 there is a gauge 6 on I such that k

I

76

Banach Space Integration

for any two &fine Ad-partitions {(ti,Ji), i = 1,. . . , Ic} and { ( S j , L j ) , j = 1,.. . , 1 } of I . Assume that {(ti,J i ) , i = 1,.. . , k ) is an arbitrary &fine Mpartition of I . By Theorem 3.6.9 we have f E M and therefore f is McShane integrable on every interval Ji, i = 1,.. . , k by the McShane variant of Theorem 3.3.4. Hence for the given E > 0 there is a gauge 6’ on I such that 6’(t) 5 6 ( t ) for t E I and such that for any #-fine M-partition { (s:z), L:!)),j = 1,.. . , of the interval Ji we have

Note that { (s:), L:)), j = 1,.. . , I(i), i = 1 , .. . , k } is a b-fine M-partition of the interval I and that for any i = 1,.. . , k we have l(4

f ( t i ) P ( J i )=

c

f ( t i ) P ( J i n L:))

j=1

and, because of the additivity of the indefinite integral F ( J ) = ( M )J J f,also

Hence

77

McShane and Henstock-Kurzweil Integrals

i=l ) ) j = 1

llx

&

k

i=l

This shows that f is strongly McShane integrable on I . 0

78

Banach Space Integration

Lemma 3.6.12. If a function f : I + X is strongly McShane integrable on I then it has the property S*M.

Proof. By definition for every

E

> 0 there is a gauge 6 on I

such that

for every &-fineM-partition { ( t i ,J i ) , i = 1,.. . , k } of I where F is the additive interval function from Definition 3.6.2. If we have two 6-fine M-partitions { ( t i ,J i ) , i = 1,.. . , k } and { ( s j ,L j ) , j = 1,.. . ,I} of I then k

k

1

l

k

k

k

1

1

1

because evidently { ( t i ,Ji n L j ) , i = 1,.. . , k , j = 1,.. . , 1 ) and { ( s j ,Jin L j ) ,j = 1,.. . ,1, i = 1,.. . , k ) are 6-fine M-partitions of I .

McShane and Henstock-Kurzweil Integrals

Hence f has the property S*M.

79

0

Remark. Note that the reasoning used to prove Lemma 3.6.12 cannot be used for strong Henstock-Kurzweil integrability and the property S*'FIK. Indeed, the strong Henstock-Kurzweil integrability of a function f does not imply that it has the property S*IFIIc. To see this take a function f : [0,13 R for which f E 'FIX but If 6 RK. (The classical example of such a function f : [0,1] + R is given by f ( t ) = 2tcos($) sin($) for t E (0,1], f(0) = 0. The function f is the derivative of the function F ( t ) = t2cos($), t E (0,1], F ( 0 ) = 0 and f is therefore Henstock-Kurzweil integrable but 1 f 1 $ 'FIX (see e.g. [LPY89], etc.).) Since f E 'FIX by definition for every E > 0 there is a gauge 6 : [0,1] --+ (0, m) such that for every &fine K-partition (ti,Ii),i = 1,.. . , p of [0,1] the inequality

I

--f

+9

holds. Using the Saks-Henstock lemma 3.4.1 in a straightforward way we get

for every 6-fine K-partition (ti,Ii),i = 1,.. . , p of [0,1] and this means that f is strongly Henstock-Kurzweil integrable. If f would have the property S*ZK then I f 1 would be Bochner (=Lebesgue) integrable by Corollary 3.6.10 but this is not the case. Hence f cannot have the property S*'FIK. Using Lemmas 3.6.12 and 3.6.11 we immediately obtain the following result characterizing strong McShane integrability. Theorem 3.6.13. A function f : I -+ X has the property S*M if and only if it is strongly McShane integrable on I .

80

Banach Space Integration

Remark. For real valued functions f : I -+ R the following is known (see e.g. [LVOO], Lemma 3.12.3): Iff : I R is McShane integrable then the function f possesses the property S * M . Using this, by Theorem 3.6.13 we get: G3 Every McShane integrable function f : I R is strongly McShane integrable and Theorem 3.6.5 yields G3 f : I -+R is a McShane integrable function if and only af it is strongly McShane integrable. In [Sol] the following well known result is shown in an elegant way based on an idea of R. Vfbornf: G3 f : I R is a McShane integrable function i f and only if f and 1 f 1 are Henstock-Kurzweil integrable. Using Lemma 3.6.12 this result leads to the following simple conclusion: ?3 If for f : I -+ X the real function llfllx : I -+ R is Henstock-Kurzweil integrable, then 11 f IIx is also McShane integrable. The converse implication t o this statement is clear by definition (and also from the previous statement above). --f

--f

--f

Theorem 3.6.14. If a function f Shane integrable on I then

:

I

-+

X is strongly Mc-

Proof. Theorems 3.6.13 and 3.6.9 imply the McShane integrability of f and Corollary 3.6.10 yields the McShane integrability of Ilf IIx. Let E > 0 be given. Then by Definition 3.2.1 there is a gauge 6 : I -+ ( O , + o o ) such that for every 6-fine M-partition (ti,Ii),i = 1,.. . , p of I the inequalities

UcShane and Henstock-Kurzweil Integrals

81

and

hold. For a fixed &fine Ad-partition (ti,Ii),i obtain

=

1,. . . ,p of I we

a= 1

SEt

Since E > 0 can be chosen arbitrarily small we obtain the statement of the theorem. 0

Looking at the definitions of the strong Henstock-Kurzweil an McShane integrals it is easy to see that the following variant of the Saks-Henstock lemma for the strong version of the integrals holds. Lemma 3.6.15. (Saks-Henstock) Iff : I -+ X is strongly Henstock-Kurzweil integrable (strongly McShane integrable) then

Banach Space Integration

82

to every E > 0 there is a gauge 6 on I such that i f { ( r j ,K j ) ,j = 1,.. . , m ) is an arbitrary 6-fine K-system (M-system) we have m

j=1

where F(Kj) = ( H K )

sKjf ( F ( K j )

=

sKjf).

(M)

Proposition 3.6.16. Assume that f : I 3 X is strongly Henstock-Kurxweil integrable (McShane integrable) with the additive interval function F : Z+ X . Then

for every J E Z

Proof. If f : I t X is strongly Henstock-Kurzweil integrable and J E Zassume that E > 0 is given. Let S be the gauge on I given by Lemma 3.6.15. If ( ( t i J, i ) } is an arbitrary b-fine K-partition of the interval J , then by Lemma 3.6.15 we have

s,

sJ

and this shows that ( H K ) f exist and ('FIK) f = F ( J ) . The proof for the case of strong McShane integrability is the same; note that any M-partition is automatically also a K-partition. 0 By Proposition 3.6.3 and Theorem 3.6.5 we have Remark. the inclusions

S M c SRK c H K

McShane and Henstock-Kurzweil Integrals

83

while Theorem 3.6.5 and Theorem 3.2.3 give

S M c M c 'FIX. Therefore functions f : I -+ X belonging to S M , SEX or M are all Henstock-Kurzweil integrable. Note that by Theorem 3.3.5 the interval function

J

E Z-+

('FIX)

1

f EX

(3.6.1)

J

is additive. Proposition 3.6.16 shows that in Definitions 3.6.1 and 3.6.2 the additive function F : Z -+ X can be replaced by ('FIX)JJ f for J E Z. According to this the special cases S M , S'FIX and M represent additional properties to the indefinite integral (primitive) (3.6.1) of a function which belongs to 'FIX. We will have a closer look at this in Chapter 7. We close this section by some convergence results for the strong versions of integrals. They follow the lines of Theorems 3.5.2 and 3.5.3.

Definition 3.6.17. A collection M of functions f : I -+ X is called strongly 'FIX - equi-integra ble (strong1y M - equi-integra ble) if every f E M is strongly Henstock-Kurzweil integrable (strongly McShane integrable) and for any E > 0 there is a gauge A such that for any f E M the inequality

i=l

holds provided { ( t i , Ii), i = 1,. . . , p } is a A-fine K-partition (M-partition) of I , F is the additive X-valued interval function corresponding to f E M . Theorem 3.6.18. A s s u m e that M = { f k : I -+X ;k E N} i s a strongly 'FIX-equi-integrable sequence such that lim

k+m

fk(t) =

f(t), t E I

84

Banach Space Integration

T h e n the function f : I integrable and

+

X is strongly Henstock- Kurzweil

lim Fk(I) = F ( I )

k-+w

holds. Fk, F are the additive X -valued interval functions corresponding t o fk and f , respectively. Proof. We present a sketch of the proof only. First let us observe that the strong 7%-equi-integrability given in Definition 3.6.17 implies immediately the 7-K-equi-integrability in the sense of Definition 3.5.1. Theorem 3.5.2 implies the Henstock-Kurzweil integrability of f as well as the relation limk-.+wF k ( J ) = F ( J ) for every interval J C I . Let E > 0 be given and letA be the gauge from the definition of strong 7-K-equi-integrability of the sequence f k . Suppose that { ( t i ,Ii),i = 1,. . . ,p } is an arbitrary A-fine K-partition of I and consider the sum P

P

Taking

so large large that that so

P

i=l

we obtain

McShane and Henstock-Kurzweil Integrals

85

and the strong Henstock-Kurzweil integrability of f is proved. 0

Remark. Analogously a similar convergence result for the strong McShane integrability of f can be proved.

3.7

Integration over unbounded intervals and some remarks

In this work we consider the case of functions defined on compact intervals in Iw" only. For the Bochner integral (Chap. 1) and the Pettis integral (Chap. 2) the usual approaches known from Lebesgue's integration theory can be used for defining the integral of a function defined and integrated over an unbounded (closed) interval I c R". In this section we recall some possibility of defining the Henstock-Kurzweil and McShane integrals for functions f : I + X where the interval I c R" is unbounded. The next definition follows the lines presented by C.-A. Faure and J. Mawhin in [FM97].

Definition 3.7.1. Let 1 c Iwm be an unbounded interval and f :I--tX. f is Henstock-Kurzweil integrable and J E X is its HenstockKurzweil integral if for every E > 0 there exists a gauge 6 : I -+ (0, +oo) and a compact interval L c I such that

cf P

II

(ti)P(Q

-

Jllx

<E

i=l

for every &fine K-partition ( t i ,Ii),i interval H c R" for which L c H and P

i=l

=

1,. , , , p of any compact

Banach Space Integration

86

for every b-fine K-partition ( t i ,Ii), i = 1,. . . ,p of any compact interval K C Rm which does not overlap with L. We denote J =

('FIX) s, f

*

If we replace K-partitions by M-partitions in this definition we get a definition of the McShane integral of f . A similar idea of defining sum integrals over unbounded intervals is given also by c h . Swartz in the books [Sol], [KS04] or in Part 2 (Chapters 15-17) of R. G. Bartle's book [BaOl] for the case m = 1. We are not going into details with this concept. We recall only that the integrals defined in this way have all the basic properties described in the previous sections in this chapter. Some results are given also in the paper [FM97] by C.-A. Faure and J. Mawhin.

Concluding remarks. Looking at the Definitions 3.2.1, 3.2.2, 3.6.1, 3.6.2 it can be seen easily that it is possible to present similar definitions also for functions assuming values in linear spaces with some topological structure. In this book we restrict ourselves to Banach spaces and we note there that e.g. there is a series of papers written by Valeria Marraffa [MV04], [MV04a], [MV04b] concerning the case of functions with values in locally convex topological spaces, F'rkchet spaces, etc. Special attention is paid to spaces when the strong version of Saks-Henstock Lemma 3.6.15 holds with the norm replaced e.g. by semi-norms or topologies. A great deal was done by S. Nakanishi in a series of papers, let us mention at least her paper "941 as a specimen for the effort in this direction showing that the strong version of the Saks-Henstock Lemma holds if the range space of a function is endowed with nuclearity. The strong integrability concepts are shifting the integrations based on Riemann type sums toward concepts which are similar to Lebesgue (=McShane) and Henstock-Kurzweil integrations for real functions. We will come back to this in Chapter 5.

Chapter 4

More on the McShane Integral

4.1

Special properties

-

Theorem 3.3.4 shows that if f : I X is McShane integrable then f is McShane integrable over every subinterval J c I . From this it follows that if E c I is a finite union of nonoverlapping intervals contained in I then a McShane integrable f : I -+ X is integrable over E . Lemma 4.1.1. Assume that f k : I t X , k E N are McShane integrable functions such that 1. f k ( t ) f ( t )f o r t I , 2. the set { f k ; k E N} forms an M-equi-integrable sequence. Then for every E > 0 there is an q > 0 such that for any finite collection { J j : j = 1,.. . , p } of non-overlapping intervals an I with p ( J j ) < 7 we have

c,"=,

for every k E

N.

Proof. Let E > 0 be given. Since fk are M-equi-integrable on I , there exists a gauge S on I such that

87

Banach Space Integration

88

whenever { ( t i ,Ii); i = 1,. . . , q } is an arbitrary 6-fine M partition of I and k E N. Fixing a &fine M-partition of I

{(ti,Ii);i = 1,.. . , q } let ko E

N be such that IIfk(ti) -

fWllx < E

for k > ko; put L = max{llf(ti)llx, IIfrc(ti)llx;1 5 i I q, k I ko} E and set 77 = L + 1' Suppose that { J j , j = 1 , .. . , p } is a finite family of nonoverlapping intervals in I such that C;=,p ( J j ) < q. By subdividing these intervals if necessary, we may assume that for each j , J j C_ Ii for some i. For each i E N,1 5 i 5 q let Mi= { j ;1 5 j 5 p with J j C Ii} and let

D

= {(ti,J j ) ; j E

Mi,i= 1,.. . , q } .

Note that D is a 6-fine M-system in I . Using the variant form of the Saks-Henstock Lemma 3.5.6we get

+ +

I E (L

P E)

Cp(Jj) <E

+ ( L + E)V < ~ ( +2 ~+1) E

j=1

for every k E N and this proves the lemma.

0

For the special case of fk = f, k E N, where f is McShane integrable we obtain immediately the following result.

More on the McShane Integral

89

Lemma 4.1.2. If f : I + X is McShane integrable on I , then for every E > 0 there is an 7 > 0 such that for any finite collection { Jj : j = 1, . . . ,p } of non-overlapping intervals in I with C;=,p( J j ) < q we have

Lemma 4.1.3. I f f : I t X is McShane integrable then (a) for any sequence {Ii : i E N} of non-overlapping intervals Ii c I , i E N the limit

exists, (b) for every E > 0 there is an 7 > 0 such that if the sequence {Ii : i E N} of non-overlapping intervals Ii c I satisfies c:,P(Ii) < 7 , then

Proof. Assume that E > 0 is given. Let 7 > 0 correspond to E by Lemma 4.1.2. Since C,"=,p(Ii)5 p ( I ) < a, there is an N E N such that for n > N we have C,"=, p(Ii) < 7 . Assume that n, m E N,N < n < m. Then by Lemma 4.1.2 we have

c:,,,

c,"=,+1

because p(I2) 5 P(Ii) < 7. This implies that CY=l(M)JIi f , n E N is a Cauchy sequence in X and (a) is proved.

90

Banach Space Integration

by Lemma 4.1.2 Since by (a) the series C,”=,(M) JIi f converges in X , we obtain

and (b) is proved.

0

Notation. To simplify writing from now we will use the notation { (ul, Ul)}for M-systems instead of {(ul, Ul);1 = 1,. . . , r } which specifies the number r of elements of the M-system. For a function f : I + X and an Ad-system {(ul, Ul)}we write Cl f ( U l ) P ( U l ) instead of f (.l)P(W etc.

xi=’=,

Lemma 4.1.4. Assume that f k : I t X , k E N are McShane integrable functions such that 1. f k ( t ) f ( t )f o r t E I , 2. the set { f k ; k E N} forms an M -equi-integrable sequence. Then for every E > 0 there exists an 7 > 0 such that (a) if F is closed, G open, F c G c I and p(G\ F ) < 7 then there is a gauge E : I -+ (0, m) such that -+

B ( t , J ( t )c) G for t E G , B ( t ,[ ( t ) )n I

cI\F

for t E I

\F

and (b) for any E-fine M-systems { (ul, Ul)},{(urn,Vm)}satisfying uZ,umE G, F

c Int

u 211 EF

Ul, F

c Int

u V,EF

V,

More on the McShane Integral

91

we have

I& (4.1.1) for every k E N.

Proof. Denote

@k(J) =

sJ

(M)

fk

for an interval J

c I (the

&

indefinite integral or primitive of f k ) and put 2= -. 10 Since f k are M-equi-integrable, the Saks-Henstock lemma 3.5.6 implies that there is a gauge A on I such that

II C [ f k ( r j ) C L ( w- @ k ( W l I l X

I 2

(4.1.2)

j

for every A-fine M-system { ( r j ,K j ) } and k E N. Assume that

{ (wp, W,)} is a fixed A-fine M-partition of I .

(4.1.3)

Let ko E N be such that IIfk(WP) -

f(wp)llx

<1

for k > ko and all p . Put m a x i 1 + Ilf(Wp)llX,

IIfrc(W,)IIx~*

(4.1.4)

IIfk(w,)llx < K for all k E N and p .

(4.1.5)

=

p,Kko

Then

Assume that 7

> 0 satisfies T p K I 2

(4.1.6)

0 < [ ( t )I A@), t E I .

(4.1.7)

and take

<

Since the sets G and I \ F are open, the gauge can be chosen such that B ( t ,[ ( t ) )c G for t E G and B ( t ,[ ( t ) )n I c I \ F for tEI\F.

92

Banach Space Integration

This is part (a) of the lemma and now we will show part (b). Since { (wp,W p ) }is a partition of I , we have U pWp= I and therefore

c

fk(Ul>P(Ul>

(4.1.8)

1

and similarly (4.1.9)

The M-syst ems

More on the McShane Integral

are A-fine and therefore, by (4.1.2), we have the inequalities

CC C p

fk(ul)p(WpnulnVm)

1,ulEF m , u m € F

n

--@k(WP Ul

n Vm) 5 F, X

Hence

and similarly also

Therefore

93

94

Banach Space Integration

-cc 1 p

fic(~m)P(Wpn~lnVm5 ) 4z

(4.1.10)

1,uLEF m,vmEF

Since system with we obtain by the properties of the gauge given in (a) and from the assump-

Further, the M-systems

Therfore by (4.1.2) we have

More on the McShane Integral

95

This yields

X

5 2z By virtue of (4.1.11) and (4.1.6) we have

X


and therefore

and similarly also

Banach Space Integration

96

From (4.1.8), (4.1.9),(4.1.10),(4.1.12) and (4.1.13) we get

and (4.1.1) is satisfied. This proves part (b) of the lemma.

0

page intentionally left blank As an immediate This corollary of Lemma 4.1.4 we obtain

Lemma 4.1.5. Assume that f : I + X is McShane integrable. Then for every E > 0 there exists an q > 0 such that (a) if F is closed, G open, F c G c I and p(G\ F ) < q then there is a gauge : I t (0, co) such that

B ( t , [ ( t ) )c G for t E G , B ( t , [ ( t ) )n I

cI\F

and (b) for a n y (-fine M-systems ( u1,Vm

E G, F

c Int

for t E I

\F

( ~ 1 ,Ul)},

U

~

1

F,

c

uleF

we have

5 E.

(4.1.14)

Proof. The lemma follows immediately from Lemma 4.1.4 if we put f k = f for every Ic E N. 0 This rather technical Lemma 4.1.5 enables us now to show easily that a McShane integrable function f : I + X is McShane integrable over each measurable subset E c I . Theorem 4.1.6. I f f : I --+X is McShane integrable then f - X E is McShane integrable for every measurable set E c I ( f is McShane integrable over E).

97

More on the McShane Integral

Proof. Let E > 0 be given and let q > 0 correspond to E by Lemma 4.1.5. Assume that E c I is measurable. Then there exist F c I closed and G c I open such that F c E c G where p(G \ F ) < q. Assume that the gauge ( : I -+ ( 0 , ~ ) is given as in Lemma 4.1.5 and that { ( U I , U l ) } , { ( U r n , Vm)} are &fine M-partitions of 1. The following implication holds: if

u1 E

E then Ul c G and F C int

u

Ul.

uiEF

(By the properties of the gauge [ we have UZc I \ F if ~1 $ F , u u ,EI\F Uz c I\F. If t E F then t $ 1\F and t f UuIET,F Ul. Hence t E UulEF Uz.) Similarly,

i'e'

if urn E E then Vmc G and F C int

u

V,.

v,EF

Hence by (4.1.1) from Lemma 4.1.5 we have

and therefore also

By the McShane version of Theorem 3.3.3 we can see that the 0 McShane integral ( M )JT f - X E = ( M )JE f exists.

Remark. Theorem 4.1.6 was proved in [FM94] (2E Theorem) by a different approach for the case when I C R. It shows that the McShane integral behaves like the Lebesgue integral even in the case of a Banach space valued function. Our proof based on Lemma 4.1.5 follows the method presented in [KS03a] and [KS03b].

98

Banach Space Integration

Note that an analogue of Theorem 4.1.6 for the HenstockKurzweil integral does not hold. To see this the classical example of the real function f : [0,1] --+ R, defined by 7l

f ( t ) = 2tcos(-)

t2

27r + -sin(-) t t2 7l

t E (O,1],

f(0) = 0 can be taken. The function f is the derivative of the function 71

F ( t ) = t2cos(t2), t

E

(0,1],

F ( 0 ) = 0 and f is therefore Henstock-Kurzweil integrable. If we take e.g. E = { t E [0,1]; f ( t ) > 0) c [0,1] then clearly E is measurable but f is not integrable over E. See e.g [Sol] (Ex. 12, p. 18) for this example.

Theorem 4.1.7. Iff : I + X as McShane integrable then for every E > 0 there is an q > 0 such that zf E c I is measurable with p ( E ) < q then

Proof. Let E > 0 be given and let q > 0 correspond to E by Lemma 4.1.2. Assume that p ( E ) < q. Then there is an open set G c I such that E c G and p(G) < q. The McShane integrability of f over I implies the existence of a gauge A : I -+ (0, +GO) such that for every A-fine M-partition {(ti,I i ) } of I the inequality

holds. By Theorem 4.1.6 the integral ( M )JI f - xexists ~ and by the definition of the integral for every 8 > 0 there is a gauge S : I t (0, +m) which satisfies B ( t ,S(t)) c G if t E G, d ( t ) 5 A(t),

More on the McShane Integral

99

t E I and

holds for any 6-fine M-partition ((zI,, V,)} of I . If u, E E c G then V, c G and ~,,vm,,p(Vm)5 7 . Since {(urn,V,); ,IZ E E } is a A-fine M-system, we have by the Saks-Henstock lemma 3.4.2 the inequality

and by Lemma 4.1.2 we get

Hence

This proves the statement because 6' trarily small.

> 0 can be chosen arbi0

Remark. Theorem 4.1.7 represents an analogue of absolute continuity of the indefinite McShane integral which was extended to measurable sets E c I by Theorem 4.1.6. The similarity with the Lebesgue integral is again clear for the ca5e of McShane integrable functions with values in a Banach space.

Banach Space Integration

ZOO

Theorem 4.1.7 can be reformulated as follows: Let f : I t X be McShane integrable. Then

where E c I are measurable sets. This means that the indefinite McShane integral is pcontinuous. Theorem 4.1.8. Iff : I + X is McShane integrable and E C I is measurable, Fi C E , i E N are closed sets with Fi C Fi+l and p ( E \ Fi)= 0 , then

Ui

Proof. First note that for a given measurable set E a sequence of closed sets Fiwith the properties given in the theorem always exists. Let an arbitrary E > 0 be given and let q > 0 corresponds to it by both Lemma 4.1.2 and Lemma 4.1.5. Let G C I be open such that E c G and p(G \ E ) < 2. 2

rl < and 2 therefore p(G\ Fko) < q and of course also p(G\ Fk) < q for all

Further, there is a ko E

k

N such that p ( E \

Fko)

2 ko.

set

) given by Lemma 4.1.5 for the Let a gauge ( : I + ( 0 , ~be Fko instead of F . In particular we have

B(t,<(t))c I

\ Fko

if t E I

\ Fko.

Assume that {(ul, U l ) } is an arbitrary &fine M-partition of I . Fix an arbitrary k 1 ko. By Theorem 4.1.6 the gauge can be chosen in such a way that we have also

More on the McShane Integral

The last two inequalities can be written in the form

This gives

Further, by the Saks-Henstock lemma 3.4.2 we have

while

101

Banach Space Integration

102

by Lemma 4.1.2,because we have q. Hence

for k

xulEE,Fk P(U1) I P(G\FkCJ

2 ko and this proves the theorem.

<

0

Theorem 4.1.9. I f f : I -+ X is McShane integrable and F1, F2 c I are closed sets with F1 n F2 = 8 then

Proof, Assume that E > 0 is given and that q > 0 corresponds to E by Lemma 4.1.5. Since Fl and Fz are disjoint closed bounded sets, we have dist(Fl, 8'2) > 0 and therefore there exist open sets G1 and Gz rl such that Fl c G1, Fz c Gz, G1 n G2 = 8, p(G1\ F l ) < 2, p(Gz\ Fz) <

is denoted.) Hence

rl -. (By dist the usual distance of two sets in R"

2

P(G1 u GZ \ (Fl u F2)) < rl.

For the open set G = G1 U Gz and the closed set F = F1 U Fz let the gauge J : I -+ (0, +GO) be given by Lemma 4.1.5. For a given J-fine M-partition { (uz, Uz)} of I we have

More on the McShane Integral

103

which means in other words

This yields

and the statement of the theorem is proved because be taken arbitrarily small.

E

> 0 can 0

Theorem 4.1.10. I f f : I t X is McShane integrable and E l , E2 C I are measurable sets with El n E2 = 8, then

Proof. By Theorem 4.1.9 the statement holds for closed sets, Theorem 4.1.8 yields the result by passing to limits for sequences 0 of closed sets contained in E l , E2. Theorem 4.1.11. Iff : I --+ X is McShane integrable and Ei C I , i E N are measurable sets with Ei f l Ej = 8 for i # j then

Proof. By Theorem 4.1.6 all the integrals ( M )JI f ( M )J" xE,, i E N exist. f

a

*

xui Ei7

104

Banach Space Integration

Let E > 0 be given; by the definition of the McShane integral there exist gauges 6 : I + ( O , + o o ) , Si : I -+ ( O , + o o ) , i E N such that

for any S-fine M-partition { ( t j , I j ) } of the interval I and

for every &-fine M-partition { ( t j ,I j ) } of the interval I , i E N. Assume now that q > 0 corresponds to the given E by Lemma 4.1.5 and that k E N is such that

Assume further that a closed set F C I is contained in the Ei ( F C Ui>kEi) while measurable union Ui
Hence we have

p ( U E i \ F ) < TJ i

and there is an open set G

F)< 7 . Let E : I

---f

c I such that Ui Ei C G and p(G \

( O , + o o ) be the gauge given by (a) from Lemma

4.1.5. Take

S,), i E N,0 = min(J, S) and let I E N be such that I > k . Put 0i = min(5,

w = min(0,

el, . . . ,el)

More on the McShane Integral

105

and take an arbitrary w-fine M-partition { ( s j , K j ) } of I . For such a partition we have

i.e. <E X

and

Therefore

5 2e and

106

Banach Space Integration

Using Lemma 4.1.2 we get

which yields

and the statement is proved because this can be done for every 1 > k.

Remark. Theorem 4.1.11 extends the statement (a) from Lemma 4.1.3 to sequences of measurable sets saying that the indefinite McShane integral of a given McShane integrable f : I + X is countably additive. Since the McShane integral is not sensitive for changes of the integrated function on sets of measure zero (Theorem 3.3.1) it is easy to see that in Theorem 4.1.11 the requirement Ein E j = 8 for i # j can be weakened to p(& n E j ) = 0 for i # j. Proposition 4.1.12. Let El, C I , k E W be a sequence of disjoint measurable sets and xk E k E N. A s s u m e that the series Xkp(Ek) is unconditionally convergent in X and define

,:c

T h e n the sequence f n I

x,

+ X,n E

N is M-equi-integrable.

Proof. Since constant functions are McShane integrable, by Theorem 4.1.6 the functions xk x ~ ~ ( kt )E , W are McShane integrable and the McShane version of Theorem 3.3.6 yields that also the functions fn : I + X , n E N given by (4.1.15) are

More on the McShane Integral

107

McShane integrable with

Given an E > 0 then for every n E (0, +a) such that

N there is a gauge 6, : I

4

1,. . . ,p } is a &-fine Ad-partition of I . Since the series ~ ~ , z k p ( & is) assumed to be unconditionally convergent in X , it is weakly absolutely convergent by if {(ti,I i ) ,i

=

Proposition B.19 from Appendix B and therefore there is an N E N such that 00

(4.1.17) k=N+1

for all x* E B(X*). Let us set

u N

HN =

k=l

u 00

E k u ( I \

Ek)

k=l

and put

Hn = E n for n E N,n > N . Since the sets E k are disjoint, the sets H N and Hn7n also disjoint and HNU Un,N Hn = I . Let us define

s(t) = min{Sl(t), . . . , s N ( t ) } for t

E HN

> N are

108

Banach Space Integration

and

6 ( t ) = min(&(t), . . . , S n ( t ) } for t

E

Hn, n > N .

S : I -+ (0, +oo) is a gauge on I . Let {(ti,Ii), i = 1,.. . , p } be a S-fine Ad-partition of I . If n 5 N then by (4.1.16) we get (4.1.18) Assume that n

> N is fixed. We will consider systems

{(ti,Ii), i = l , . .. , p , ti E H j } for j 2 N . Note that fn(t) We have

=

n-1

0 if t E Hj with j > n.

P

r

/ . - I

By the choice of the gauge S the system {(ti,Ii), i = 1,.. . , p , ti E H j , j 2 n } is a &-fine Ad-system in I . Hence by the Saks-Henstock lemma 3.4.2 and by (4.1.16) we obtain

109

More on the McShane Integral

Since for j further

< n and ti

E

Hj we have fn(ti) = f j ( t i ) ,we obtain

X

(4.1.21) Now,since{(ti,Ii), i = l , . . . , p , t i ~ H j } f o r j = N , . . . ,n - l i s a Sj-fine M-system in I , we obtain by the Saks-Henstock lemma 3.4.2 and by (4.1.16)

and

110

Banach Space Integration

By (4.1.21) we obtain from this inequality the relation

<

- 2N+1

+

22

j=

[ ( M ) Ii/ f j

N i=1,ti E Hj

Observe that

and therefore for j

Hence

Ii

( M ) /Ii

. (4.1.22)

fn] X

Let us consider the sum

J

-

k=l

< n we have

More on the McShane Integral n-1

n-1

p

n

P

72

j = N k=j+l

n

i=l,tiEHj

k-1

P

k=N+1 j = N

n

k=N+1

Consequently

i=l,tiEHj k-1

P

j=N i=l,tiEHj

111

112

Banach Space Integration

by by (4.1.17). (4.1.17). This inequality together with (4.1.22) yields

2

/

[ f n ( t i ) p ( I i )- (M)

j = N i=l,tiEHj

I,

5 2N+1 + E . &

fn] X

By (4.1.19), using (4.1.20) and (4.1.23), we obtain for n> N the inequality

and this together with (4.1.18) proves the equi-integrability of 0 the sequence f n , n E N.

Theorem 4.1.13. Let El, C I , k E N be a sequence of pairwise disjoint measurable sets and xk E X , k E N. Assume that the series x k p ( & ) is unconditionally convergent in X . Then the function f : I --+ X defined by

xE1

k=l

is McShane integrable and

Proof.

By Proposition 4.1.12 the sequence

k=l

is M-equi-integrable and it is easy to see that

lim fn(t)= f ( t ) for t E I .

n+m

More on the McShane Integral

113

Hence the Convergence theorem 3.5.3 applies and f is McShane integrable with r

r

n

00

Remark. Theorem 4.1.13 was proved by R. A. Gordon in [G90] (Theorem 15) in a slightly different situation of a onedimensional interval I . The proof of Proposition 4.1.12 and that of Theorem 4.1.13 use the ideas from [G90].

4.2

An equivalent definition of the McShane integral

The notion of the McShane integral of a function given in Definition 3.2.1 is based on the concept of M-partitions of the interval T

1.

Let us introduce the following notions.

Definition 4.2.1. A system (finite collection) of pairs { ( t i ,Ei), i = 1,.. . , p } with Ei C I measurable, Ei n Ej = 0 for i # j is called an M*-system in I . P

An M*-system in I is called an M*-partitiono f 1 if

U Ei= I . i=l

Given a gauge A : I --+ (0, +m), an M*-system {(ti,Ei), i = 1,. . . ,p } in I is called A-fine if

Ei

c B(ti,A ( & ) )i, = 1,.. . , p .

Definition 4.2.2. A function f : I t X is McShane* integrable and J E X is its McShane* integral over I if for every E > 0 there exists a gauge A : I + (0, +m) such that for every A-fine M*-partition ( s i ,Ei),i = 1,. . . ,p of I the inequality

114

Banach Space Integration

holds. We denote J = ( M * )JI f. It is clear that 63 i f f : I + X is McShane* integrable then f is McShane integrable in the sense of Definition 3.2.1. We will show that the concept of the McShane integral from Definition 3.2.1 is not more general than that of the McShane* integral from Definition 4.2.2 To this aim let us prove a lemma. In the sequel by a figure we mean a finite union of compact non-degenerate intervals in Rm. Lemma 4.2.3. Assume that A c I is a figure and that E > 0 is given. Let 6 : I + (0, +GO) be a gauge and let {(ti,Ei), i = 1,.. . , k } be such that t i E I , Ei C A are measurable sets with 1 Ein E j = 8 for i # j and Eic B ( t i ,-&(ti)). 2 Then for i = 1,. . . , k there exist figures Ci C A such that p(Ci n C j )= 0 for i # j and

p(Ei A Ci) < E for i = 1,.. . , k (Ei A Ci = (Ei \ Ci) U (Ci \ Ei)is the symmetric difference of the sets Ei and Ci),

cz c B(t2,6(t,)). Proof. We prove the statement by induction. Assume that k = 1, i.e. we have tl E I and El c A . Let X > 0 be arbitrary. Then there is a set G c A which is open in A such that G c B(t1,6 ( t 1 ) ) , El c G, p(G \ E l ) < A, and a figure C1 such that C1 c G and p(G \ C1) < A. We have

5 P(G \ C l ) + P(G \ El)5 2X &

and the statement holds in this case if we put X = - because 2 c 1 c G c B(tl,&(tl)).

More on the McShane Integral

115

Coming to the induction step, assume that the statement of the lemma holds for sonie k E N and let ( t o , Eo),(tl, E l ) ,. . . , ( t k , Ek) be k 1 point-set pairs satisfying the assumption. Let A > 0 be arbitrary, By the first part of the proof there is a figure Co contained in A, such that

+

and

co c "0,

d(t0))

hold. Put A* = cl(A \ Co), (cl(A \ Co) is the closure of the set A \ Co)and let E; = Ei n A* for i = 1,. . . , k . A* is evidently a figure while {(ti,E:), i = 1, . . . , k } satisfies the assumptions of the lemma. By the induction hypothesis there exist figures C1,. . . , Ck contained in A* such that p(Ci n C j )= 0 for i # j and

p(E; n CZ) < A,

(4.2.2)

Ci c B(ti,&(ti)), i = 1,.. . , k . We also have p(Con Ci) = 0 for i = 1,.. . , k

because Ci c A* = cl (A \ Co). For i = 1, . . . , k we have Eo n Ei= 0 and therefore

Ein Co = Ein (Co\ E o ) ,

Since Eic E,* U Ei n Co we get p(ci

\ Ei) = p(C2 \ El) < A.

(4.2.4)

116

Banach Space Integration

On the other hand, by (4.2.2) and (4.2.3) we have,

p(Ei\

ci)I P(E~*\ CJ + p ( ( E in co)\ Ci)I 2x

and this together with (4.2.4) shows that for i = 1,.. . , k we have

p(Ei n CZ) 5 3x. Taking into account (4.2.1) we obtain the result because A > 0 0 can be taken arbitrarily small. Theorem 4.2.4. I f f : I is McShane* integrable and

+X

is McShane integrable then f

Proof. Let E > 0 be given. By the Saks-Henstock lemma 3.4.2 there exists a gauge 6 : I + (0, +GO) such that for every &fine M-system (rj,K j ) ,j = 1,.. . , q of I the inequality

holds. This implies that if { ( r j ,C j ) , j = 1,.. . , q } , rj E I , Cj are non-overlapping figures contained in I with Cj c B ( r j ,S ( r j ) ) then

1 Assume that { (si,Ei), i = 1,.. . , p } is an arbitrary -6-fine M*2 partition of I . By Theorem 4.1.7 there is an q > 0 such that if E C I is measurable and p ( E ) < q then

(4.2.6)

More on the McShane Integral

117

By Lemma 4.2.3 there exist figures Ci c I such that p(Ci n C j ) = 0 for i # j with

and

for i

=

l , ,. . , p .

We have (4.2.S)

Since

118

Banach Space Integration

we have

and because

we obtain by (4.2.6).

(4.2.10) Using the figure version (4.2.5) of the Saks-Henstock lemma we obtain finally from (4.2.8), (4.2.9) and (4.2.10)

and this shows that f is McShane* integrable and that 0 ( M * ) f = ( M ) f holds.

s,

s,

Hence we arrive at the following result.

Theorem 4.2.5. A function f : I + X is McShane integrable if and only i f f is McShane* integrable and the integrals f , J’f coincide.

sI*

Remark. The concept of McShane* integrability was considered, in a more general setting of a a-finite quasi-Radon measure space as the space over which we integrate, in F’remlin’s paper [F95] (1A Definitions). See also [FM94] (2H Lemma) for the case of I = [0,1] c R or [DPMuOl].

More on the McShane Integral

4.3

119

Another convergence theorem

In this section we will show that for Banach space-valued functions a Vitali type convergence theorem is valid. Let us start with the following simple lemma.

Lemma 4.3.1. Let f : I X be McShane integrable and Ilf(tllx 5 K f o r t E I . Assume that H c I is a measurable set. Then ---$

Proof. By Theorem 4.1.6 f is McShane integrable over H and the function K - X H : I t R being a simple function is also McShane integrable to the value K . p ( H ) . For an arbitrary M-partition {(ti,Ii)} we have

i

Given an arbitrary

E

> 0 take a gauge 6 : I

and

for any 6-fine M-partition { ( t i ,Ii)}. Then we have

-+

( 0 , ~ such ) that

120

Banach Space Integration

2E

+K

*

p(H)

and the statement follows because E > 0 can be taken arbitrarily small. 0 Definition 4.3.2. Let M be a family of McShane integrable functions f : I -+ X. If for every E > 0 there is a S > 0 such that for E c I measurable with p ( E ) < 6 we have 11 ( M )JE fllx < E for every f E M , then the family M is called uniformly absolutely continuous (with respect to the measure p ) . Theorem 4.3.3. Assume that M = { f k : I

--+

X ;k

E

N} is a

uniformly absolutely continuous family such that lim f k ( t )= f ( t ) ,t E I

k-cc

Then the family M is M-equi-integrable. Proof. First assume that the LetO<&
fk,

k E R? are measurable. E

n;sup IIfi(t) - f(t)ll < i>n

P(I)

and for k E N put Ak = { t E I ; r ( t )= k } . Since r is measurable, each set Ak C I is measurable, Ak are pairwise disjoint and I = Ak. If t E Ak and m _> k , then

uEl

IIfm(t>llx 5 IIfm(t>- f(t>Ilx+ Ilf(t>llx

(4.3.1)

More on the McShane Integral

121

If H Ak is measurable and m, n k then

for t E H and Lemma 4.3.1 yields

(4.3.2) By the uniform absolute continuity of M , for every k choose 61, > 0 such that

for all n E N when p ( E ) < 6k. Pick GI, open in I , A k C Gk min(&k,6,). Let A, be a gauge such that

,

such that p(Gk\Ak) <

when {(ti,I i ) } is a Ak-fine M-partition of I . Define a gauge A such that A(t)

= min(A,(t), . . . , A,(t))

and

B ( t ,A ( t ) )n I for t E Ah. Suppose {(ti,Ji); i

=

c Gk

1,.. . , N } is a A-fine M-partition of I .

122

Banach Space Integration

which implies

Evidently, we also have

Therefore, if m

and

2 k , by (4.3.1)

1

Now,

(4.3.5)

More on the McShane Integral

123

Since {(ti,Ji) : ti E Ak) k 2 m } is Am-fine, for the second term on the right hand side of (4.3.5) we have

by the Saks-Henstock Lemma 3.4.2. To estimate the first term on the right hand side of (4.3.5) we have

124

Banach Space Integration

Now, by (4.3.4),

By (4.3.3),

II

Ilm-1

By (4.3.2), since JF

c Ak,

More on the McShane Integral

Since { (J!, ti) : i E

~

k

125

is} &-fine, we get

by the Saks-Henstock Lemma 3.4.2 and therefore m- 1

Cc[(M)/

m- 1 f k - .fk(ti)p(JI")]

I x&/2k < &.

Finally, II

Ilm-1

1) k=l m-1

Thus,

iELk

JJX

126

Banach Space Integration

and

Hence the left hand side of (4.3.5) is less than 1 0 ~ . To remove the measurability assumption, define T and AI,as before, but T need not be measurable so the A k need not be measurable. Since A k c I , we have p * ( A k ) < 00, denoting by p* the outer Lebesgue measure. For each k , pick a measurable V k 3 A k such that p ( V k ) = p * ( A k ) (see e.g. [WZ77],(3.32) Theorem). Then (4.3.1) remains valid. Condition (4.3.2) is replaced by:

H c V k measurable

=+

(4.3.6)

if m , n 2 k . To see this, first note that p ( H ) = p*(Ak n H ) . Indeed, if p * ( H n A k ) = p ( H ) - 7 with 7 > 0, there exists a measurable B 3 H n AI, such that p ( B ) = p * ( H n A k ) = p ( H ) - 7 . NOW Ak

c (Vk\H) u (Hfl A,) c (Vk\H) u B

and therefore P * ( A k ) = P ( V k ) I P(Vk\H)

+ P ( B ) = P(Vk\H) + P ( H ) - rl,

which implies 7 = 0. For x* E B(X*) let Ak(X*) =

{ t ; Iz*(fn - f m ) ( t ) l

I Eh(t),n,mL

k}

c A,;

each A ~ ( x *is) measurable since z*(fn- fm) is Lebesgue integrable.

More on the McShane lnntegral

127

We have

P(H\A~(x*)) = p ( H ) - p ( A I , ( X * ) n H )I p ( H ) - p * ( A I , n H ) = 0 by the observation above. Thus ,

5 SUP ( M ) / 11x*111~

Eh

I ( M )L c h .

HflAk(2')

Now choose GI,open in I such that GI,2 VI, and m(GI,\Vk) < min(EI,,SI,)and define A as before. The argument then carries through as in the case of measurability of the functions f k . 0

Remark. For the case of a sequence of real valued functions f k , Ic E N Theorem 4.3.3 was proved in [KS03a] using Egoroff's Theorem 1.1.7 essentially. The proof of our Theorem 4.3.3 is adapted from the paper [RS04] of R. Reynolds and Ch. Swartz. In that paper the authors prove Theorem 4.3.3 for the case of a one-dimensional interval I which can be infinite. Also norm convergence in the space of integrable functions is included in [RS04].

Theorem 4.3.4. Assume that f k : I + X ,k E N are McShane integrable functions such that 1. fI,(t)-+ f ( t )f o r t E I , 2. the set { f k ; k E N} is M-equi-integrable. Then fI, - X E , k E N is an M-equi-integrable sequence for every measurable set E c I . Proof. Let E > 0 be given and let q > 0 corresponds to E by Lemma 4.1.4. Assume that E c I is measurable. Then there exist F C I closed and G c I open such that F c E c G

128

Banach Space Integration

where p(G \ F ) < q. Assume that the gauge J : I t (0, GO) is given as in Lemma 4.1.4 and that {(ul,Ul)},{ ( U r n , V,)} are J-fine Ad-partitions of I . By virtue of (a) in Lemma 4.1.4 we have if

ul

E E then

Ul c G, F c int

u Ul

u1 EF

and if v, E E then V, C G, F

c int

u

V,.

V,EF

Hence by (b) from Lemma 4.1.4 we have

1c

fk(%)P(Ul)

1,UlEE

-

c

fk(vm)P(vm)l

5E

m,u,EE

and therefore also

This is the Bolzano-Cauchy condition from Theorem 3.5.5 for equi-integrability of the sequence f k - X E , k E N and the proof is complete. 0

x,

Proposition 4.3.5. Assume that f k : I 3 k E McShane integrable functions such that 1. f k ( t ) f ( t )for t E I , 2. the set { f k ; k E N} is M-equi-integrable. Then for every E > 0 there is an q > 0 such that if E measurable with p ( E ) < q then

N are

cI

as

for every k E N.

Proof. Let E > 0 be given and let q > 0 correspond to E by Lemma 4.1.1 and assume that p ( E ) < q. Then there is an open set G c I such that E c G and p(G) < q.

More on the McShane Integral

129

The equi-integrability of fk implies the existence of a gauge A : I --+ (O,+m) such that for every A-fine M-partition {(ti,Ii)} of I the inequality

holds for every k E N.. By Theorem 4.3.4 the integrals ( M ) J I f k- X E , k E N exist and for every 19 > 0 there is a gauge & : I --t ( O , + o o ) which satisfies B ( t ,b ( t ) ) c G if t E G, b ( t ) I: A(t) for t E I and

holds for any &-fineM-partition {(urn,Vm)}of I and every k E N. If E E C G then Vm c G and E mr m , ,,p(Vrn) Iq. Since {(urn,Vm);u, E E } is a A-fine M-system, we have by the Saks-Henstock Lemma 3.5.6 the inequality

and by Lemma 4.1.1 we get

Hence

Banach Space Integration

130

This proves the statement because Q > 0 can be chosen arbitrarily small. Using Theorem 4.3.3 and Proposition 4.3.5 and the concept of uniform absolute continuity of a sequence of functions given in Definition 4.3.2 we obtain Theorem 4.3.6. Assume that f k : I --+ X , k E N are h f c Shane integrable functions such that f k ( t ) -+ f ( t )for t E I . Then the set { f k ; k E N} f o r m s an equi-integrable sequence if and only if { f k ; k E N} is uniformly absolutely continuous.

By Theorem 4.3.6 and by the convergence Theorem 3.5.3 we arrive immediately at the following result. Theorem 4.3.7. Assume that f k : I -+ X , k E N are h f c Shane integrable functions such that f k ( t ) 4 f ( t )for t E I and that { f k ; k E N} is uniformly absolutely continuous. Then the function f : I --+ X is McShane integrable and

holds.

If f k ( t ) + f ( t )almost everywhere in I then there is a set 2 c I with p ( 2 ) = 0 such that f k ( t ) 4 f ( t ) for t E I \ 2. Defining g k ( t ) = f k ( t ) , g ( t ) = f ( t ) for t E I \ 2 and g k ( t ) = O , g ( t ) = 0 for t E 2 we have g k ( t ) g ( t ) for t E I . Using Theorem 3.3.1 we can see easily that if { f k ; k E N} is uniformly absolutely continuous then also { g k ; k E N} is uniformly absolutely continuous and by Theorem 4.3.7 the function g : I --+ X is McShane integrable and --+

k-cc

More on the McShune Integrul

131

because g ( t ) = f ( t )a.e. in I . This yields the following.

Theorem 4.3.8. Assume that f k : I 3 X ,Ic E N are MeShane integrable functions such that f k ( t ) + f ( t ) almost everywhere in I and that { f k ; k E N} is uniformly absolutely continuous. Then the function f : I -+X is McShane integrable and k+cc lim

(M)Jh

=

(M)lf

holds. Theorem 4.3.8 is a Vitali type convergence theorem for the McShane integral and, as Theorem 4.3.6 shows, it is of the same power as the easily provable convergence Theorem 3.5.3.

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Chapter 5

Comparison of the Bochner and McShane Integrals In this chapter we compare the concepts of Bochner and McShane integrals introduced in Chapters 1 and 3.

5.1

Strong McShane integrability and the Bochner integral

Lemma 5.1.1. Assume that f : I t X is Bochner integrable and let E > 0 be given. Then there is a gauge w : I t (0, +m) and q E ( 0 , ~ such ) that the following statement holds.

If is an w-fine M-system for which P

m=l then

Proof. For j = 1 , 2 , . . . let us set

133

Banach Space Integration

134

Since llfllx is integrable by Lemma 1.3.6, the sets E j are measurable and Ei n E j = 8 for i # j , while

We also have

and therefore W

W

n

n

For j = 1 , 2 , . . . there is an open set Gj and

cI

for which Ej

c Gj

and this together with the convergence of the series C,”=, jp(Ej) yields M

M

M

j=1

j=1

j=1

Assume that

EO

> 0 is given. Then there is an r

E

N such that

j=r+l

If t E I then there is exactly one j Let us choose a gauge w so that

=j(t) E

N such that t E Ej.

Comparison of the Bochner and McShane Integrals

135

If now { ( H m ,t,), m = 1,. . . ,p } is an u-fine M-system, then we have t, E Ejm, Hm

c B ( t m , w ( t m ) )c Gj,

and

Ilf(tm)Ilx < j m for m = 1,.. . , p . Note that for a given tm it is possible to have various intervals H which are pairwise non-overlapping and contained in the same open set Gj,. Hence P

P

P

C Ilf(tm)llxP(Hrn) 5 m=l,jm
P

Taking

j m ~ ( ~ m )

m=l,jm>r

m=l

&

< - and q <

&

we obtain the result. 2 2r 1 Proposition 5.1.2. I f f : I + X is Bochner integrable t h e n f has the property S*M from Definition 3.6.7 and EO

~

+

(5.1.1)

Proof. Assume that E > 0 is given. Let f q , q = 1 , 2 , .. . be an L-Cauchy sequence of simple functions which converges to f almost everywhere in I , i.e.

for almost all t E I . Let 7 E (0, E ) and let the gauge w : I -+ (0, GO) be given by Lemma 5.1.1. Take a E (0, -). By the fundamental Lemma 2P(I) 1.3.2 the sequence f q , q = 1 , 2 , .. . can be chosen in such a way

'

Banach Space Integration

136

a

that there exists a measurable set 2, c I with p(Za) < - for 2 which the sequence f, converges to the function f uniformly on I \ 2,. For the measurable set 2, there is an open set G, c I such that 2, c G, and p(G,) < a. Let us define the closed set

F,=I\G,cI\Z,. Thus we have the following result. For a > 0 there is a closed set Fa c I such that P t I \ Fa) = PtG,)

and there is an n, E

< QI

N such that Ilfqtt)

- ftt)Ilx < Q!

for q 2 n, and t E Fa. Assume that q 2 n,. Since f q is a simple function, there is a finite sequence E,, C I , m = 1,.. . ,p , of measurable sets such =:, E,,, where that Eqmn Eql = 0 for m # I and I = U

f,(t) =,y,

EX

for t E E,,, m = 1 , .. . , p , .

By measurability of the sets EQmthere exist closed sets F,, with Fqm c Eqmand

p(Eq,

rl \ Fqm)< for m = 1,.. . ,p,.

2P,

Hence

Define further

Aqm= F,

n Fqm,m = 1,.. . ,p,.

The sets A,, are closed and A,, n A,, = 0 for m # 1. Therefore the distance of any two different sets A,, is positive, i.e. there

Compan'son of the Bochner and McShane Integrals

is a p

> 0 such that if t E A,,,

sE

137

A,, and m # 1, then

dist(t,s) > p. Further we have Po

Pa

Pa

m=l

m= 1

m= 1

and therefore

m= 1 Pa

P Let us take a gauge S on I for which d(t) < min(w(t), -) for 2 t E I and

w,d(t>>n I c I \ u A,, P,

m= 1

provided t E I

\

P4

U Aqm. This can be done because the set m= 1

I

P,

\ U A,, m= 1

is open.

Assume that {(ti,J i ) , i = 1,.. . , k} and { ( s j , L j ) , j = 1,.. . , 1 ) are 6-fine M-partitions of I . By the choice of the gauge d given above we obtain the following properties of a d-fine M-partition {(ti,Ji),i = 1,. . . , k } of I :

Banach Space Integration

138 P,

If ti

E

u A,,,

then there is r E {1,. . . , p , } such that ti E

m=l

P we have B(ti,S(ti))n A,, = 0 provided A,,; since S(ti) < -, 2 m # r and therefore also Ji n A,, = 0 for m # r .

If ti 4.

P,

u A,,,

then

m= 1

P,

C B(ti,&(ti))C I

Ji

U A,,,

\

(5.1.2)

m=l

i.e.

Ji n A,,

=

8 for

m

=

1,.. . , p , .

Moreover, since

P4

Uf=l,ti$” A,,

Ji

CI

\ U

A,,, we get

m=l

p(

6

P, Ji>

i=l,ti$UA,,

5 p ( \~IJA,,) < 7 . m=l

Similar properties hold also for the partition { ( s j , Lj)}. ~~

Assume now that ti,sj E

P,

U A,,.

m= 1

necessarily dist (ti, s j )

Ji

n Lj # 0,

then

< p because dist (ti, Ji) < -P , dist (sj, L j ) < 2

f by the choice of the gauge

2

If

S and

dist(ti, s j ) 5 dist(ti, a )

+ dist(sj, a ) < p

where a E JinLj. In this situation there is an T- E { 1,.. . , p , } for which ti, sj E A,,. Indeed, if ti and s j belonged to different A,,, then we would have dist(ti,sj) > p and this would contradict the inequality given above. Hence fq(ti>= f q ( s j ) = yq,

because A,, c E,,. At the same time we also have A,, and therefore IIf,(t) - f(t)Ilx < a for t E A,,. This yields

Ilf(tJ - f(Sj)llx

c Fa

(5.1.3)

Comparison of the Bochner and McShane Integrals

P,

P,

U Aqm,sj

If at least one of the inclusions ti E

139

U A,,

E

m= 1

does

m=l

P,

not hold, i.e. if we have for example sj E I

(5.1.2) we have

u

\ U Aq,, m=l

then by

P,

Ji n Lj

c Lj c I \

A,,

m= 1

and the tagged interval ( s j , Ji n L j ) is &fine. Similarly also the tagged interval (ti,Ji n L j ) is 6-fine. The other possible cases lead to the same conclusion. For showing that the function f has the property S*M we need an estimate for the sum k

1

The set

M = { ( z , j ) ; i = 1, . . . , k , j

=

1,. . . , 1 )

can be split into

u P,

M l = { ( Z , j ) E M ; t i , s j ~ Aq,)andM2=M\M1. m=l

Then

140

Banach Space Integration

Hence by Lemma 5.1.1 we get

I

C

(idEM2

+

C

I I ~ ( ~ ~ > InI ~ L P~ ( )J ~ II~(s~>II~ nPL (~JI~) 2 E . (i ,AE M2

Altogether we have, by the choice of q > 0, the inequality

s < 2ap(I)+ 2E and this yields the property S*M of the function f by definition. It remains to show that for the integrals the equality (5.1.1) holds. Suppose that E > 0 is given. Assume that E c I is an arbitrary measurable set. Let us put F = I \ E ; then evidently I = E U F . In this situation there exist open sets G and H such that E c G, F c H and

Comparison of the Bochner and McShane Integrals

Let us define a gauge S : I

--$

141

(0, +oo) such that the implications

t

E E ==+

B(t,6(t))n I

cG

t

E

F ==+

B ( t ,S ( t ) ) n I

cH

and

hold. Let {(ti,J i ) } be an arbitrary 6-fine M-partition of I and assume that X E is the characteristic function of the set E . Then k

k

i=l

i=l&EE

and similarly k

i=l

Further we have

and also

k

k

i=l

i= I

XI =XE

+ X F . This yields

k

k

k

i=l

i=l

i= 1

> P ( I ) - ( P P ) + 4 = P(E) - E . This inequality together with (5.1.4) implies I k

I

Banach Space Integration

142

sIxE

and therefore ( M )

= p ( E ) . Since by definition we have

we have also

Since and

XE

E B , by Proposition 5.1.2 X E has the property

S*M

If 1~ E X, then the function y . X E : I + X belongs to B. Therefore y X E has the property S*M and

On the other hand, looking at the integral sums we obtain also

and

This immediately implies that

for an arbitrary simple function g : I +. X (g E J’, see Definition 1.1.1). Without any loss of generality we may assume that for a sequence (fq) of simple functions which determines f E B (see Definition 1.3.4) the inequality

143

Comparison of the Bochner and McShane Integrals

holds for almost all t E I . (Indeed, it is possible to define g q ( t ) = f q ( t )if Ilfq(t)IlxI Ilf(t)Ilx 1 and g q ( t ) = 0 otherwise; gq is the desired bounded determining sequence of simple functions for the function f . ) Since

+

there is a q E N,q

> n, such that

and for the simple function f q the equality

holds. By virtue of Lemma 5.1.1 there is a gauge w : I and 7 E (0, E ) such that

-+

(O,+oo)

P

m=l

if { ( t mH , m ) ,m = 1,.. . , p } is an w-fine M-system for which C:=l p ( H m ) < V. Assume that 6 is a gauge on I for which S ( t ) < w ( t ) if t E I and

for every 6-fine M-partition {(ti,J i ) ,i = 1,.. . , k}. For such a S-fine M-partition {(ti,J i ) ,i = 1,. . . , k } we have

144

Banach Space Integration

II i=l i=l IIX We need an estimate for the sum on the right hand side of this inequality. To this aim we split the sum into two parts, one with

ti E

P,

U A,, m= 1

and the other with ti $!

P,

U Aqm,i.e. m=l

Comparison of the Bochner and McShane Integrals k

k

m=l

m=l

by Lemma 5.1.1 because p ( U k

i=l,ti@

u

pq

A,,

Ji) < q in this case.

m=l

If ti E

P,

U

145

A,, then because q > n, we have

m=l

and

Putting together all these estimates, we finally obtain

< 2.5 + 3 E + &/%(I) = E(5 +/%(I))

146

Banach Space Integration

for every &fine M-partition { ( t i Ji), , i = 1,. . . , Ic} and this implies

i.e. (5.1.1) is satisfied. 0

We have shown that if X is a general Banach space then f E B implies that f has the property S*M and therefore also

fEM. On the other hand, the following statement holds. Proposition 5.1.3. I f f : I f is Bochner integrable and

--+

X has the property S*M then

Proof. Assume that f has the property S * M . By Theorem 3.6.9 f is McShane integrable and therefore for every m = 1 , 2 , .. . there is a gauge 6, on I such that

for every 6,-fine M-partition { ( t i ,I i ) , i property S*M also

=

1,.. . ,p} and by the

for any two 6,-fine M-partitions { ( t i ,J i ) , i = 1,.. . , k } and { ( S j , L j ) , j = 1 , .. . , 1 } of I . Assume without loss of generality that

6,+1(t) 5 6,(t) for t E I , rn = 1 , 2 , .. . . Let { (tj”’, J,!”)), i I.

=

1,2, . . . , k,}

be a 6,-

fine M-partition of

Comparison of the Bochner and McShane Integrals

Assume further that the next b,+l-fine

147

Ad-partition

{(t""+",J,(m+''),i= I, 2 , . . . , IC,+~} is a refinement of the partition {(ti"', J,!"'), i = 1 , 2 , . . . , k m } in the following sense: for every i = 1 , 2 , . . . , km+' there is a j E { 1 , 2 , . . . , km} such that J,!"") c i.e. every interval J,"") intersects the interior of only one of the intervals J,!"'), i = 1 , 2) . " , k,. Define

~3'"'

fm(t>= f(t,'")) for t

E int

J,!"', i

=

1 , 2 , . . . , IC,

f m ( t )= 0 otherwise (int J,!"' is the interior of the interval Jj"'). The functions fm : I + X ,m E N are evidently simple. Denote

w(m,i> = { j E { 1 , 2 , . . . , km+l};

~,(mf')c J,!")>.

Since the partition { (t,'""), J,!"")), i = 1 , 2 , . . . , ICm+l} is 6,= Ji ("+I) fine and Ji("+I) n J'") 3 for j E W ( m ,i ) , we have by (5.1.7)

For the given

E

> 0 let us take N

E

N such that

148

Banach Space Integration

This implies that ( f m ) is an L-Cauchy sequence of simple functions and by Lemma 1.3.2 it contains a subsequence (we denote it again (f m ) ) which converges pointwise almost everywhere to a certain function g : I + X . By Definition 1.3.4 we have g E B and the function g has the property S*M by Proposition 5.1.2. Hence

n

=

J, f m

lim ( M )

m-+m

=

lim

m--

(a)

because the McShane and Bochner integrals of simple functions coincide. If E c I is a measurable set then it is easy to see that xE f m , m E N is a sequence of simple functions which is L-Cauchy and therefore

By the definition of f m we further have

for every interval J c I . Using the Saks-Henstock lemma 3.4.2 we get by (5.1.6)

Comparison of the Bochner and McShane Integrals

149

and this shows that n

for every interval J Hence

cI

for every interval J

c I and

I)

for every interval J c I . Note that the function f - g has the property S*M. Let us show now that this implies that f = g almost everywhere in I . Assume the contrary, i.e. that there is a measurable set E C I , p ( E ) > 0 such that f ( t ) - g ( t ) # 0 for t E E. Looking at the sets E j = {t E E ; j-1 5 Ilf(t)-g(t)llx < j } , j E N we can see that there is a K > 0 and a measurable set E, c E such that

and p ( E K ) > 0. Then there is a closed subset F c EK with p ( F ) > 0 and we define for t E I \ F a gauge b l ( t ) > 0 in such a way that B ( t ,& ( t ) )c I \ F . Since f - g is strongly McShane integrable and

there is a gauge 6 on I such that 6 ( t ) < &(t)for t E I

\F

and

150

Banach Space Integration

if { ( t i ,Ji);i = 1,. . . ,p } is a &-fineAd-partition of I . By the definition of the gauge 61 we have F C UtitF Ji and therefore p ( F ) 5 p(UtiEFJi). This leads t o the contradictory inequality

and therefore f = g almost everywhere on I . The Bochner integrability of g yields the Bochner integrability of f and the 0 statement of the proposition holds. Using Proposition 5.1.2, Proposition 5.1.3 and Theorem 3.6.13 we obtain

Theorem 5.1.4. A function f : I + X is Bochner integrable if and only if f has the property S*M (see Definition 3.6.7) or, equivalently, i f and only i f f is strongly McShane integrable (see Definition 3.6.2). In the paper [H92] Ch. S. Honig proved this result in a different framework. The result of Honig is repeated in the paper [Fed041 of M. Federson because the reference [H92] is not easily available. Further, by Theorem 3.6.5 we get the following result.

Theorem 5.1.5. Iff : I + X is Bochner integrable then f is McShane integrable, i.e. we have B c M .

5.2

The finite dimensional case

Now we will show that the following statement holds.

Proposition 5.2.1. If X is afinite dimensional Banach space, then a function f : I + X is McShane integrable i f and only i f it has the property S * M .

Comparison of the Bochner and McShane Integrals

151

Proof. Since X is finite dimensional, we can assume without loss of generality that d i m X = 1. Otherwise it is possible to work componentwise. So assume that f : I --+ R and f E M . Let E > 0 be given. By Theorem 3.3.3 there is a gauge 8 on I such that

I

j=1

*

for every &fine Ad-partitions {(ti,J i ) , i = 1,.. . , k } and { ( s j , L j ) , j= 1,...,I}. Clearly { ( s j , J i n L j ) , = i 1, . . . ,k , j = 1,.. . ,Z} and { ( t i ,Ji n Lj),i= 1,.. . , k , j = 1,.. . , l } are also 8-fine M-partitions of I . Further we have

Denote by Ad+ the set of indices ( i , j ) , i for which

and by which

=

1,.. . , k , j = 1,.. . , Z

M- the set of indices ( i , j ) , i = 1,.. . , k , j

By the Saks-Henstock lemma 3.4.1 we get

= 1 , .. . , Z

for

Banach Space Integration

152

and similarly also

I

(ij)€ M-

Hence k

l

and f has the property S*M. Conversely, if f has the property S*M then f is McShane integrable by Theorem 3.6.9. 0

By Proposition 5.2.1 and Theorem 5.1.4 we obtain the next. Theorem 5.2.2. If X is a finite dimensional Banach space t h e n a function f : I -+ X as McShane integrable if and only i f f is Bochner integrable, i.e. we have M = S M = B in this case . From Remark 1.3.14 and Theorem 5.2.2 we obtain also the following well-known fact.

Theorem 5.2.3. A function f : I + R", n E integrable ijf and only i f f is McShane integrable.

N is Lebesgue

Comparison of the Bochner and McShane Integrals

153

5.3 The infinite dimensional case In the sequel we will use the Dvoretzky-Rogers theorem B . l l and its Corollary B.12 from Appendix B to show that the result of Theorem 5.2.2 does not hold for infinite-dimensional Banach spaces, i.e. that there is a function f E M which is not Bochner integrable.

Lemma 5.3.1. Suppose that zi E X , X i E [0,1] for i 1,.. . , k . Assume

=

II C ZiIIX < 1 iEQi

for any subset

Ql

of { 1 , 2 , . . . , k ) with 1 elements where 1 I k .

Then k

II

1Xjzjllx < m m j I 1. 3

j=l

Proof. Without loss of generality assume that 05

5 A2 5

* ' *

5 x k 5 1.

Then k

k

j=1

j=2

Banach Space Integration

154

and therefore k

k

k

j=1

j=1

j=2

Proposition 5.3.2. If X is an infinite-dimensional Banach space then there exists a function f : I + X which is McShane integrable but not Bochner integrable (f E M , f $ a). 00

C .zj

Proof. Assume that

is an unconditionally convergent

j=1

series for which

j=1

Such a series exists by Corollary B.12 in Appendix B. Let Kj c I , j = 1 , 2 , .. . be open intervals such that KjnKi = 0 for i # j . We have 00

j=l

Denote

uK ~c, 00

K

=

j=1

The set K

cI

is open. Let us set

=I

\K

Comparison of the Bochner and McShane Integrals m

03

The series

155

C y j p ( K j ) = j=1 zj

unconditionally converges to a

j=1

sum s E X while

j=1

Let

E

> 0. Take m E N such that (5.3.1)

and (5.3.2) for any finite set Q c { m

+ 1,m + 2,. . . } and define

f ( t )= O for t

E

f ( t ) = yj for t E K j , j Assume that 6 : I

t

C, =

I, 2 , . . . .

( 0 , ~ is) a gauge on I such that

B ( t ,6 ( t ) )n I C Kj for j = 1 , 2 , .. . and t E Kj. Let &

O
and let G

cI

5 IlYjllx>

j=1

be an open set for which

C =I

\ K c G and p(G) < p(C) + 11.

For t E C assume that

B(t,6(t)n ) I

c G.

156

Banach Space Integration

Let { ( t i , J i ) , i = 1,.. . , k } be a 6-fine M-partition of I . Then by (5.3.1)

Denote

uK j m

K,

=

u Kj 03

and K,,

=

j=m+l

j=1

and split the sum k

k

i=l

i=l,ti EK

into two parts

m

k

03

j=1 i=l,tiEKj

m

k

j=m+l i=l,tiEKj

k

03

k

Then we obtain

X

Comparison of the Bochner and McShane Integrals

157

The last term in this inequality consists of a finite number of nonzero terms only and we have

i.e.

c k

P(JZ) = W K j >

i=l,ti EKj

where

Aj

E [0,1]. By (5.3.2) and Lemma 5.3.1 we get

since ti E Kj the sum on the left is finite. k

It remains t o give an estimate for [[

C,”=,yj( C p(Ji)i=l,tiEKj

P ( K j ) > l l X . w e have

231,(c k

j=1

and

i=l,ti E Kj

PL(Ji) - P U ( W X

Banach Space Integration

158

Since

we get

0 5 p(Kj \

U

Ji) = p ( K j ) - p(

tiEKj

IJ

Ji)

U

F P(K \

tiEKj

Ji)


tiEK

for every j = 1 , .. . , m. Therefore

Finally, using (5.3.1) we obtain

and this means that the integral ( M )JI f exists and

00

i.e. f E M . On the other hand, since the series

C pjp(Kj) j=1

does not converge absolutely, the Bochner integral not, exist because

Proposition 1.4.5).

(a)JI f

does

(see also 0

Using Theorem 5.1.5, Theorem 5.2.2 and Proposition 5.3.2 we arrive at the following result.

Comparison of the Bochner and McShane Integrals

159

Theorem 5.3.3. Given a Banach space X then every Bochner integrable function f : I -+ X is McShane integrable. The class of Bochner integrable functions B coincides with the class M of McShane integrable functions if and only if the dimension of the Banach space X is finite.

An concrete example showing that there is a function f : [O,1] + 12 which is McShane integrable but not Bochner integrable is presented e.g. by M. Federson in [FedO4]. Using the sets of integrable functions we can present the following general scheme. Theorem 5.3.4. Given an arbitrary Banach space X we have

B=SMcM. The relation B = M (and B = S M = M ) holds if and only if the dimension of the Banach space X is finite.

5.4

An example

In this section we will show by an example what the differences between the integrals defined using Riemann type sums are for the case of measurable functions. We are obliged to Ch. Swartz for the idea of the example as well as for some of its details. See the recent paper [SO41 where the example is presented for a function f : [0, GO) + X . Assume that ( z k ) is a sequence of elements belonging to X and define the function f : [0,1] + X as follows: (5.4.1) Proposition 5.4.1. rfll&zkl[x < B , B > 1 then for the function f : [O, I] + X given b y (5.4.1) the following holds: (a) f is Henstock-Kurzweil integrable if and only if the series CFZl is convergent. (b) f is McShane integrable i f and only if the series ELl S z k is unconditionally convergent.

3.k

160

Banach Space Integration

(c) f is strongly McShane integrable if and only i f the series CEl & z k is absolutely convergent.

~((6,

Proof. Let us mention that F 1 ]= ) F1 . First we focus on the cases (a) and (b). Let 0 < E < B. If a) the series $ z k converges then there is an no E such that

xEl

N

for n > m 2 no and if b) the series C z l & z k converges unconditionally then there is an no E N such that

for every sequence a = ( a i ) E 1, with Ilalll, 5 1 (see Theorem B.5 in Appendix B). Define a gauge A : [0,1] + (0, +GO) such that 1. if t E (&, then B ( t , A ( t ) )c k E N, € N, 2. A(&) < &, 3. A(0) < where no E N is given by a) or b). Assume that 0 = a0 < a1 < < am-1 < am = 1 and put Ii = [ Q ~ - ~ , c Li~= ] , 1 , ., . ,m. Suppose that { ( t i ,Ii), i = 1,. . . , m} is a A-fine partition of [0,1] (at this moment we do not distinguish between a K - or Ad-partition). Consider the first point-interval pair ( t l ,1 1 ) = ( t l ,[0,all) belonging to the partition {(ti,&), i = 1, . . . , m}. Then necessarily tl = 0. Indeed, if tl # 0 then either tl E (&, or tl = 2 1-1 for some k E N. In the first case we have by the property 1. of the gauge [0,a11 C (&, &) and in the second case it is

A)

(3,A),

9

.

.

A)

Comparison of the Bochner and McShane Integrals

161

1 [0,al] C [m - $q,& TI + &B] by the property 2. of the gauge A. The first possibility is evidently impossible while for 1 1 1 22k < zk-l the second we have 0 < F 5 0 and this is again impossible. by the choice of the gauge A (cf. 3 ) , there Since a1 < is an mo E N,mo 2 no such that a1 E -1. 1 Consider now { ( t i ,Ii),ti E then

(A, (h,A)},

For {(ti, Ii),ti

= &i we} have

2&

<--22kB

-

€

22k-1B’

(5.4.3)

Let us consider the integral sums corresponding to the A-fine partition {(ti,Ii),i = 1,.. . , m}. First we have

and by ( 5 . 4 . 2 ) we get

Banach Space Integration

162

(m+...> (&

&

&

= 2k

-

_5 _E 42k

+ -)2 k&+ 2 &

< 2-.

2k

&

=2k

(1

1 +J

(5.4.4)

Secondly, by (5.4.3),

(5.4.5)

Since f(0)

= 0,

we have

m

m

i=l

i=2

because, as it was shown above, the first point-interval in the partition is ( O , l 1 ) = (0, [0, all). We know that a1 E and the system of intervals &, i = 2, . . . , m covers the interval [ q1 ,1. Therefore the corresponding tags t i , i = 2 , . . . , m contain all the points 2"1-',

(A, h]

k = 1 , .. . , mo and at least one of the tags t i , i = 2 , . . . , m belongs to the interval ($, &), k = 1 , .. . , mo and no tags ti, i = 2 , . . . , m belong to [0,a l ) . Hence

Comparison of the Bochner and McShane Integrals

+zc k=l t . 2

f ( t i ) P ( l i )-

"

1

k=mo+l

1

-

c

163

m

Consider the case (a) and the situation of a) for the series c

E

1 &zk

Assume that {(ti,Ii),i = 1,.. . , m } is a A-fine K-partition of [0,1]. Then by (5.4.4), (5.4.5) and (5.4.6) we obtain

A

mo

52 c 5+ k=l

&

c& + mo

k=l

E

< 2~ + 2~ + E

=5

~ .

164

Banach Space Integration

Hence f is Henstock-Kurzweil integrable and ('FIX)

Jif =

c p = 1 3.k.

For the case (b), the situation of b) for the series CE1+ z k and a A-fine Ad-partition {(ti,Ii),i = 1,.. . , m} of [0,1] the circumstances are different. Namely, some of the intervals 12 contained in &) can have the point 0 as their tag while the tags of the remaining intervals of this sort belong t o with mo 2 k 2 no. some intervals of the form ($, Hence we have

(2h,

A)

m

f ( t i ) P ( 1 i )-

" 1 3 2 k = k=l

i=l

k=l t "gT .- 1

By b) we have

c m

f(tz)P(Iz) -

i=2

" 1 3 " k k=l

k=mo+l

Y

(5.4.7)

Comparison of the Bochner and McShane Integrals

165

because IP(UtiE(&,&) 12)- 11 5 1. Hence, by (5.4.4), (5.4.5) and (5.4.7), we obtain (similarly as in the case (a))

Jt c;=,

and f is McShane integrable and ( M ) f = g1z k . In this way we have shown that the conditions in (a) and (b) are sufficient. We now consider the necessity of conditions (a) and (b). For (a), by Hake’s Theorem 3.4.5, we have

czl

and this shows that the series f z k converges. By Theorem 4.1.11 the indefinite McShane integral is countably additive. Thus,

&/*

e ( M ) k=l

f

=

- 1 -zk 2% i=l

=

(M)

and since any rearrangement of the intervals [$,&]satisfies the same condition, the series $& converges unconditionally. Concerning (c) observe that by Theorem 5.1.4 our function f is strongly McShane integrable if and only if it is Bochner integrable. By Proposition 1.4.5 this happens if and only if the series $ z k is absolutely convergent and (c) is proved. 0

xFl

czl

Banach Space Integration

166

The techniques of the proof of Proposition 5.4.1 will be exploited for showing the following result. Proposition 5.4.2. If I I $ z k ll X < B , B > 1 and the series C;=,& z k is unconditionally convergent then the function f : [0,I] + X from (5.4.1) is strongly Henstock-Kurzweil integrable. Proof. Define F : [0,1] --+ X as follows:

for t E

(&,I

n E N and F ( 0 ) = 0 E x.

It is easy to see that F ( & ) = C;=, & z k and that the function F is continuous on [O,1]. Since the series Cp=l$ z k converges unconditionally there is an no E N such that (5.4.8)

for every sequence a = (ai) E 1, with IJalll, 5 1 (see Theorem B.5 in Appendix B). Define the gauge A : [0,1] t ( 0 , ~ in ) the same way as in the proof of the previous Proposition 5.4.1 and assume that {(ti,[ai-l,ail), i = 1,.. . , r n } is a A-fine K-partition of [O,1] with 0 = a0 < al < < a, = 1. In the proof of Proposition 5.4.1 the properties of such a partition are described in detail. &) for some j E N If (ti,[ai-l,ail) is such that ti E then [&$-I, ail c by the properties of the gauge A and it is easy to compute that

(5,

(5,A)

f ( t Z ) ( Q i - ai-1) - [F(az)- F(CLi-l)] = 0.

If (ti,[ai-l,ail) is such that ti

=

& for some j

(5.4.9)

E N then

Comparison of the Bochner and McShane Integrals

167

and

=

1 (ai- - ) Z j

1 - (ai - -)Zj-1

23-1

=

23-1

1 (a2 - + ( Z j 23-

- zj-1).

Hence

1

+ 2j-l)B

< (ai - -)(2’ 23-1

1 -)B 2

&

< -2’(1+ 22jB

because IIzlcllx 5 2 k B by the assumption and -

1 1 - < A(?)

23-1

-

2.1-

by the definition of the gauge A.

&

<22jB

=

3 & --

2 23

168

Banach Space Integration

For the case of the first point-interval pair (0, [O, all) we have for some mo E N, f(0) = 0 , F ( 0 ) = 0 and a1 E 2momo 2 no and by 5.4.8 we get

(A,

Ilf(tl>(.Il- a o )

=

-

[Wd- F(ao>lllx

1 (a12m0-1)5Zmo

+

"

c

(5.4.11)

1

<E

Uk---Zk 2k

- 1 5 1 and mo 2 no. since 0 < Hence by (5.4.9), (5.4.10) and (5.4.11) we obtain

and f is strongly Henstock-Kurzweil integrable.

0

The properties of the function f from (5.4.1) presented in the previous Propositions 5.4.1 and 5.4.2 show the following: By the Dvoretzky-Rogers theorem B.11 and its Corollary B.12 in Appendix B, in every infinite-dimensional Banach space W

X there is an unconditionally convergent series

C zk, z k E X k=l

that is not absolutely convergent. Putting z k = 2 k z k for k E N we obtain by (a) and (b) from Proposition 5.4.1 that the corresponding function f : [0,1] + X is McShane integrable but not strongly McShane integrable and therefore the inclusion S M c M is proper. On the other hand, Proposition 5.4.2 shows that our function f is strongly Henstock-Kurzweil integrable and therefore also the inclusion S M c SEX is proper.

Comparison of the Bochner and McShane Integrals

169

Basically the example from Proposition 5.4.2 (with some small technical differences) was presented in the paper [DPMOaa] by L. Di Piazza and V. Marraffa saying that there is a function f : [0,1] --+ X variationally Henstock (=strongly Henstock-Kurzweil) integrable but not variationally McShane (=strongly McShane = Bochner) integrable.

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Chapter 6

Comparison of the Pettis and McShane Integrals

In this chapter we compare the concepts of Pettis and McShane integrals described in Chapters 2 and 3.

6.1

McShane integrable functions are Pettis integrable

Proposition 6.1.1. I f f : I t X is McShane integrable with ( M )JI f E X then for every x* E X* the real function x*(f ) : I --+ R is McShane integrable and

Similarly, i f f : I + X is Henstock-Kurzweil integrable with ('FIXJI) f E X then for every x* E X* the real function x*(f) :

I

--+

R is Henstock-Kurzweil integrable and

('FIX)/x*(f) I

lf).

= x*((7-IK)

Proof. By Definition 3.2.1 for every + (0, +a) such that

E

> 0 there is a gauge

S :I

for every &fine Ill-partition { ( t i ,Ii), i 171

=

1,. . . ,p } of I .

172

Banach Space Integration

X*then by the previous inequality we have

If x* E

for every 6-fine M-partition {(ti,I i ) , i = 1,.. . , p } of I . This proves the first part of the proposition. For the second part the same reasoning works if we use K-partitions instead of M partitions. 0

Remark. Note that by Remark 1.3.14 and by Theorem 4.3.4the McShane and Lebesgue integrals of x*(f ) : I -+ R coincide and therefore we can replace in Proposition 6.1.1 the McShane integrability of x*(f ) by its Lebesgue integrability. consequently, we also have that the function f : I X is weakly measurable. -+

Theorem 6.1.2. I f f : I -+ X is McShane integrable with ( M ) f E X then f is also Pettis integrable and

sI

(6.1.1)

for every measurable E c I . Hence we have M c P . Proof. According to the previous Remark the function f : I X is weakly measurable. By Theorem 4.1.6 for every measurable set E c I the function X E is McShane integrable and ( M ) f X E = ( M )JE f E X by definition. --f

sI

f

a

*

Hence by Proposition 6.1.1 for every x* E tion x*(f X E ) is McShane integrable and

X*the real func-

( M ) / Ix * ( f * X E = ) ( M ) k x * ( f=) x * ( ( M ) k f ) .

Comparison of the Pettis and McShane Integrals

173

By Definition 2.2.1 (or 2.2.2) this implies that f is Pettis inte0 grable and (6.1.1) holds.

6.2

The problem of P

cM

In the previous section we have proved that every McShane integrable f : I + X is Pettis integrable, i.e. M c P. Now we will consider the converse inclusion P c M which is rather complicated and depends on the properties of the Banach space X in which the functions f take their values. The first result in this direction belongs to R. A. Gordon [G90] (Theorem 17) and reads as follows.

Theorem 6.2.1. Let f : I -+X be measurable. Iff i s Pettis integrable o n I t h e n f is McShane integrable o n I .

Proof. By Proposition 1.1.9 the measurability of f implies that there is a bounded measurable function g : I + X and a measurable function h : I -+X of the form

n= 1

with xn E X , En c I , n E N, En being pairwise disjoint measurable sets such that

f ( t )= g ( t )

+ h@),t E I .

The function g is bounded and measurable, therefore it is Bochner integrable by Theorem 1.4.3 because I is assumed to be a compact interval. Therefore g is McShane integrable by Theorem 4.3.4 and by Theorem 6.1.2 it is also Pettis integrable. Since f is assumed to be Pettis integrable, the function h = f - g must be Pettis integrable and therefore by Proposition z, . p ( E n )converges unconditionally in X. 2.3.3 the series C,"=, By Theorem 4.1.13 the function h is McShane integrable and therefore f is also McShane integrable. 0

174

Banach Space Integration

The next theorem is a corollary of Theorem 6.2.1. It is also mentioned in Gordon's paper [G90], p. 566.

Theorem 6.2.2. A s s u m e that the Banach space X is separable and that f : I X i s Pettis integrable. T h e n f is McShane int egra b 1e . --+

Proof. The Pettis integrability o f f assumes that f is weakly measurable. By Proposition 1.1.10 the function f is measurable 0 and Theorem 6.2.1 gives the McShane integrability of f. Using Theorem 6.2.2 and Theorem 6.1.2 we obtain immediately the following result (see also [FM94], 2D Corollary).

Corollary 6.2.3. A s s u m e that the Banach space X is separable. T h e n f : I X is Pettis integrable if and only if it is McShane integrable, i.e. M = P holds in this case. ---f

Now we give another interesting relation between P and M due to D. H. F'remlin [F94]. Let us start with the following lemma (see [F94], (6. Lemma)).

Lemma 6.2.4. Let g : I --+ R be a real function. Let b : I (0, +GO) be a gauge o n I and let 7 > 0 be such that

t

W

i=l

for every 6-fine K - s y s t e m {(ti,Ii), i = 1,.. . , p } in I .

Then

~ U n( I

U int

B ( t , b ( t ) ) )5

dt)2E

for every

E

2&

> 0.

Proof. Assume that F c I n int B ( t ,b ( t ) ) is an arbitrary closed set. Then there is a b-fine K-system { ( t i , Ii), i = 1,.. . , p } with ti E F such that F c Ii. Of course we have

Ui

Comparison of the Pettis and McShane Integrals

175

g ( t i ) 2 E for i = 1,. . . ,p . This implies

i

i

and by the assumption

i

'I Hence p ( F ) 5 -. &

Since the closed set F C Ir7Ug(t)2E int B ( t ,d ( t ) ) can be taken arbitrary (closely approximating the set InU,,,),, int B(t,6 ( t ) ) ) we get

Theorem 6.2.5. Assume that f : I -+X is Pettis integrable and Henstock- Kurzweil integrable. Then f is McShane integrable and

Proof. It is easy to see by Definition 3.2.2 that if x* E X * then x*(f) : I + R is Henstock-Kurzweil integrable and

holds. Since f is Pettis integrable, the function x*(f .xE)is McShane integrable, and therefore also Henstock-Kurzweil integrable for every measurable E c I and x* E X*. This implies that for any x* E X*we have

176

Banach Space Integration

and therefore (6.2.2) Let us set

c = {x*(f); x* E B(X*)}= T ( B ( X * )c) L1. The operator T : X * t L1 is given by T ( x * )= x*(f)for x* E X * (see (2.1.2)). By Theorem 2.3.7 a) the operator T is weakly compact and therefore the set C = T ( B ( X * )is) weakly compact. This implies that for every rl > 0 there is a finite collection hl, . . . , h, E C such that for any h E C there is an hi such that ( M )JI Ih - hi1 I q. Let E > 0. For k E N set C

rlk =

2y2e

+ 12k) > 0.

(6.2.3)

Choose hk,l,. . . , h k , , ( k ) E C such that for every h E C there is an index 1 E (1,. . . , r ( k ) ) such that

(Note that we use the McShane integral here which is equivalent to the Lebesgue integral as we know by Theorem 5.2.3.) By the Henstock-Kurzweil integrability off and the McShane integrability of any element of C , for every k E N there is a gauge d k : I --f (0, +GO) such that

177

Comparison of the Pettis and McShane Integrals

if {(ti,Ii), i

if ((ti,Ii),i

=

1,. . . , p } is a Sk-fine K-partition of I , and

=

1,.. . , p } is a Sk-fine M-partition of I and I E

(1, ' . ,+)). For k E N set *

We have UkcNA k = I . Define S ( t ) = S k ( t ) for t E Ak.

Our aim is to show that f is McShane integrable with ( M ) f = ( P )J, f and the gauge S will do the job. Assume that {(ti,I i ) , i = 1,. . . , n } is an arbitrary &fine Adpartition of I and take any h E C , i.e. h = z*(f)for some

s,

z*

E B(X*). Let k E N be fixed. Let Jk =

{i; i E (1,.. . , n } ,ti E

Ak}

and

sI

Take I E (1,. . . , r ( k ) }for which ( M ) Then

Ih-hk,lI 5

qk

by (6.2.4). (6.2.7)

178

Banach Space Integration

and

by the Saks-Henstock lemma 3.4.2 because {(ti,Ii); ti E I k } is a bk-fine M-system and (6.2.6) holds. Let { (v,,V,), T = 1,. . . , m} be an arbitrary &-fine K-system in I and put

H=UV,. r

Then, by (6.2.1), ('FtIc) JH f = ( P )JH f and the Saks-Henstock lemma 3.4.1 implies by (6.2.5) the inequality

Hence for every x* E B ( X * )we have

Comparison of the Pettis and McShane Integrals

179

and this yields

because both h and

hk,l

are of the form x*(f) for some x* E

B(X*). Since I(M)J H ( h- h k , l ) l 5 gives

r]k

by (6.2.4), the last inequality

E ( h ( u r )- hk,l(%))P(K))5

3Vk.

r

Define

Using Lemma 6.2.4 we obtain (6.2.9) Let us consider the set

i E J k , h ( t i ) - - h k , l (ti)>&

Since i E J k implies ti E A k , we have 6 ( t i ) = &(ti) and li C B(ti,b,(ti)). It is not difficult to check that

Therefore by (6.2.9)

180

Banach Space Integration

and similarly also

This gives

Now we have for i Jk

IE

c

P(L)

l2kVk +7 .

i E Jk

The inequalities (6.2.7), (6.2.8) and (6.2.10) give

Comparison of the Pettis and McShane Integrals

=

12k ( 2 + -)qk

-k & p ( H k ) .

&

12k By (6.2.3) we have (2+-)qk & we obtain

=

c 5+ 1

Ep(I) = ( 1

k

Since h E C was arbitrary we have

for any z* E

X*and this implies

(6.2.11)

& 3 and by the estimate (6.2.11)

Summing over k we get

=E

181

+

P(I))E.

182

Banach Space Integration

This inequality is fulfilled for every &fine M-partition and therefore f is McShane integrable and

as required.

{ ( t i ,I i ) }

I7

Since by Theorem 6.1.2 every McShane integrable function f is Pettis integrable and by Theorem 3.2.3 also Henstock-Kurzweil integrable, we can state the following corollary of Theorem 6.2.5. Theorem 6.2.6. A function f : I X is McShane integrable if and only af it is Pettis integrable and Henstock-Kurzweil integrable, i.e. we have M = P n 7-K. --f

This is the result of D. H. F'remlin given in [F94], (8. Theorem). Our proof of Theorem 6.2.5 follows closely the idea from [F94]. Using Theorem 6.2.6 we obtain immediately the following corollary. Corollary 6.2.7. Assume that f : I -+ X is Pettis integrable. Then f is McShane integrable if and only if it is HenstockKurzweil integrable.

F'remlin's result from Theorem 6.2.6 leads to the following interesting characterization of McShane integrable functions, see 9. Corollary in [F94]. Theorem 6.2.8. Let f : I t X be a function. Then f is McShane integrable if and only if for every measurable set E c I the function f . X E : I -+ X is Henstock-Kurxweil integrable.

Proof. If f is McShane integrable then by Theorem 4.1.6 the function f - X E is McShane integrable for every measurable E c I and Theorem 3.2.3 implies Henstock-Kurzweil integrability of f X E for every measurable E c I . If on the other hand f ' X E is Henstock-Kurzweil integrable for every measurable E c I with (XK)JI f - X E = ('FIK)JE f E X, the second part of Proposition 6.1.1 gives that the real function

Comparison of the Pettis and McShane Integrals X* (f X E ) = X* (f) X E is Henstock-Kurzweil

measurable E

c I and

183

integrable for every

for any x* E X*. For every x* E X* the real function x*(f) being HenstockKurzweil integrable is measurable (see e.g. Theorem 6.5.5 in [LVOO] or Theorem 9.12 in [G94]) and therefore also Lebesgue (=McShane) integrable while

for any x* E X* and E c I measurable. Thus f is Pettis integrable and by Theorem 6.2.5 it is also McShane integrable. 0

6.2.1

Functions weakly equivalent to measurable ones

Let us start with some definitions.

Definition 6.2.9. A function cp : I --+ X is called scalarly negligible if x * ( f ( t ) )= 0 for every x* E X*and almost all t E I . Two functions f , g : I --+ X are called weakly equivalent if their difference f - g is scalarly negligible. Proposition 6.2.10. If cp : I t X is scalarly negligible t h e n cp is Pettis integrable and ( P ) cp = 0 for every measurable

E

c I.

SE

Proof. Since x*(cp) = 0 a.e. in I for any x* E X * , we have ( M )JE x* ( c p ) = 0 for every measurable E c I . Hence cp is Dunford integrable with (D) JE cp = 0 E X and this proves the proposition (cf. Definition 2.2.1). 0

Let us consider the case of a Pettis integrable f : I which is weakly equivalent to a measurable function g : I

--+ --+

X X.

184

Banach Space Integration

+

Then f = g f - g where f - g is scalarly negligible. By Proposition 6.2.10 f - g is Pettis integrable and therefore also g is Pettis integrable. By Gordon's Theorem 6.2.1 the function g is McShane integrable and the problem of McShane integrability of f reduces to the problem of McShane integrability of the scalarly negligible difference f - g. Let us focus on the problem when a scalarly negligible cp : I + X is McShane integrable. Proposition 6.2.11. A s s u m e that cp : I ligi b1e. Set

A,

=

X as scalarly neg-

{ t E I ; 3x* E B ( X * ) such that x*(cp(t))= 0 }

c

.*(cp(ti))P(Ii)

i

t

cc 03

=

n = l i,tiEN,

X*(P(ti))P(A)

Comparison of the Pettis and McShane Integrals M

co

n=l

n=l

185

Y

and this means that the set {x*(cp);x* E B(X*)} is M-equiintegrable. By Proposition 3.5.4 we obtain the McShane integrability of cp and of course ( M )J' cp = 0. 0

Remark. The statement of Proposition 6.2.11 concerns the special case when there is a universal set A c I with p ( A ) = 0 such that z*(cp(t)) = 0 for all t E I \ A and x* E X*. It is clear that in the general case the set A , from Proposition 6.2.11 can be of positive measure and Proposition 6.2.11 cannot be used in such a case. Lemma 6.2.12. Assume that cp : I + X is scalarly negligible. Then for every sequence xk E B(X*), rn E N the set {x&(f); m E N} is M-equi-integrable. Proof. Let

E

> 0 be given. Let us set Am

=

{ t E I ; Xk(cp(t))# 0)

for m E N. Since cp is scalarly negligible we have p(A,) = 0 for m E N and therefore for A = UmENAmwe have p ( A ) = 0. For n E N define

Nn

=

{ t E A; n - 1 5 Ilpllx < n}.

Since p ( A ) = 0 we have also p ( N n ) = 0 for every n E N and therefore there exist open sets Gn C I such that Nn c Gn and & P(Gn) < Define a gauge S : I --+ (0, +oo) such that S(t) = 1 for t E I \ A and B ( t ,S(t))c G, for t E N,. We have U n Nn = A and therefore 6 is defined for all t E I . If {(ti,I i ) } is and arbitrary &fine M-partition of I then for every m E N we have

186

Banach Space Integration

n=l i,tiEN,

n = l i,tiEN,,

n=l

n= 1

and this means that the set {xk(cp); x* E B ( X * ) }is M-equiintegrable.

Lemma 6.2.13. Let cp : I t X be scalarly negligible and assume that the ball B ( X * )c X * is weak* separable. Then the set {x*(cp);x* E B ( X * ) } is M-equi-integrable and therefore cp is McShane integrable b y Proposition 3.5.4.

Proof. Since B ( X * )C X * is assumed to be weak* separable, there exists a sequence { x k E B ( X * ) ; m E W} such that for every x* E B ( X * ) there exists a subsequence { x k k = xz E B ( X * ) ;k E N} of {x; E B ( X * ) ;m E N} such that xL(x) -+ x * ( x )for every x E X if k

+ 00.

(6.2.12)

Assume that x* E B ( X * )is given. Then by (6.2.12) we have

xi(cp(t))+ x*(cp(t))for every t E I if k

--f

00.

(6.2.13)

By Lemma 6.2.12 the set {xk(cp);m E W} is M-equi-integrable and therefore also {z;(cp);

k E N} is M-equi-integrable.

(6.2.14)

Assume now that E > 0 is arbitrary. Then by (6.2.13) we obtain that for any t E I there is a j o = j g ( ~ , t )E N such that

Comparison of the Pettis and McShane Integrals

187

Since (6.2.14) holds, by Definition 3.5.1 there is a gauge b : 1 3 (0,m) such that

for every Ic E N provided D = { ( t i ,I i ) } is a &fine Ad-partition of I . Let D = { ( t i ,I i ) } be an arbitrary &fine M-partition of I and let k E W be such that k > max(jo(E,ti)). Then using (6.2.15) and (6.2.16) we obtain

P

< E C p ( I , ) + E = +(I)

+ 1).

i=l

Since x* E B(X*) and E > 0 have been taken arbitrarily, we 0 obtain the M-equiintegrability of {x*(cp);z*E B ( X * ) } .

Theorem 6.2.14. I f f : I t X is Pettis integrable on I , f is weakly equivalent to a measurable function g : I X and the ball B(X*)c X * is weaP separable then f is McShane integrable on I . ---f

Proof. By definition the function f and by Lemma 6.2.13 the set

-g

is scalarly negligible

{ x * ( f - 9 ) ; x* E B(X*)}

188

Banach Space Integration

is M-equi-integrable. Hence by Proposition 3.5.4 the function f - g is McShane integrable and this implies the McShane integrability of f = g (f - g), because g is McShane integrable by Theorem 6.2.1.

+

Remarks. The result of Theorem 6.2.14 was obtained in [YSO2], Theorem 23 in a slightly different form. The assumption of weak equivalence of the Pettis integrable function f : I -+ X to a measurable function in Theorem 6.2.14 is another restriction on the Banach space X. In [E77] G. A. Edgar proved that f is weakly equivalent to a measurable function if and only if the image measure f(p) is tight in the space X equipped with the weak topology (see 5.2 Theorem in [E77]). In 5.4. Proposition of [E77] it is shown that every weakly measurable f : I -+ X is weakly equivalent to a measurable function if and only if the space X equipped with the weak topology is measure-compact. These results can be used to reformulate Theorem 6.2.14 by replacing the assumption of weak equivalence of f to a measurable function by the appropriate equivalents given by Edgar. 6.2.2

P

C

M does not hold i n general

By Theorem 6.1.2 we know that every McShane integrable function is also Pettis integrable. The first example showing that there are Pettis integrable functions that are not McShane integrable was given by Fkemlin and Mendoza in [FM94], 3C Example. Their example is as follows: Assume that Em, rn E N is a stochastically independent se1 quence of measurable sets in [0,1] C R with p ( E m ) = m+l for rn E N. The the set {xE,; rn E N} is stable in the sense of Talagrand. Define f : [0,1] -+Z,(N) as follows:

f ( t ) ( m )= 1 if t E Em, f(t)(rn)

=0

if t E [0,1] \Em.

Comparison of the Pettis and McShane Integrals

189

Since this function is properly measurable it is Talagrand integrable and therefore it must be Pettis integrable. The value of the integral is

In [FM94] it is then proved that the assumption of McShane integrability of this function f leads to a contradiction. This result of Fremlin and Mendoza shows that the possible hypothesis of P C M is false for the case of general Banach spaces X. Another example is given by L. Di Piazza and D. Preiss in [DPP03] showing that at least under the Continuum Hypothesis there is a scalarly negligible function f : [0,1] -+lm(wl) which is not McShane integrable, where w1 is the first uncountable ordinal. Note that by Proposition 6.2.10 the function f must be Pettis integrable with the Pettis integral equal to 0 E X. Let us present the example of Di Piazza and Preiss: Let N,, a E wl,C,, a E w1 be two collections of subsets of the interval [0,1] such that p ( N , ) = 0 for every a E w1, if a < ,O then N, C Np, for every subset E C [0, I] with p ( E ) = 0 there is an a E w1 such that E C N,, C, is countable for every Q E w1, if Q < ,O then C, c C p , for every countable subset E c [0,1] there is an Q E w1 such that E C C,. Define

f ( t ) ( a )= 1 if t E N,

\ C,, f ( t ) ( a )= 0 if t E [0,1]\ ( N , \ C,).

For this function f : [0, I] --3 Zw(wl) it is shown that it is scalarly negligible but not McShane integrable. Both of the examples open the problem of characterizing Banach spaces for which P C M holds. Moreover, the example of Di Piazza and Preiss opens also the

190

Banach Space Integration

problem of characterizing Banach spaces for which every scalarly negligible function is McShane integrable. For a long time the only sufficient condition for P c M was the separability of the space X (Theorem 6.2.2). For the second problem a partial answer is given by Lemma 6.2.13. Concerning the above problems we have to focus on nonseparable Banach spaces. A crucial step in this field was achieved by L. Di Piazza and D. Preiss in [DPPOS]. They proved the following

Theorem 6.2.15. If X is a super-reflexive space or i f X = c 0 ( r ) for some set I' then every Pettis integrable function f : [0,1] + X is also McShane integrable.

A proof of this result is fairly beyond the scope of our text. It uses deep results concerning nonseparable Banach spaces and their geometry. The extraordinarily compendious monograph [F-ZOl] is the best source for understanding this. Let us only recall that a Banach space X is said to be superreflexive if every Banach space finitely representable in X is reflexive. A Banach space Y is finitely representable in X if for every finite-dimensional subspace F c Y and any E > 0 there is a linear one-to-one operator T : F +. T ( F ) c X such that llTll IIT-q < 1 E . For the proof of Theorem 2.14 in [DPP03] the authors have used the characterization of super-reflexive spaces X as spaces having an equivalent uniformly convex norm and the fact that these spaces admit so called long sequences of projections on X . *

+

Chapter 7

Primitive of the McShane and Henstock-Kurzweil Integrals

In this chapter we discuss the primitives (indefinite integrals) of McShane and Henstock-Kurzweil integrable functions described in Chapters 3 and 4 as well as some connections of the integrals with the concept of derivative of a Banach space-valued function. In the subsequent parts of the book we consider mostly functions defined on one-dimensional intervals, i.e. the situation of I = [a,b] c R. Given a function F : [a,b] -+X we may define for J = [c,d] c [a,b], J E Z an additive interval function : Z + X (1is the set of all compact subintervals in [a,b ] ) by the relation

F

F ( J ) = F ( d )- F ( c ) . On the other hand if a function F : [a,b] + X by the relation

F : Z+ X is given, we define

In this way we have an evident one-to-one correspondence between point-functions and additive interval functions defined on [a,b], Z,respectively. If there is no confusion and things are clear from the context, we will identify point-functions with the additive interval functions which correspond to them and vice versa. 191

Banach Space Integration

192

7.1

Absolutely continuous functions and functions of bounded variation

First let us recall some elementary facts concerning continuity of Banach space-valued functions.

Definition 7.1.1. (a) F : [a,b] -+ X is said to be (strongly) continuous at t o E [a, b] if

(b) A function F : [a,b]t X is said to be weakly continuous at t o E [a, b] if lim Iz*(F(t)- F ( t 0 ) ) J= O

t+to

for all z*E X*is satisfied. (c) if F is continuous (weakly continuous) at each point t E [a,b]then F is called continuous (weakly continuous) on [a,b].

Remark. If F : [a,b] --+ X is continuous at t o E [a,b] then F is weakly continuous at to E [a,b]. The converse is not true. The first statement is easy to prove. For the second we show the following example. Example 7.1.2. Let F be a function from [0,1]into 12 defined as follows:

where el = ( 1 , 0 , 0 , . . .), e2 = (0, 1,0,0,. . .), etc. Then F ( t ) t F ( 0 ) = 0 E 12 weakly for t -+ 0. But, because llF(:)lll, = l(n E N),we get F ( t ) -H F ( 0 ) = 0 E 12 in the strong sense for t + 0. This means that F is weakly continuous at t o = 0 but not strongly continuous at to = 0.

Remark. There is a function F : [a,b] + X which is weakly continuous everywhere on [a,b] but not strongly continuous on [a,bl.

Primitive of the Henstock and the McShane Integrals

193

Example 7.1.3. Let F be a function from [0,1] into L2[0,27r] given as follows: F ( 0 ) = 0 E L2[0,27r]

E for O < t 5 1, J E [O,27r] [ F ( t ) ] ( J= ) sin t

The Riemann localization lemma shows that F is weakly continuous at t o = 0 but F is not strongly continuous at this point.

Theorem 7.1.4. If E c [u,b] is a closed set and F is continuous o n E , then F is uniformly continuous o n E . The proof of this theorem is completely similar to the case of real-valued functions, so we do not present it. Now let us introduce the concepts of absolute continuity and of bounded variation of functions for the case when they are Banach space-valued. The concepts of AC, BV, BVG and ACG functions are wellknown for the case of real functions mapping [u,b]into R. For details we refer e.g. to [LPY89],[DL89],[G94]or [S37]. Let us present first the definitions for the case of real functions.

Definition 7.1.5. Let F : [u,b]--+ R and let E be a subset of the interval [a, b] c R. (a) The function F is BV (or BV*)on E if s u p x i ( F ( d i )F(ci)l (or sup{Ci w ( F ,[ci,d i ] ) } ) is finite where the supremum is taken over all finite sequences { [ci,d i ] } of non-overlapping intervals that have endpoints in E and

w ( F , [C,dI)=

SUP

lF(P)- F ( 4

lff,PlCIC>dl

is the oscillation of F on [c,d]. (b) The function F is AC (or AC*) on E if for each E > 0 there exists q > 0 such that IF(di) - F(ci)l < E (or C i w ( F ,[ci,di])< E ) whenever {[ci,di]}is a finite sequence of

xi

194

Banach Space Integration

non-overlapping intervals that have endpoints in E and satisfy C i ( d 2 - C i > < rl. (c) The function F is BVG (or BVG*) on E if E can be expressed as a countable union of sets on each of which F is BV (or BV*). (d) The function F is ACG (or ACG*)on E if F is continuous on E and if E can be expressed as a countable union of sets on each of which F is AC (or AC*). Similar definitions can be presented for the case of Banach space-valued functions.

Definition 7.1.6. Let F : [a,b] + X and let E be a subset of [a,bl. (a) F is said to be of weakly bounded variation (or w B V ) o n E if for every x* E X * the numerical function x * ( F ) is of bounded variation (or B V ) on E . (b) F is said to be of bounded variation (or * B V ) o n E if sup 1) C i [ F ( d i )- F(ci)]J J Xis finite where the supremum is taken over all finite sequences { [ci,di]} of non-overlapping intervals that have endpoints in E . (c) F is said t o be of strongly bounded variation (or B V ) o n E if sup{Ci IIF(di) - F ( c i ) l l x } is finite where the supremum is taken over all finite sequences { [ci,d i ] } of non-overlapping intervals that have endpoints in E. (d) F is BV* on E if sup{Ci u(F, [ c i ,d i ] ) } is finite where the supremum is taken over all finite sequences { [ci,di]}of nonoverlapping intervals that have endpoints in E ,

UP, [Ci,diI)=

SUP

[a,PIc[cz ,&I

IIF(P) - F(a)Ilx

is the oscillation of F on [ci,di].

Definition 7.1.7. Let F : [a,b] + X and let E be a subset of [ a ,bl. (a) F is said t o be weakly absolutely continuous (or WAC)o n E if for every x* E X * the numerical function x * ( F )is absolutely continuous (or AC) on E ;

Primitive of the Henstock and the McShane Integrals

195

(b) F is absolutely continuous (or *AC)on E if for each E > 0 there exists q > 0 such that 11 C i [ F ( d i ) - F ( c i ) ] I l x< E whenever { [ci,di]} is a finite sequence of non-overlapping intervals that have endpoints in E and satisfy Ci(di- ci) < q. (c) F is said to be strongly absolutely continuous (or AC) on E if for each E > 0 there exists q > 0 such that CiIIF(di) - F ( c i ) l l ~< E whenever {[ci,di]} is a finite sequence of non-overlapping intervals that have endpoints in E and satisfy Ci(4- Ci) < 7. (d) F is AC* on E if for each E > 0 there exists q > 0 such that Ci w ( F , [ci,di])< E whenever { [ci,di]}is a finite sequence of non-overlapping intervals that have endpoints in E and satisfy Cz(di- C i ) < q.

Remark. It is easy to see that if e.g. F is *ACon a set E c [a,b] and Eo c E then F is also *AC on Eo. Similarly for WAC,AC, AC*,wBV, * B V ,B V , BV*. If F : [u,b]+ R in Definition 7.1.6 and Definition 7.1.7 is a real-function, then *BV = BV and *AC = AC. More generally the concepts coincide for the case of functions F : [a,b] + X whenever dimX < 00. But for the case of infinite dimensional Banach space-valued functions the concepts of BV and BV*, AC and AC* are different in general. Remark. If a function F : [a,b] -+ X is wBV, * B V , BV or BV* on [a,b] then it is bounded. For example, for a wBV function F on [a,b] the numerical function x * ( F ) is bounded. In fact, because F is of weakly bounded variation on [a,b], we have sup Ix*(F(t))l< +GO for any x* E X * . tE[a,bl

It follows immediately from the Banach-Steinhaus uniform boundedness principle that SUpt+b] IIF(t)IIx < +00. For functions which are *BV or BV the result can be obtained directly from the definitions.

196

Banach Space Integration

Remark. From the above Definitions 7.1.7 and 7.1.6 we can see that

AC* =+ AC

+ *AC3 WAC

BV* + BV

+ *BV j

and

wBV

for a given subset E c [a,b] where the implications mean that if a function has a given property then it has all the properties to the right in the chain of implications. In general, the reverse implications do not hold. The following example shows e.g. that there is a function F such that it is of bounded variation on [0, I] (*BV on [0,1]) but not of strongly bounded variation on [0,1] (not BV on [0,1]).

Example 7.1.8. Let F be a function from [0,1] into L,[O, 11 defined as follows:

F ( l ) ( J ) = 1 t E [o, 11. Then F ( t ) is of bounded variation on [0,1] but not of strongly bounded variation on [0,1].

Proof. According to the hypothesis of the example, for any finite collection of disjoint intervals ( a k , p k ) , k = 1,2, . . . , n we have

So F(t)is of bounded variation on [0,1]. On the other hand, for aj # pj we have llF(pk) - F ( a k ) I I L m [ O , l ] = 1. Thus F ( t ) is not of strongly bounded variation on [0,1]. 0

Primitive of the Henstock and the McShane Integrals

197

Theorem 7.1.9. A function F : [a,b] -+ X is of bounded variation on [a,b] if and only if F is of weakly bounded variation on [a,b].

Proof. (Necessity). By the inequality

5

[Iz* 11 11

zIF(Pk) k

-

F(ak)l

IIX,

x*,

which holds for all z*E where [ a k , P k ] , k = 1 , 2 , . . . , n is a finite sequence of non-overlapping intervals in [a,b], it follows immediately that if F is of bounded variation then F is of weakly bounded variation. (Sufficiency). If F is of weakly bounded variation on [a,b], then for every finite sequence of non-overlapping intervals [ a k , P k ] C [a,b] and z*E X * we have

I

Define a family 1 of bounded linear functionals on

X*

where [ a k , B k ] is a finite sequence of non-overlapping intervals in [a,bl. We have

198

Banach Space Integration

for all x* E X*. By the Banach-Steinhaus uniform boundedness principle, we obtain immediately

So F is of bounded variation on [a, b] ("BV on [a, b]).

0

To justify that under some circumstances it suffices to consider only closed sets E in Definitions 7.1.6 and 7.1.7 we prove the following theorem. Theorem 7.1.10. Let F : [a,b] -+ X and let E be a subset b]. If F is *AC of [a, b]. Suppose that F is continuous on [a, (WAC,AC, AC* wBV, * B V ,B V , BV*) on E , then F is *AC ( W A C ,AC, AC* wBV, * B V , B V , BV*) on the closure F of E.

Proof. Suppose e.g. that F is *AC on E . Then for every E > 0, there exists 7 > 0 such that for every finite sequence of non] bi - ai 1 < 7 overlapping intervals { [ai , bi)} with ai, bi E E and we have

xi

C F ( b i )- F ( u ~ ) < E . i

X

Now, let {lei,d i ] } be any finite sequence of non-overlapping intervals with ci, di E F and Idi - cil < 7 . For each i , there Iwi - u il < q such that exist ui,vi E E with ui < vi and

xi xi

llF(ui) - F(ci)llx < ~ / 2 aand llF(vi) - F(di)llx < ~ / 2 ~ . Observe that [ui, vi]may not be non-overlapping intervals. However, dividing the system of intervals {[ci,di]}into two parts, where in each part the intervals are disjoint, we can choose [ui, vi]t o be disjoint. Hence we may assume [ui, vi] to be nonoverlapping. As a result, we have

Primitive of the Henstock and the McShane Integrals

< & +& +& Therefore, F is AC on similar way.

E.

199

= 3&.

The other cases can be treated in a

Remark. We can see from Theorem 7.1.10 that if F is continuous on [a,b] then F is *AC on E if and only if F is *AC on E and vice versa. Similarly also for all cases listed in Theorem 7.1.10. This means that if it is a priori known that F is continuous on [a,b] then the set E can be assumed to be closed in Definitions 7.1.6 and 7.1.7. Theorem 7.1.11. Let F : [a,b] + X and let E be a closed subset of [a,b] with bounds c and d . Let G : [c,d] --f X be the function that equals F o n E and is linear o n the intervals contiguous to E. If F is BV (wBV, * B V , BV*, WAC, *AC, AC, AC*) on E , then G is BV (wBV, * B V ,BV, BV*, WAC, *AC, AC, AC*) o n [c,d ] . The proof of the theorem is similar to the case of a real function and it is not difficult. So we do not present it. Using the idea from [LPY89], pp. 27-28 the following can be proved.

Lemma 7.1.12. Let E be a closed set in [a,b] and let ( a ,b) \E be the union of open intervals (ck,dk) for k = 1,2, .... Suppose that F : [a,b] -+ X is continuous o n [a,b]. Then the following statements are equivalent:

Banach Space Integration

200

(i) F is AC* on E , (ii) F is AC on E and

k=l

(iii) (d) in Definition 7.1.7 holds with ai or bi belonging to E for every i. Let us mention that under the continuity condition, to proceed from AC on E to AC* on E we change either from the difference F ( y ) - F ( x ) to the oscillation w(F,[x,y]) or from the two endpoints of the intervals belonging to E to the possibility that at least one endpoint belongs to E .

7.2

Generalized absolute continuity and functions of generalized bounded variation

Definition 7.2.1. Let F : [a,b] --+ X and let E be a subset of [a,b]. The function F is wBVG (or *BVG, BVG, BVG*) on E if E can be expressed as a countable union of closed sets on each of which F is wBV (or * B V ,B V , S V * ) . Definition 7.2.2. A function F : [a,b] + X is wACG (or *ACG,ACG, ACG*)on E c [a,b]if F is continuous on E and if E can be expressed as a countable union of closed sets on each of which F is WAC (or *AC,AC, AC*). Remark. Similarly to the Remarks after Definitions 7.1.6 and 7.1.7 and according to the above Definitions 7.2.1 and 7.2.2 we can also see that

ACC

+ ACG +

*ACG+ wACG

and

BVG* + BVG

+ *BVG + wBVG.

Primitive of the Henstock and the McShane Integrals

201

In general, as the Example 7.1.8 shows, the reverse of above inclusions does not hold. Recall that a portion of a set E c R is any subset of E of the form E n ( a ,p) with a , p E R and a < p.

Theorem 7.2.3. Let a continuous F : [a,b] + X be given. The function F is *ACG (or wBVG, *BVG, BVG, BVG*, wACG, ACG, A C P ) on [a,b] i f and only i f every closed set contains a portion on which F is *AC (or wBV, * B V , B V , BV*, WAC,AC, AC*).

Proof. The condition is necessary. Since F is *ACG on [a,b], then [a,b] can be written as a countable union of closed sets E, such that F is *AC on each En. Let E c [a,b]is a closed set. By Baire’s Theorem ([DS],Theorem I.6.9), there exists at least one En0 which contains a portion P = E n I of E , I c [a,b]is an interval. Since F is *AC on En0,F is *AC on P . This means that F is *AC on the portion P of E . For showing sufficiency suppose that every closed set contains a portion on which F is *AC. Let A = {I,} be the sequence of all open intervals in [a,b]that have rational endpoints, and each I, can be written as a union of a countable closed sets E r ) such that F is *AC on each E p ) . This means that I, = U k E r ) ( n = 1 , 2 , . ) and F is *AC on each E p ) . Obviously, A # 8. Let H = [a,b] \ UJ., Then H is a closed set and

-

[a, b] 7H

U (UnI,)= H

u (u,u k E f ) ) .

(7.2.1)

Now we prove that H = 8. Suppose that H # 8, then there exists a portion P = H n I # 8 of H such that F is *AC on P . Without loss of generality we may assume that the open interval I has rational endpoints. By (7.2.1), we have

Banach Space Integration

202

So, I E A and therefore there is a m such that I shows

HnI,

=P

=

I,.

This

nIf 0

and F is *AC on H n I , = P n I . This is a contradiction since the set H by definition has no points in common with any of the intervals I , and therefore the sufficiency is proved. Similarly, we can prove the cases of wBV, *BV,BV, BV*, WAC,AC, AC*. 0

7.3

Differentiability

At first we introduce the concepts of differentiation. Definition 7.3.1. A function F : [a,b] + X is said to be weakly differentiable (or Gciteaux differentiable) at t E [a,b] if there is a x E X and for each x* E X*we have ( x * ( F ) ) ’ ( t= ) lim 6-0

1

+

x*(F(t 6))- x*(F(t)) = x*(x). 6

We denote the weak derivative of F at t by F,$(t) = 2 . If F is weakly differentiable at each t E [a,b] then F is called weakly differentiable on [a,b].

Definition 7.3.2. A function F : [a, b] X is said to be differentiable (or strongly differentiable, Fre‘chet differentiable) at t E [a, b] if there is a x E X such that ---f

i.e.

F(t

+6) - F(t) +x

strongly for 6 -+ 0.

6 We denote x = F’(t) the derivative of F at t.

Primitive of the Henstock and the McShane Integrals

203

If F is differentiable at each point t E [a,b]then F is called differentiable on [a,b]. The following shows the relationship between weak diff erentiability of a Banach space-valued function F ( t ) and the differentiability of the numerical function x * ( F ) for x* € X * . By definition it is evident that if F : [a,b] .--) X is weakly differentiable at t o E [a,b], then for every x* E X*the numerical function x * ( F ) of F is differentiable at to E [a,b] and

(x*(F))’(to)= x*(FL(tO)) for each x* E X * . For the reverse we have the following. Theorem 7.3.3. If F : [a,b] --+ X , the Banach space X is weakly sequentially complete and for every x* E X * the real function x * ( F ) is diflerentiable at to E [a,b] then F is weakly differentiable at t o E [a,b] and

(x*(F))’(to)= x*(FL(to)) for z*E X * . Proof. For any sequence t , E [a,b], n E N with tn + to let

Since the numerical function x * ( F )is differentiable at t o E [a,b], we have

for all x* E X * . So the sequence x,, n E

N is weakly Cauchy.

Banach Space Integration

204

Since X is assumed to be weakly sequentially complete, there exists xo E X such that

Let t6, n E W be another sequence with t6 t to. The hypothesis that the numerical function x * ( F ) is differentiable at t o E [u,b]implies that

for n t 00 and for all x* E X*. This means that

So we have

F(t’,) - F(t0) t xo weakly for n t:, - t o

-+

00.

Hence F is weakly differentiable at t o E [a,b] with xo = F&(to) and the theorem is proved. 0

Remark. For the statement of Theorem 7.3.3 the hypothesis that the Banach space X is weakly sequentially complete cannot be omitted. The following example shows this. Example 7.3.4. Let F be a function from [0,27r]into c defined as follows

Then for each x* E c* = ZI, the numerical function x * ( F )of F is differentiable on (0,2n),but F is not weakly differentiable on (0,

ad.

Primitive of the Henstock and the McShane Integrals

205

Proof. Since c* = 11, for each x* E c* there is a unique (bn) E l1 such that x * ( F ( t ) )=

c n

b, cos nt , t E [0,27r]. n

Since Enb, sin nt is uniformly and absolutely convergent on [0,27r], it follows immediately that (x*(F))’(t)= Enb, sinnt on [O, 27~1. Now suppose that the weak derivative of F at a point to E [0,27~]exists and &(to) = E c. By Definition 7.3.1, we have

(e:)

1b, sinnto

= (x*(F))’(to) = ~ * ( F ’ ( t o= ))

n

1bnc:, n

for all x* = (bn) E 11. From the formula above we obtain that (5;) = (sin kto). Note that limk-+msin kto does not exist. This contradicts the hypothesis (5:) E c. Hence, F is not weakly differentiable at any point of (0,27r). Now we discuss the relation of strong and weak differentiability. From the corresponding definitions it is easy to get the following theorem.

Theorem 7.3.5. I j F : [a, b] + X is diflerentiable at t o E [a,b] then the function F is weakly diflerentiable at t o E [a,b] and

F’(to) = F&). Remark. The reverse of Theorem 7.3.5 does not hold. This means that there is a function F which is weakly differentiable at to E [a, b] but not strongly differentiable at to E [a,b]. Example 7.3.6. Let F be a function from ( - 1 , l ) into ,Z > 1 given by

p

if s # i, s E (-1,l); F ( s ) = ( o , . . . , o , ~ , o , . . . ) , i f s = ,;1 ( n = 1 , 2 , . . . ).

Banach Space Integration

206

Then F is weakly differentiable at s = 0 but not strongly differentiable.

Proof.

So, it follows from I;

= I,

that

F ( s ) - F ( o ) t 0 weakly for s

--+

0.

S

This means the weak derivative of F at s = 0 equals 0 E I,. On the other hand, since

by the second part of the Theorem 7.3.5, we get that F is not strongly differentiable at s = 0. Finally, let us have a look at the relationship between the strong continuity and the weakly differentiability of a function on [ a ,b]. Theorem 7.3.7. Let F : [a,b] + X and t o E [a,b] be a n arbitrary point. If F is weakly diflerentiable at to t h e n F is continuous at to.

Proof. Taking a sequence 6,, n E N of positive real numbers, 6, + 0 for n -+00, define functionals x r on X * by X;(x*) = x*

for x* E X*

Obviously, x?, n E N is a sequence of bounded linear functionals on X * . Since F is weakly differentiable at t o , we obtain that for

Primitive of the Henstock and the McShane Integrals

207

every z* E X*the limit

exists. Hence there is a constant

P such that

Therefore

IIF(to + Sn) - J'(t~)llxI P and F is continuous at to.

*

IJnl

+

0 ( n 4 00) 0

Remark. The fact that a function F is continuous at t o E [a,b] does not imply in general that F is weakly differentiable at to. The following example shows this. Example 7.3.8. Let F : [0,1) + l1 be a function given as follows :

F(s)=

i"

if s # $,,n E N,s E [0,1]; i f s = :, n E N .

1

(O,... ,O,;,O,..-),

Then F is continuous at s

=0

Proof. Since IIF(s)((l,= 0 for s 1 we have n when s = i,

but not weakly differentiable.

# $, s E

F ( s ) -+ F(O) = O if s

[0,1]and IIF(s)((I1 =

+0

and F is strongly continuous at s = 0. Taking x: = (1,1,1,. . ) E ZT =,,Z we have

and

Banach Space Integration

208

Therefore the weak derivative of F ( s ) cannot exist at s

= 0.

0

Remark. Even if F is strongly absolutely continuous on [a,b], this still cannot imply that F is weakly differentiable on [a,b]. Example 7.3.9. Let F : [0, I] -+ L1[0, I] be the function given for 0 5 t 5 1 by

Then F is strongly absolutely continuous (and also of strongly bounded variation) on [0,1]but not weakly differentiable almost everywhere on [0, I].

Proof. We have

s,

P

llF(P>- F ( 4 l l L l

=

dt = (P - a>,

for any interval [cxl P] c [0,1]. Hence, for every sequence ( ( a k , P k ) ) of disjoint subintervals of [0,1] we have

c

llF(Pk) -

k

F ( a k ) ) ) h= c

( P k -ak). k

It follows immediately that F is strongly absolutely continuous on [O, 11. Let us observe that L1([0,1])* = L,[O, 11. Then for every x* = b ( J ) E L,[O, 13 we have rl

so d

- ( x * [ F ( t ) ] )= b ( t ) for almost all t E [O, 13 dt Now suppose that F is weakly differentiable almost everywhere on [0, 11 with the derivative y ( t ) E L1[0,1], i.e. F&(t)= y(t) a.e. in [O, I].

Primitive of the Henstock and the McShane Integrals

209

By Theorem 7.3.3, we obtain

for all x* = b ( ( ) E L,[O, I]. Combining the above equalities, we obtain

[ y ( t ) ] ( ( ) b ( ( ) d (= b ( t ) for almost all t E [0,1] 10

and for any x* = b ( J ) E L,[O, 11. Therefore, for each point to E [0, I] for which the above equality holds, take go = y(to) (of course, yo E L1[0,l]),we have

This contradicts the absolute continuity of integral of yo((). Hence, F is not weakly differentiable almost everywhere on [0,1]. 0

Let us close this section by the following important result which is well known for the case of real valued functions and which states that every derivative on a one-dimensional interval is Henstock-Kurzweil integrable even in the case of Banach spacevalued functions.

Theorem 7.3.10. If the function F : [a,b] X is diifferentiable on [a,b] with F'(t) = f ( t ) for t E [a,b] then f : [a,b] t X is Henstock-Kurzweil integrable and --f

s"

('Ftlc)

a

f = (m)/+ F'

= F(b)- F(a).

a

Proof. Assume that the function F : [u,b] -+ X is differentiable on [a,b] with F'(t) = f ( t ) for t E [a,b]. By the Definition 7.3.2 of the derivative this means that to a given E > 0 and t E [a,b]there is a A(t)= A(t,E)> 0 such that

210

Banach Space Integration

for s E [a,b], 0 < 1s - tl < A(t) it is

i.e. we have

for every

Hence, if

then

<

- &()a2 -

t ]+ It

- all) = &(a2 - a1).

Assume that

{(ti,[az-l,ail), 2

=

1 . .. , k }

is a A-fine K-partition of the interval [a,b]. By (7.3.2) we get

c k

F(b) - F ( a ) -

f(tj)(aj - aj-1)

k

=

C[F(Qj)

- F ( a j - 1 ) - f(tj)(Qj

- aj-l)]

Primitive of the Henstock and the McShane Integrals

211

k

5 EZ(CYj- CYj-1)

= E ( b - a).

j=l

Since E > 0 can be chosen arbitrarily small, this inequality shows that the function f = F' : [a,b] -+ X is Henstock-Kurzweil integrable and ('FIX) f = ('FIX) F' = F ( b ) - F ( a ) . 0

s,"

7.4

s,"

Primitives

In the subsequent parts of the text we will consider primitives (indefinite integrals) corresponding to functions f : [a, b] + X with respect to McShane and Henstock-Kurzweil integrations as well as to their strong versions. If f : [a,b] +. X is integrable in some sense X , X E {'FIK,M,SIFIK,SM}mentioned above, then the function F : [a, b] -+ X given by

is the primitive to f with respect to the given integration process. To the primitive we can refer also as to an interval function in spite of the correspondence presented at the beginning of this chapter. As it was noted in the Remark after Proposition 3.6.16 the Henstock-Kurzweil integral F ( t ) = ('FIX) J: f for t E [a,b] generates the interval function in the strong versions of McShane and Henstock-Kurzweil integrals and also for the McShane integral. The next result points out the basic property of all the primitives to functions integrable in the above mentioned senses.

Theorem 7.4.1. I f f : [a, b] --+ X is Henstock-Kurzweil integrable on [a,b] then the primitive F o f f is continuous on [a,b].

212

Banach Space Integration

Proof. The continuity follows from Saks-Henstock Lemma 3.4.1 and the following inequality

According to the above mentioned fact this result shows that all primitives we are dealing with are continuous. In the following subsections we deal with the primitives in all the above mentioned senses. 7.4.1

The strong Henstock-Kuraweil integral

Concerning differentiability of the primitive we have the following result.

Theorem 7.4.2. If f : [a,b] + X is strongly HenstockKurzweil integrable o n [a,b] with the primitive F , then F is differentiable almost everywhere and F’(t) = f(t) a.e. on [a,b].

Proof. Let E be the set of points t at which either F’(t) does not exist or, if it does, is not equal to f ( t ) . We shall prove that p ( E ) = 0. From the definition of E we see that for every t E E there is an q ( t ) > 0 such that for any 6 > 0 either there is a point u with 0 < t - u < 6 and IIF(t> - F ( u ) - f ( W - 4llx

or there is a point v with 0 < v llF(v> -

-t

W )- f ( t > ( v

-

> r](W- u>

< 6 and

t>llx> rl(t)(v- t>.

Fix n E N and let En denote the subset of E for which q ( t ) 2 1, n Then the above family of closed intervals [u, t]and [t,v]covers En in the Vitali sense. Applying the Vitali covering theorem, given E > O we can find [uk, vk] for k = 1,Z..., rn with uk = t k or

Primitive of the Henstock and the McShane Integrals ZI= ,tl, such

213

that

b ] , by Since f is strongly Henstock-Kurzweil integrable on [a, b] such that for any &fine Lemma 3.6.15 there is a gauge 6 on [a, [ui, 243)) of [a,b] we have K-partition {(ti,

c

llF(.i) - F ( U i ) - f (ti>(%- u2)llx < E .

i

When forming the above family of closed intervals, we may assume 6 < 6(t,) for k = 1 , 2..., m; then we have

2

dEn) <

llF(4 - F b k ) - f ( t k ) ( V k

i= 1

- Uk>llX

+

q(tk)

because 1 < n. Since E is arbitrary, the outer measure of En “l(tk) is 0 and so is E . 0 Theorem 7.4.2 is given in the paper [C92] of S. S . Cao. Our proof is adapted from the book [LPY89] of Lee P.-Y. (Theorem 5.7); another proof can be seen in the book [BaOl] of R. G. Bartle (Theorem 5.9) for the case of real valued functions. The result of the next theorem is known, see e.g. Proposition 4 in [DPM02a]. Theorem 7.4.3. If f : [a,b] -+ X is strongly HenstockKurzweil integrable o n [a,b], then the primitive F off is ACG* o n [a,b]. Proof. Since f is strongly Henstock-Kurzweil integrable on [a,b], for every E > 0 there is a function b : [a,b] -+ (0,m) such that for any &fine K-partition { (&, [uj, wj])} of [a, b] we have

c

llF(Vj)

j

- F b j ) - f(fj)bJj- uJIx

< &.

214

Banach Space Integration

We may assume that S ( ( ) 5 1 for [ E [a,b]. Put

I t [ u + - , io-+1- l }

n

i

n

for n = 2 , 3..., i = 1 , 2.... Fix Eni and let { [ a k , b k ] } be any finite collection of non-overlapping intervals with a k , b k E Eni for all k. Then { ( a k , [ a h , b k ] ) } is a &fine K-system in [u,b]. Furthermore, if ?&, z& are arbitrary points with a,+ U k < w k < b k , then { ( a k , [ a k , u k ] ) } and { ( b k , [ U k , b k ] ) ) are S-fine K-systems in [a,b]. Since, by Theorem 7.4.1, the primitive F is continuous on [a,b], there exist U k , U k with U k 5 U k 5 Wk 5 b k such that for the oscillation w(F,[ a k , b k ] ) o f F on [ a k , b k ] we have

<

w ( F ,[ a k , b k ] ) = I I F ( [ u k , v k ] ) l l X Thus,

because

Primitive of the Henstoclc and the McShane Integrals

k

215

k

< E -k x n ( b k - ak) k

and similarly for the other terms. x k ( b k - a k ) < 7. Then we get

Choose q 5

&

and let

c u ( F , [ak,b k ] ) 5 3&+ &. k

(z). Consequently,

Therefore, F is AC*(Eni)and also AC* F is ACG* on [a,61.

Theorem 7.4.4. Let f : [a,b] -+ X. Suppose there exists a function F : [a,b] -+ X which is ACG* on [a, b] such that F ' ( t ) = f ( t ) a e . on [a, b], then f is strongly Henstock-Kurzweil integrable on [a, b] with the primitive F .

Proof. Let S c [a,b] be of measure zero such that for t E [a,b] \ S we have F ' ( t ) = f ( t ) . For E [a,b] \ S , given E > 0 there is a 6(<) > 0 such that whenever E [u,v] c (< - 6(<), + 6(<)) we have

<

<

<

llN% 4)- f(E>(v- 4Ilx I EI'U - 4. Since F is ACG*, there is a sequence of closed sets ( E i )such b] and F is AC* on Ei for each i. that UiEi = [a, Let Y1 =

for i

= 2 , 3 , ....

El,

y, = Ez \ (El u Ez u ... u Ei-1)

Denote Sij the set of points t E S n j

-

such that

1 I Ilf(t>llx< j .

Obviously, Sij, j E N are pairwise disjoint and their union is the set S.

Banach Space Integration

216

Since F is also AC* on Sij, there is a qij

&

j2i+j

such that for any sequence of non-overlapping intervals ( I k ) with at least one endpoint of I k belonging to Sij and satisfying

k

we have

Choose Gij to be the union of a sequence of open intervals such that p(Gij) < qij and

Sij

c Gij.

<+

<

Now for E S i j , i , j = 1 , 2 , ...) put (< - S(<), 6(c)) c Gij. Hence we have defined a positive function S ( t ) on [a,b]. Take any &fine K-partition {(&, [ul, wl])). Splitting the sum El over 1 into two partial sums CFeS and CSES we obtain

c

Ilf(tl>(vl- U l ) -

F~[ul,wll)llx

1

< &(b- a) + 2 E . This proves that f is strongly Henstock-Kurzweil integrable to 0 F ( [ %bl) on [a,bl.

Primitive of the Henstock and the McShane Integrals

217

Combining Theorem 7.4.1, Theorem 7.4.3 and Theorem 7.4.4, we obtain the following.

Theorem 7.4.5. The function f : [a,b] t X is strongly Henstock-Kurzweil integrable on [a,b] with the primitive F if and only if F : [a,b] t X is continuous and ACG* o n [a,b] such that F’(t) = f ( t ) a.e. in [a,b]. Remark. Theorem 7.4.5 is a certain descriptive definition of the strong Henstock-Kurzweil integral which, in the case of realvalued functions, coincides with the well-known descriptive definition of the restricted Denjoy (= Henstock-Kurzweil) integral. For the case of vector-valued functions let us mention the following concept presented by A. P. Solodov in [SoOl]. Definition 7.4.6. A function f : [a,b] --+ X is called DenjoyBochner integrable (in the restricted sense) if there is a function F : [a,b] + X which is differentiable a.e. in [a,b] and ACG* on [a,b] such that F’(t) = f ( t )a.e. in [a,b]. Using this definition Solodov proves in [SoOl] the following.

Theorem 7.4.7. The function f : [a,b] t X is DenjoyBochner integrable if and only i f f is strongly Henstock-Kurzweil integrable on [a,b]. This Theorem is a reformulation of Theorem 7.4.5 given above. Let us close this subsection by the following result (see Theorem 9 in [ C E ] ) .

Theorem 7.4.8. If the function f : [a,b] t X is strongly Henstock-Kurzweil integrable on [a,b] then f is measurable. Proof. For x* E X * the real function x*(f) : [a,b] t IR is Henstock-Kurzweil integrable. Since real-valued HenstockKurzweil integrable functions are measurable (see e.g.Theorem 5.10 in [LPY89]or [G94])we have the measurability of x*(f ) for every x* E X * , i.e. f is weakly measurable. The primitive F ( t ) = (7-K)s,” f is continuous (by Theorem 7.4.1) and because this the set { F ( t ) ; tE [a,b]} C X is compact and therefore separable. If L c X is the closed linear subspace

218

Banach Space Integration

spanned by the set { F ( t ) ;t E [a,b]} c X , then L is separable. By the definition of the derivative of F at some point t E [a,b] we have F ' ( t ) E L if the derivative exists. Hence by Theorem 7.4.2 we have f ( t ) = F'(t) E L a.e. in [a,b] and this means that f is almost everywhere separable valued. Hence by the Pettis 0 measurability theorem 1.1.6 the function is measurable. 7.4.2

The McShane and the strong McShane integral

Using Theorem 7.4.2 and the fact that S M c S'FIK: (see the remark after Proposition 3.6.16) we obtain immediately the following result concerning differentiability of a strongly McShane integrable function. Theorem 7.4.9. If f : [a,b] -+ X is strongly McShane integrable o n [a,b] with the primitive F , then F is diflerentiable almost everywhere and F'(t) = f ( t ) a e . o n [a,b]. Theorem 7.4.10. If f : [a,b] -+ X is McShane integrable with the primitive F , then F is absolutely continuous (is *AC) on [a,b].

Proof. By Lemma 4.1.2 we obtain that for every exists q > 0 such that

E

> 0 there

whenever ( [ci,di]) is a finite sequence of non-overlapping intervals in [a,b] satisfying C i ( d i - c i ) < q. This shows that F (given 0 by ( M )s," f for t E [a,b]) is *AC on [a,b]. In the more special case of the strong McShane integral we have more. Theorem 7.4.11. Iff : [a,b] + X is strongly McShane integrable on [a,b] with the primitive F , then F is strongly absolutely continuous (is A C ) o n [a,b].

Primitive of the Henstock and the McShane Integrals

219

Proof. Let E > 0 be given, Since f is strongly McShane integrable on [a, b ] , there exists a gauge S on [a, b] such that a

i=l

whenever {(ti,[ui, wi]);i = 1,. . . , q } is an arbitrary S-fine M partition of [a,b]. Fix a 6-fine M-partition of [a, b]

{(ti, 1% put K

4);i = 1,. . , d, *

& 1 1 ~ 1; 5 i 5 q } and set 7 = K+1' { [aj,4 1 : j = 1,.. . ,p } is an arbitrary

= max{ Ilf(ti)

Suppose that finite collection of non-overlapping intervals in [a,b] such that - a j )< 7.By subdividing these intervals if necessary, we may assume that for each j , [ a j,Oj] , c [ui, wi]for some i . For each i , 1 5 i 5 q let

xy=,(P,

Mi = { j ; 1 5 j 5 p

with [q, p j ] C [ui, wi]}

and let

D = { ( [ c u j , @ j ] , t i ) : j E Mi,i = 1,* . . , q } . Note that D forms a 6-fine Ad-system in [a,b]. The closure [u,b] \ U j [ a j,4] , consists of a finite number of disjoint closed subintervals MI, c [a,b]. Taking an arbitrary 6fine M-partition DI, of MI, we take D U UkDI, to get a S-fine Ad-partition { (71, [q,d l ] ) } of [a,b]. Then

and

Banach Space Integration

220

j=1

j=1

5 E +K

P

E(@j

- aj)

< E + K q < 2~

j=1

and this proves the theorem.

0

Theorem 7.4.12. If f : [a,b] t X is McShane integrable on [a,b] wiih the primitive F , then for each x* E X* the real function x * ( F ) : [a,b] -+ R is differentiable almost everywhere on [a,b] and ( x * ( F ) ) ’ ( t = ) x*(f ( t ) )a. e. in [a,b]. If X is weakly sequentially complete then F is weakly differentiable a.e. in [a,b] and for the weak derivative we have F h ( t ) = f ( t ) a. e. in [a,b]. Proof. Since f is McShane integrable on [a,b] with the primitive F , the real function x*(f) is McShane integrable on [a,b] and

Sb

(M)

x * ( f )= z * ( F ( [ ab, ] ) )= x * ( F ( b )- F ( 4 )

for each x* E X*. By the properties of McShane integral of a real function, we obtain that x * ( F ) is differentiable almost everywhere on [a,b] and ( x * ( F ) ) ’ ( t )= x*(f ( t ) )a.e. The last statement follows from Theorem 7.3.3. For the derivative of the primitive of a strongly McShane integrable function we have the following result.

Theorem 7.4.13. If F is AC on [a,b] and F’(t) = f ( t )almost everywhere on [a,b], then f : [a,b] t X is strongly McShane integrable on [a,b] with the primitive F .

Primitive of the Henstock and the McShane Integrals

221

Proof. Let S be the set of points t at which either F’(t) does not exist or, if it does, is not equal to f ( t ) . Then S is of measure zero. For ( E [a, b] \ S, given E > 0 there is a 6([) > 0 such that whenever ( E [u, v] C ( E - d((), E 6(6)) we have

+

Let Sj denote the set of points t E S such that j - 1 5 Ilf(t)lIx < j , j E N. Obviously, Sj,j = 1 , 2 , ... are pairwise disjoint and their union is the set S. Since F is continuous and E2-j

AC on [a,b ] , F is also AC on Sj. So there is a rlj < - such j

that for any collection of non-overlapping intervals { I k } with at and satisfying least one endpoint of I k belonging to

sj

we have

Choose Gj to be the union of a sequence of open intervals (an open set) such that p(Gj) < qj and Sj

< E Sj, j

c Gj.

1,2, ..., put S(() > 0 such that (( S(<), S(<)) C Gj.Hence we have defined a positive function on [a, bl. Take any &fine M-partition {(&, [u~, wl])} of [a, b]. Splitting the sum C over D into two partial sums with $ S and E S respectively, we obtain Now for

<+

=

222

Banach Space Integration

< &(b- a ) + 2 ~ . This shows that f is strongly McShane integrable to the value F ( [ a b, ] ) = F ( b ) - F ( a ) E X on the interval [a,b]. 0 Theorem 7.4.14. A function f : [a,b] -+ X is strongly McShane integrable on [a,b] i f and only i f there is a function F : [a,b] t X which is AC on [a,b] such that F’(t) = f ( t ) almost everywhere on [a,b].

Proof. This result is obtained immediately by Theorem 7.4.2 and Theorem 7.4.13. 0 Theorem 7.4.14 is again a certain type of descriptive definition of the strong McShane integral. Using the fact that the Bochner and strong McShane integrals coincide by Theorem 5.1.4 we obtain by Theorem 7.4.14 immediately the next result. Theorem 7.4.15. A function f : [a,b] t X is Bochner integrable on [a,b] i f and only zf there is a function F : [a,b] t X which is AC on [a,b] such that F’(t) = f ( t ) almost everywhere on [a,b].

Primitive of the Henstock and the McShane Integrals

223

In this way Theorem 7.4.15 gives a descriptive definition of the Bochner integral which is completely analogous to the wellknown descriptive definition of the Lebesgue integral of realvalued functions and justifies the following alternative definition of the Bochner integral used e.g. by R. A. Gordon in [G89], Definition 20, (c).

Definition 7.4.16. A function f : [a,b] -+ X is Bochner integrable on [a, b] if there exists F : [a,b] -+ X which is AC on [a,b] and F’(t) = f ( t ) almost everywhere on [a,b]. According to the facts mentioned above a similar definition can be given by replacing Bochner integrability by strong McShane integrability in Definition 7.4.16.

Theorem 7.4.17. If the Banach space X is finite dimensional, then f : [a,b] -+ X is McShane integrable on [a,b] with the primitive F [a,b] -+ X i f and only if the function F is AC on [a,b] such that F’(t) = f ( t ) almost everywhere on [a,b]. Proof. Since X is finite dimensional, by Theorem 5.2.2, the McShane integral and the strong McShane integral coincide. So, 0 Theorem 7.4.14 yields the result. 7.4.3

The Henstock-Kurzweil integral

Theorem 7.4.18. Iff : [a,b] --+ X is Henstock-Kurzweil integrable on [a,b] then the primitive F o f f is *BVG on [a,b]. Proof. Since f is Henstock-Kurzweil integrable on [a, b], for ) that for any 6-fine every E > 0 there is a 6 : [a,b] 3 ( 0 , ~such K-partition { ( t j ,[uj, q ] ) we } have

For convenience, we may put

E

= 1 and

6 ( t ) 5 1.

Banach Space Integration

224

+ y, a + i] r l [a,b]

Let E,i denote the set of all points t E [a such that 1 1 Ilf(t)llx 5 n, < b(t) I n n - 1'

Obviously, the union of E,i (i = 1 , 2 , . - . , n = 2 , 3 , . . . ) is the whole interval [a,b]. Take any finite collection of nonoverlapping intervals { [ a h , b k ] } with a k , b k E E,i for all k . Then { ( a k , [ a k , b k ] ) } is a b-fine K-system in [a,b]. So, we have

<

+

c

Ilf(ak)llX(bk - a k )

< 1 + n(b - a ) .

k

This shows that F is *BV on E,i. Hence F is *BVG on [a,b]. 0

Theorem 7.4.19. If f is Henstock-Kurzweil integrable on [a,b] then the primitive F of f is *ACG on [a,b].

Proof. The proof is in fact technically similar to the proof of Theorem 7.4.18. Since f is Henstock-Kurzweil integrable on [a,b], for every E > 0 there is a S : [a,b] --f (0,1] such that for uj])}we have any b-fine K-partition { ( t j , [uj,

Let E,i denote the set of all points t E [a such that

+ 5, a+

n [a,b]

Primitive of the Henstock and the McShane Integrals

225

The union of E,i (i = 1 , 2 , - - , n = 2 , 3 , . - ) is [a,b]. Take any finite collection of non-overlapping intervals { [ak,b k ] } with a k , b k E Eni for all Ic. Then { (ak, [ak,b k ] ) } is a &fine K-system in [a,b] and

k

k

Choose 7 5

&

n(b - a )

and

xk(bk

-

ak) < 7 , then

Hence F is *AC on E,i. Consequently, F is *ACG on [a,b]. 0 Concerning the differentiation of the primitives of HenstockKurzweil integrable functions we have the following simple result.

Theorem 7.4.20. I f f : [a,b] --+ X is Henstock-Kurzweil integrable on [a, b] with the primitive F then for each x* E X* the real-function x * ( F ) is differentiable for almost all t o E [a, b] and

(x*(F))'(to)= x*(f ( t o ) )for almost all t o E [a, b]. If X is weakly sequentially complete then F is weakly difb] and for the weak derivative we have ferentiable a. e. in [a, Fh(t) = f ( t )a. e. in [a, b].

226

Banach Space Integration

Proof. By the Henstock-Kurzweil integrability of f on [u,b], we know that the real valued function x*(f) is HenstockKurzweil integrable on [a,b] for each x* E X * and the primitive (WC) s," x*(f) of x*(f) equals x * ( ~ ( t )for ) t E [a,b] (see Proposition 6.1.1). So the primitive x * ( F ) of x*(f) is ACG* on [u,b] (see e.g. [LPY89], Lemma 6.19) and ( x * ( F ) ) ' ( t= ) x * ( f ( t ) )a.e. for each x* E X * (see e.g. [LVOO], Theorem 3.8.2). The second statement is a consequence of Theorem 7.4.12. 0

7.5

Variational measures and primitives for SMandS'FIK:

In this section we consider again the case of a compact mdimensional interval I c R". As before let Zdenotes the family of all compact subintervals J c I . Given an interval function F : Z+ X, an arbitrary set A c I and a 6 : A + (0, +oo) (a gauge on A) let us set

where the supremum is taken over all S-fine M-systems { ( t i ,Ii)} in I with ti E A (the system is anchored in A) and similarly

where the supremum is taken over all 6-fine K-systems { ( t i ,Ii)} in I with ti E A. We put

VMF(A)= inf V ( F ,A, S, M ) , 6

VHKF(A) = i;f V ( F ,A, 6, K ) ,

Primitive of the Henstock and the McShane Integrals

227

where the infimum is taken over all functions S : A t ( O , + c o ) (gauges on A ) . By B. Thomson's results from [T91] or [TO21 it is known that the set functions V H K ( - ) and V M ( . ) are Bore1 metric outer measures on I. VHKF and VMF are called the Henstock-Kurzweil and McShane variational measures of A generated by the function

F:Z+X. Variational measures have been used extensively for studying the primitives (indefinite integrals) of real functions. See e.g. the paper [DPOl] of L. Di Piazza, [LTYOS] of Lee Tuo-Yeong, the book [PfOl] of W. F. Pfeffer for relations to integration and the fundamental general work [T84] of B. S. Thomson. For Banach space-valued functions there are some recent results of Lee Tuo-Yeong [LTYO4]. Let us present the following result based on the work of L. Di Piazza [DPOl] (see also Theorem 3.1 in [LTYO4]).

Theorem 7.5.1. Let f : I t X be stronglp Henstock-Kurzweil (McShane) integrable and let F : Z + X be given by F ( J ) = ('FtVJJ f P(J ) = ( M )J J f ) for J E Z. Then

VHKF(N)= 0 (VMF(N)= 0 ) f o r every N

c I with p ( N ) = 0.

Proof. Assume that N Nn

= {t E

c I and p ( N ) = 0.

Define for n E N

N ; n - 1 5 Ilf(tllx < n}.

Then the sets N, are pairwise disjoint, U,"==,Nn = N and p(N,) = 0 for n E N. Fix an arbitrary E > 0 and for n E N take an open set G, c R" such that N, c G, and p ( G n ) < &. By Lemma 3.6.15 there is a gauge 60 on I such that i

228

Banach Space Integration

for every &-fine K-system (M-system) {(ti,Ii)} in I . For t E Nn take &(t) > 0 such that B ( t ,6,(t)) c G, and put

6 ( t ) = min(&(t), 6n(t)) > 0 for t E N,. In this way a gauge is given on N. Assume that { ( s j ,Jj)}is a 6-fine K-system (M-system) with sj E N . Note that if s j E N, then J j c G, and therefore P(UsjtNn Jj) < &*we have

c j

IlJYJJlX

=

cIHJd

- f(%)P(JJ

+ f(SAP(Jj)llx

j

Hence V(F, N , 6 , K ) < 2~ (V(F, N , 6 , M ) < 2 ~ and ) also VHKF(N) < 2~ (VMF(N) < 2 ~ ) . Since E > 0 was arbitrary we get VHKF(N)= 0 (VMF(N) = 0) and the theorem is proved. 0 Lemma 7.5.2. If f : I primitive F to f we have

t

X belongs to S M then for the

JJ Ilf

VMF(J) 5 ( M )

IIX

for every interval J E Z.

Proof. Assume that J E Z. The function 11 f IIx is McShane integrable (cf. Corollary 3.6.10). From Theorem 3.6.14 it follows easily that

Primitive of the Henstock and the McShane Integrals

229

for every interval K E Z.Hence for any finite system of nonoverlapping Kj E Zwe have

and this yields the result.

0

Defining strong absolute continuity of F : Z t X on I by the property: for each E: > 0 there exists q > 0 such that l l F ( J i ) l l ~< E whenever { J i ) is a finite sequence of non-overlapping intervals in Z with p( Ji) < 7 , we get the following analog of Theorem 7.4.11 for functions defined on m-dimensional intervals.

xi

xi

Corollary 7.5.3. Iff : I + X belongs to S M then the primitive F to f is strongly absolutely continuous on I . Proof. By Lemma 7.5.2 the Borel outer measure V M Fis finite. Hence Theorem 7.5.1 and Theorem 6.11 in [R74] imply that ) E for any given an E > 0 there exists 7 > 0 such that V M F ( E < Borel set E c I with p ( E ) < 2q. Assume that Ji, i = 1,. . . , p is a finite sequence of nonoverlapping intervals in Zwith p( Ji) < 7 . Then there exists a gauge 6 on U iJi such that V ( F , Ji, 6, M ) < E (because U iJi can be covered by an open G with p(G) < 2q). By the Cousin Lemma 3.1.1 for every i = 1,.. . , p there is a 6-fine M-partition { ( t i ,K j ) } of Ji. Since F is additive, we have

xiUi

C i

IIF(Ji>IIX =

C II C i

j

FCK,Z)IIX

5

CC i

j

IIF(K,Z)IIX

230

Banach Space Integration

and this shows that F is strongly absolutely continuous on I . 0 In connection with Lemma 7.5.2 let us recall the following result of Lee T.-Y. from [LTYO4] (see also Theorem 3.4.1 in [LVOO] for the case of real functions).

Lemma 7.5.4. I f f : I primitive F to f we have VMF(J) f o r every interval

+

=

X belongs to S M then for the

(MI

/ J

llfllx

J EZ

Proof. Let E > 0, J E Zbe given. Since f E S M there exists a gauge 61 on J such that

for every &-fine M-partition {(ti,J i ) , i = 1,. . . , k } of J . By Corollary 3.6.10 11 f ( ( xis McShane integrable and therefore there is a gauge 62 on J such that

for every &-fine M-partition {(ti,Ji),i = 1,. . . , k } of J . By Lemma 7.5.2 V M F ( J )is finite and by definition there exists a gauge S3 on J such that

for any &-fine M-partition ((ti,Ji), i = 1,.. . , k } of J . Using Lemma 7.5.2 and the inequalities presented above we have

Primitive of the Henstock and the McShane Integrals

231

k J J

i=l I

k

I i=l

I

for any min(S1,&, &)-fine Ad-partition {(ti,J i ) , i = 1,. . . , k } of 0 J and this yields the result.

7.6

Controlled convergence

There is a considerable work dealing with convergence theorems for Henstock-Kurzweil and McShane integrals based on the properties of the indefinite integrals (primitives) of the sequence of integrable functions which converge pointwise (a.e.) to a given function (see e.g. [H88], [LPY89], [LPY88], [KJ91], etc.). Let us start with the following definitions.

Banach Space Integration

232

Definition 7.6.1. A sequence of continuous functions Fh : [a, b] --+ X is said to be uniformly diflerentiable to f k on a set A c [u,b]if for every E > 0 there exists a gauge 6 ( t ) > 0, t E A such that if 0 < 1s - tl 5 s(t),s E [a,b],t E A then

for all Ic E N.

Definition 7.6.2. A sequence of continuous functions Fk : [a,b] + X is said t o be of uniformly negligible variation on a set 2 c [a, b] if for every E > 0 there exists a gauge 6 ( t ) > 0, t E 2 such that if {(ti,[ui, vi])} is a &fine K-system with ti E 2 then

for all Ic E

N.

We have the following simple result (see also 8.15 Theorem in the book [BaOl] of R. G. Bartle).

Theorem 7.6.3. Let f k : [a, b] X , k E N be a sequence of Henstock-Kurzweil integrable functions such that limk4- f k ( t ) = f ( t )for all t E [a, b], and let Fk(t) = s,” f k . If there is a set 2 c [a, b], p ( 2 ) = 0 such that (a) the sequence Fh is uniformly diflerentiable to f k on [a, b] \ Z and (b) the sequence Fk is of uniformly negligible variation on Z , then the function f : [a, b] --+ X is Henstock-Kurzweil integrable and --+

(xx)

rb

Proof. Define

rb

Primitive of the Henstock and the McShane Integrals

f f ( t ) = o for t

E

233

Z.

Then limk,mffrcz(t) = f z ( t ) for every t E [a,b] where f Z ( t )= f ( t ) for t E [a,b] \ 2 and f”(t) = 0 for t E 2. Since ff’ = fk ff - fk and ff - fk = 0 a.e. on [a,b] we have by Corollary 3.3.2 the Henstock-Kurzweil integrability of ff and Fk(t) = (‘Ftlc)s,” fk = (‘Ftlc)Jat fkZ for t E [a,b].

+

The gauge s(t) given for t E [a,b]\ 2 by the uniform differentiability from (a) and for t E 2 by the assumption (b) g’ives a gauge on [a,b] . Assume that {(ti,[ui, vi])}is a &fine K-partition of [a,b] and consider

5 Y

Banach Space Integration

234

Hence

5 2E

C(Ui

- Ui)

+

E

= 2&(b- u)

+

&

i

for any k E N. By Definition 3.5.1 this means that the sequence f f ) k E N is 'FIX-equi-integrable and Theorem 3.5.2 gives the Henstock-Kurzweil integrability of the function fZ and b Z limk+,('FIX) $ff = ('FIX) f Since ff', f differ from f k , f on the set 2 with p ( 2 ) = 0 0 only, Corollary 3.3.2 gives the result.

s,

*

Remark. Note that in Theorem 7.6.3 (using Corollary 3.3.2 in a straightforward way) the pointwise convergence limk,, fk ( t )= f ( t ) for all t E [a,b] can be replaced by the convergence of the sequence fk to f almost everywhere in [a,b]. Now we will present a convergence result for a sequence of pointwise convergent Henstock-Kurzweil integrable functions using conditions on the sequence of their primitives. This result is given in Theorem 7.6.14 and it is based on the convergence theorem 3.5.2 which uses 'FIX-equi-integrability.

Definition 7.6.4. If { ( t i , I i ) } and systems in [a,b]and P ( UA i

then

{ ( t i ,I i ) } , {(sj,

A

u4)I

{(sj,Jj)}

are two K-

rl

j

J j ) } are said to be q-close.

Note that by A the symmetric difference of two sets is denoted.

Primitive of the Henstock and the McShane Integrals

235

Definition 7.6.5. Let M c [a,b]. A function G : Z -+ X is said to be (strongly) ACV(A4) if for every E > 0 there exists 6 : M + (0,m) and q > 0 such that

for any two q-close &fine K-systems {(ti,Ii)}, { ( s j , J j ) } with the tags ti, sj E M . A sequence GI, : Z --+ X is said to be unzjormly (strongly)ACv(M)if for every E > 0 there exists 6 : M -+ (0, m) and Q > 0 such that

for any two q-close &fine K-systems {(ti, I ; ) } , {(si,J j ) } with

ti,s j E M and all k E N. For these concepts see the paper [KJ91]. One of the systems (e.g. { ( s j , J j ) } ) can be empty in Definition 7.6.5 and then we have

for a uniformly A C ' J ( M ) sequence GI, and

236

Banach Space Integration

for a uniformly strongly ACv(M) sequence P(U,Ii>< rl. Let us start with a few lemmas.

G k

provided

Lemma 7.6.6. Assume that p : [a,b] --+ ( O , o o ) , Z c [ a , b ] , p ( 2 ) = 0 and E > 0. Then there is a 6 : [a,b] t ( 0 ,m) such that i

provided {(ti,I i ) } is a 6-fine K-system with ti E 2.

Proof. Put z k =

{t E 2;k

-

15 p(t) < k}.

Since p ( z k ) = 0, there exist open sets G k such that z k C G k and p ( G k ) < &. There exists 6 : [a,b] -+ (0,m) such that B ( t , 6 ( t ) )c G k for t E z k , k E N. Assume that {(ti,I i ) } is a 6-fine K-system with ti E 2. The system {(ti,&), ti E Z k } is a &fine K-system. w e have

u Ii c

G k

tiEZk

and

Hence

and the Lemma is proved.

0

Corollary 7.6.7. Let fk : [a,b] t X , k E N be such that limk+'x f k ( t ) = f ( t ) E X for every t E [a,b], Z c [a,b ] , p ( Z ) = 0.

Primitive of the Henstock and the McShane Integrals

T h e n f o r every

E

237

> 0 there is a 6 : [a,b] + ( 0 , ~ such ) that i

f o r k E N provided { ( t i ,Ii)} is a 6-fine K-system with ti E 2.

Proof. For t E [a,b] define p ( t ) = SUpkeI, I I f k ( t ) I l x . Since the sequence fi converges pointwise in X we have p : [a,b] + ( 0 , ~ ) . Using Lemma 7.6.6 we get

for k E N provided 6 : [a,b] -+(0, GO) is the gauge from Lemma 7.6.6 and { ( t i ,Ii)} is a 6-fine K-system with ti E 2. 0

Lemma 7.6.8. Let C1 C [a,b] be closed, C2 c [a,b] and let the sequence of additive functions Gk : Z t X ,k E N be uniformly ACV(C,) f o r m = 1 , 2 . T h e n the sequence Gk, k E N is uniformly ACV(C1 U (22). Proof. By Definition 7.6.5 for every [a,b] + (O,GO) and qm > 0 such that

E

> 0 there exists 6,

:

for any two r7,-close 6,-fine K-systems { ( t i ,Ii)}, { ( s j ,J j ) } with

ti, sj E C, and all k E N,m = I,2. Put 7 = $min(q1,q2) and 0

< 6 * ( t ) < min(6l(t),62(t)) for

t E [a,b]. Let H c R be open such that C1 c H while p ( H \ C1) < q. Let 6 be a gauge such that 0 < 6 ( t ) < 6*(t) and B ( t , 6 ( t ) ) c H for t E C1,

B(t,6 ( t ) )n C1 = 0 for t E [a,b] \ C1. Let { ( t i ,Ii)} and { ( s j ,J j ) } be two 7-close &,-fine K-systems with ti, sj E C1 U C2.

238

Banach Space Integration

We have

tiECl

Symmetrically also

and

Hence

which implies

Analogously we get

s j ECl

Primitive of the Henstock and the McShane Integrals

239

and the additivity of the interval function Gk implies

which completes the proof.

0

Lemma 7.6.9. Let Gk : Z -+X ,k E N be additive interval functions. Assume that there is a sequence T/i c [a,b], 1 E N of measurable sets such that & = [a,b] and the sequence Gk is unzformly ACV(T/i) for 1 E N. Let Z c [a,b], p ( 2 ) = 0 . Then for every E > 0 there is a gauge 6 : [a,b] + (0, GO) such that

uz

if {(ti,I i ) } is a 6-fine K-system with ti E 2.

> 0 be given and assume without loss of generality that the sets (L are pairwise disjoint. Proof. Let

E

By Definition 7.6.5 (taking { ( s j , J j ) } empty) there exists 61 : [a,b] + (0, co)and vl > 0 such that

for any &-fine K-system with ti f 6 and p(Ui Ii) < qi. Since p(V, n 2)= 0, there exist open sets I-IL c R, 1 E that r/; n 2 c Hl and ,u(Hl) < 71 for 1 E N. There is a gauge 6 : [a,b] 4 ( 0 , ~such ) that

t E 2 n V, implies B(t,6 ( t ) ) c Hl and

t E Zn

implies 6 ( t ) 5 6,(t).

N such

240

Banach Space Integration

Assuming now that {(ti,I i ) } is a 6-fine K-system with ti E 2 we note that since UtaEK Ii c H2 we have p(UtiEKIi) < q and

co

&

' & = E I= 1

and the statement is proved.

0

Lemma 7.6.10. Let f k : [a,b]-+ X , k E N be a sequence of Henstock- Kurxweil integrable functions with the primitives F k . Suppose that limk-co f k ( t ) = f ( t ) for t E [a,b]. Assume that there exists a sequence of measurable sets C [a,b], 1 E N such that r% = [a,b] and F k is uniformly ACV(K)

u2

for 1 E N.

Then there exists a sequence of closed sets such that ~ ( [ bl a ,\

UQ i >

Ql,

Q1

C

Q1+1

=0

1

and for every 1 E N the sequence f k converges uniformly to f on and the sequence F k is uniformly ACV(Q1) for 1 E N. Proof. For 1 E N there exist closed sets W L c , ~& with W L c ,~ W,r+l,7- E It and P ( K \ W2,r)= 0. Put

u,

fi

=

W1J u WZ,l u w3,1 u

* * *

u W&1 E N.

Since the sets l&'~, C are closed Lemma 7.6.8 implies that is uniformly AC" (8) for 1 E N and

P ( b , bl \

u8)

=0

1

and we have Pl

c fi+l.

Fk

Primitive of the Henstock and the McShane Integrals

241

By Egoroff's Theorem the pointwise convergence of the sec [a,b], quence f k to f implies that there exist closed sets C y+1,p([a,b] \ U1 = 0 such that for every 1 E N the sequence f k converges uniformly to f on ll. Setting QZ= 4nu, for I E N we get the statement of the Lemma. 0

x

x)

Lemma 7.6.11. Let q > 0 , Ql c [u,b] closed, Hl c [a,b] open, 1 E N such that Q1 c Q1+1, Q1 c Hl and p ( K \ Q 1 ) < 2. Let 6 : [a,b] -+ (0, m) be a gauge such that if t E Q l \ QI-1 and the point-interval pair ( t ,I ) is 6-fine (i.e. t E I c B ( t ,6 ( t ) ) ) then I c \ & I p 1 . Assume that 0 < 6 * ( t )5 6 ( t ) for t E [a,b]. If {(ti,Ii)} is a 6-fine K-system with ti E QL\Ql-l then there is a P-fine K-system { ( s j ,J j ) } with sj E Q l \ Q l - I such that the systems {(ti,I i ) } and { ( s j , J j ) } are q-close. Proof. Assume that { ( t i I, < ) }is a 6-fine K-system with Qi

\ Qi-1.

Consider the family Fi of intervals [t - a , t

ti

E

+ a] such that

t E Q 1 n I i ,[ t - a , t + a ] c I i a n d [ t - a , t + a ] c ( t - 6 * ( t ) , t + 6 * ( t ) ) .Taking F = Ui Fi we obtain a Vitali covering of the set Q1 nInt(Ui Ii) C Q 1 n (Hl \ Q L - ~c) Ql \ Q L - and ~ therefore there is a finite S*-fine K-system and

f i r t her we have

This inclusion implies

{(sj, Jj)}

such that

uj

Jj

C

uiIi

Banach Space Integration

242

and therefore the systems { ( t i ,I i ) } and { ( s j , J j ) } are q-close. 0

Theorem 7.6.12. Let f k : [a,b] + X ,k E N be a sequence of Henstock-Kurxweil integrable functions such that lim

k+cc

fk(t)

= f ( t ) ,t

E [a,b].

Let F k , k E N be the (Henstock-Kurzweil) przmztiues of f k . Assume that there exists a sequence of measurable sets fl C [a,b], 1 E N such that fl = [a,b] and F k is uniformly ACV(&) for 1 E N. T h e n the sequence f k : [a,b] + X, k E N is HenstockKurzweil equi-integra bl e.

u1

Proof. By Lemma 7.6.10 there exists a sequence of closed sets Q 1 , Q1 C Q1+1 such that p( [a,b] \ Ul Q1) = 0 and for every 1 E N the sequence fh converges uniformly to f on Q1 and the sequence F k is uniformly ACV(Q1) for every 1 E N. Denote A = [a,b] \ U1Q1; then p ( A ) = 0. Let E > 0 and put E L = 21+4(f+b-a) for 1 E N. By the uniform convergence f k t f on &1, for every 1 E N there exists rl E N such that (7.6.1) Ilfdt) - fm(t>Ilx< El whenever k , m 2 r1 and t E Q 1 . It can be assumed that r1+1 > q , 1 E N. Since the functions f k are Henstock-Kurzweil integrable, the Saks-Henstock Lemma 3.4.1 implies that there is a gauge 6: : [a,b] + (0,m) such that

for k = 1,.. . , rl provided { ( t i ,Ii} is an arbitrary $-fine Ksystem. Since the sequence p k is uniformly ACV(K) for 1 E N, there exist by definition a gauge 6; : [a,b] + ( 0 , ~ and ) 71 > 0

Primitive of the Henstock and the McShane Integrals

243

such that

for k E N if {(ti,Ii}, { ( s j , J j } are arbitrary q-close @-fine Ksystems with ti, s j E Q i . Set QO= 8 and by Lemma 7.6.11 for each 1 E N choose an open set Hl such that Q1 c H l , p(Hl\ Q i ) < There is a gauge S* : [a,b] -+ (0,oo) such that

F.

B(t,s*(t))c Hl

\ Qi

for t E

Qi

\ Qi-I,

1E

N

and

J

c B ( s , S * ( s ) )if

(s, J ) is 6*-fine.

By Corollary 7.6.7, since p ( A ) = p([a,b] \ gauge 63 : [a,b] ( 0 , ~ such ) that

Ul Ql) = 0, there is a

---f

i

for k E N provided {(ti,Ii} is an arbitrary s3-fine K-system with ti E A. By Lemma 7.6.9 there is a gauge 64 : [a,b] -+ (0,oo) such that

c

5

II~k(~i)llX

;

(7.6.5)

i

for k E N if {(ti,Ii} is an arbitrary &-fine K-system with ti E A. Let us now take S ( t ) > 0 for t E [a,b] such that S(t)

I min(#(t),

S:(t), S * ( t ) ) for t E Qi

\ &I-I

and S ( t ) 5 rnin(&(t),&(t)) for t E A.

Suppose that {(ti,Ii}is an arbitrary S-fine K-partition of [a,b] and k E N.

244

Banach Space Integration

We will prove that (7.6.6)

By (7.6.4) and (7.6.5) we have (7.6.7)

X

Let us fix 1 E N. If k 5 rl then by (7.6.2) and the SaksHenstock Lemma we have

because 6 ( t ) 5 6; for t E Ql

\ Q1-l.

Primitive of the Henstock and the McShane Integrals

6:

245

Suppose k > rl. By the Saks-Henstock Lemma find a gauge [u,b]-+ (0,m) such that

:

c

tieQi\Qi-i

W i ) - CW,) 3

I El

(7.6.10)

X

(7.6.11) Now we have using (7.6.10), (7.6.11), (7,6.1), (7.6.9), (7.6.8) the following inequality

246

= El(5

Banach Space Integration

+ 2(b - a ) ) = 21+4(1+5 E b - a )

Hence by (7.6.7) we obtain

+

2&(b- a ) 7e < -21+4* 21+4(1+ b - a )

Primitive of the Henstock and the McShane Integrals

=

7E 1 EXzi + 5 < +-2 + -2 = E

E

247

E

&

1

and the theorem is proved.

0

Remark. Looking at Lemmas 7.6.8 and 7.6.9 and their proofs it can be checked easily that the statements hold also for the case of uniform strong ACV. The same holds also for Lemma 7.6.10 and Theorem 7.6.12 where in addition Henstock-Kurzweil integrability is replaced by strong Henstock-Kurzweil integrability. In this way we obtain the following. Theorem 7.6.13. Let f k : [a,b] t X,k E N be a sequence of strongly Henstock-Kurzweil integrable functions such that lim

k-cc

fk(t) =

f ( t ) ,t

E

[a,b].

Let Fk) k E N be the (Henstock-Kurxweil) primitives of f k . Assume that there exists a sequence of measurable sets c [a,b], 1 E N such that L$ = [a,b] and Fk is uniformly strongly ACV(L$) for 1 E N. Then the sequence f k : [a,b] + X , k E N is strongly 7-Kequi-integrable.

u1

Using the convergence Theorems 3.5.2 (and 3.6.18 for the case of the strong Henstock-Kurzweil integral) we can now state the following consequences of the previous Theorems 7.6.12 and 7.6.13.

x,

Theorem 7.6.14. Let f k : [a,b] --+ k E N be a sequence of Henstock-Kurzweil integrable functions such that lim

k4cc

fk(t) =

f ( t ) ,t

E

[a,b].

Let F , , k E be the (Henstock-Kurzweil) primitives of Assume that the following condition

fk.

( C ) there exists a sequence of measurable sets & c [a,b], 1 E N such that Ul L$ = [a,b] and Fk is uniformly ACV(&)for 1 E N is satisfied.

Banach Space Integration

248

Then the function f : [a,b] --+ X is Henstock-Kurzweil integrable and

JI:

b

lim (HK)

k+co

fk =

(EX)

J” f . a

x,

k E N be a sequence of Theorem 7.6.15. Let f k : [a,b] + strongly Henstock-Kurzweil integrable functions such that lim

fk(t)

k-w

= f ( t ) ,t E [ ~ , b ] .

Let Fk, k E N be the (Henstock-Kurzweil) primitives of Assume that the following condition

fk.

( C S ) there exists a sequence of measurable sets Vi c [a,b], 1 E N such that U,Vi = [ q b ] and F k is uniformly strongly A C V ( 6 ) for 1 E N is satisfied. Then the function f : [a,b] Kurzweil integrable and

+

X is strongly Henstock-

lim F k ( I ) = F ( I )

k+cc

holds. Fk, F are the additive X-valued interval functions corresponding to f k and f, respectively. Remark. Theorems 7.6.14 and 7.6.15 represent convergence results formulated in terms of the primitives Fk. The pointwise convergence and condition (C) ((CS)) guarantee the usual convergence result since they imply the pointwise convergence and the (strong) Henstock-Kurzweil equi-integrability of the sequence fk by Theorem 7.6.12 (7.6.13). For the proof of Theorem 7.6.12 the ideas presented in the paper [KJ91] of J. Kurzweil and J. Jarnik are used. It is worth to mention at this place that in the paper [KJ91] for real valued functions it was also proved (in a very abstract setting) that the pointwise convergence and the HenstockKurzweil equi-integrability of the sequence f k implies pointwise convergence and condition (C) presented in Theorem 7.6.12. In

Primitive of the Henstock and the McShane Integrals

249

other words for pointwise convergent sequences of functions fk the 'FliC-equi-integrability is equivalent to the condition (C) from Theorem 7.6.12. Moreover in this paper [KJ91] of J. Kurzweil and J. Jarnik the equivalence of equi-integrability and of condition (C) is presented for functions defined on intervals I C Rm with m 2 1. The same can be done also for our case of X-valued functions. Let us close this section by mentioning a recent result of Paredes, L. I., Lee P.-Y. and Chew. T. S. presented in [PLCOS]. Similar ideas are also in the work [LPY88] of Lee P.-Y.; they have been a motivation for the paper [KJ91] of J. Kurzweil and J. Jarnik. b] a funcFollowing [PLCOS], Definition 1.7, for a set A4 c [a, tion G : [a,b] t X is said to be AC,**(A4)if for every E > 0 there exists a gauge 6 : [a,b] -+ (0, oo) and q > 0 such that for any two &fine K-systems 7rl = {(ti,Ii)},7r2 = { ( s j ,J j ) } with the tags ti, S j E A4 such that any interval J j lies in some interval li,we have that

c

x1\..z

P(I)

< 7 ==+

c

IIG(l)llx < E

7rl\7r2

where xi \ x2 = {(ti, Ii \ Uj,JjcIi J j ) } . If I = Ii \ U ~ , J ~J jC then G ( I ) = G ( &\ J j ) = G ( 4 ) - Cj,JjcIi G ( J j ) and P V ) = P ( I i \ Uj,JjCIi J j ) = P ( I J - C j , J j C I iP(Jj>Furthermore, G is ACG:*(A4)if [a, b] = Mi such that G is AC,**(Mi), i E N. A sequence of functions Gh : [a,b] t X is said t o be UAC,**(M)if for every E > 0 there exists a gauge 6 : [a,b] t (0,oo) and q > 0 such that for any two &fine K-systems 7rl = { ( t i , I i ) } ,7r2 = { ( s j , J j ) } with the tags ti,sj E A4 such that any interval J j lies in some interval Ii, we have that

Uj,JjcIi

Uzl

r1 \7rz

for every k E

N.

A1 \7rz

I ~

Banach Space Integration

250

Furthermore, Gk is UACG,**if [a,b] = Uz1Misuch that G is UAC;*(Mi),i E N. It follows that G is strongly ACV(1M) provided it is A(?,**( M ) and a sequence Gk is strongly ACV ( M ) provided it is UAC,**(M). Using this notions Theorem 7.6.15 can be used for obtaining the following.

Theorem 7.6.16. Let fk : [a,b] +. x,k E N be a sequence of strongly Henstock-Kurzweil integrable functions such that lim

k+m

fk(t) = f ( t ) ,

t E [a,b].

Let F k , k E N be the (Henstock-Kurzweil) primitives of f k . Assume that the primitives Fk are UACGi*. Then the function f [a,b] + X is strongly HenstockKurzweil integrable and 6

k+cc lim

[.ti=

Jf a

holds. Theorem 7.6.16 is the controlled convergence theorem 4.8 from [PLCO3] in which an additional condition, namely the uniform convergence of the primitives pk, is assumed. It can be seen that this condition is superfluous in [PLCOS].

Chapter 8

Generalizations of Some Integrals

In this chapter we will mention some of the possible generalizations of integrals of X-valued functions. For making the explanations simple, we assume as before that the functions studied are defined on one-dimensional intervals, i.e. we consider functions f : [a,b] 3 X .

8.1

Bochner integral

The classical approach to Bochner integration was presented in Chapter 1. In Theorem 7.4.15 the following was stated.

Theorem 8.1.1. A function f : [a,b]-+X is Bochner integrable on [a,b] if and only if there is a function F : [a,b] -+ X which is AC on [a,b] such that F'(t) = f ( t ) almost everywhere on [a,b]. This result leads to the following alternative descriptive definition (see Definition 7.4.16) of the Bochner integral.

Definition 8.1.2. A function f : [a,b] -+ X is Bochner integrable on [a,b] if there exists F : [a,b] t X which is AC on [a,b] and F'(t) = f ( t )almost everywhere on [a,b]. We have (B)J: f = F(b) - F ( a ) in this situation. The function F : [a,b] --$ X is the (Bochner) primitive to f. In Definition 7.4.6 the following concept was presented. 25 1

252

Banach Space Integration

Definition 8.1.3. A function f : [a,b]+ X is called DenjoyBochner integrable (in the restricted sense) if there is a function F : [a,b] + X which is differentiable a.e. in [a,b] and ACG* on [a,b] such that F’(t) = f ( t )a.e. in [a,b]. We denote F ( b ) - F ( a ) = ( R D B )Jab f . The function F : [a,b] + X is the (restricted Denjoy-Bochner) primitive to f . We denote by R D B the set of functions f : [a,b] 4 X which are Denjoy-Bochner integrable in the restricted sense. Looking at this definition we can see that for X-valued functions the lines of the descriptive definition of the restricted Denjoy integral known for real-valued functions is imitated. See the classical book [S37] of S. Saks or Gordon’s [G94]. In Theorem 7.4.7 we have shown the next result.

Theorem 8.1.4. The function f : [a,b] + X is in the restricted sense Denjoy-Bochner integrable if and only if f is strongly Henstock-Kurzweil integrable o n [a,b], i.e. we have R D B = s7-K Let us introduce the concept of the approximate derivative of a function F : [a,b] + X at a point t E [a,b].

Definition 8.1.5. Let F : [a,b] 4 X and let t E [a,b]. An element z E X is the approximate derivative o f F at t if there exists a measurable set E c [a,b] that has t as a point of density such that

We will write F&(t) = z and say that F is approximately differentiable at the point t E [a,b]. The function F is approximately diferentiable on [a,b] if F is approximately differentiable at every point t E [a, b]. Concerning approximate derivatives, we have the following property.

Generalizations of Some Integrals

253

Theorem 8.1.6. Let F : [a,b] + X be ACG on [a,b] and suppose that F i s approximately diflerentiable almost everywhere o n [a,b]. If F&,(t) = 0 almost everywhere o n [a,b] then F is constant on [a,b].

Proof, Suppose that F is not constant on [a,b]. Then there exist points tl and t2 in [a,b] and x* E X*such that x * ( F ) ( t l )# x*(F)(tZ). Since F is ACG on [a,b] and F&,(t) = 0 almost everywhere on [a,b], then x * ( F ) is ACG on [a,b] and since (x*(F))&,(t)= 0 almost everywhere on [a,b]. So the function z * ( F ) is constant on [u,b] by Corollary 1 of Theorem 25 in [CD78]. This contradiction completes the proof. Now we give the definition of Denjoy-Bochner integral.

Definition 8.1.7. A function f : [a,b] + X is DenjoyBochner integrable o n [a,b] if there exists an ACG function F : [a,b] -+ X such that F is approximately differentiable almost everywhere on [a,b] and F&(t) = f ( t ) almost everywhere on [a,b], F is said to be the (Denjoy-Bochner) primitive of f. In this case we write F(b) - F ( a ) = (DB) Jab!. The function f is Denjoy-Bochner integrable on the set E c [a,b] if the function f x is~Denjoy-Bochner integrable on [a,b] ( X E is the characteristic function of E ) . In this case we write ( D B ) f = (DqJab f X E . By DB the set of Denjoy-Bochner integrable functions f : [a,b]+ X is denoted.

s,

We can see that the Theorem 8.1.6 guarantees the uniqueness of the Denjoy-Bochner integral. Remark. For the case X = R Definition 8.1.2 represents the descriptive definition of the Lebesgue integral. Definition 8.1.3 is the descriptive definition of the classical restricted Denjoy integral which is known to be equivalent to the Perron and HenstockKurzweil integrals (see Chapter 11 in [G94]) and according to our Proposition 3.6.6 also of the strong Henstock-Kurzweil integral. Definition 8.1.5 is for the case of real-valued functions the

Banach Space Integration

254

classical definition of the approximate derivative (see e.g. [S37] or [G94]). The Denjoy-Bochner integrability given in Definition 8.1.7 is the descriptive definition of the classical (wide) Denjoy integral given in [S37].In the book [G94] in Chapter 15 Gordon uses the term Khintchine integral for this sort of integration. The following theorem was proved by R. A. Gordon in [G89].

Theorem 8.1.8. If a function f : [a,b] t X is DenjoyBochner integrable o n [a,b], then f is measurable.

Proof. Since f

: [a,b] t X is Denjoy-Bochner integrable on [a,b],then there exists an ACG function F : [a,b] -+ X such that F is approximately differentiable almost everywhere on [a,b] and F& = f almost everywhere on [a,b]. So for each x* E X* x*(f) is Denjoy integrable on [a,b] and therefore x*(f) is measurable. Let F ( t ) = ( D B )Jatf. Since F is continuous, the set { F ( t ): t E [a,b ] } is compact and hence separable. Let Y be the closed linear subspace spanned by { F ( t ) : t E [a,b ] } . Then Y is separable and Y contains the set { f ( t ) : F&(t) = f ( t ) } . Hence, the function f is essentially separably valued. It follows from the Pettis measurability theorem 1.1.6 that f is measurable. 0

8.2

Dunford and Pettis integral

In Chapter 2 the following two definitions have been presented.

Definition 8.2.1. If f : [a,b] t X is weakly measurable and such that the function x*(f) : [a,b] + R is Lebesgue integrable for each x* E X* then f is called Dunford integrable. The Dunford integral (27) JE f of f over a measurable set E c [a,b] is defined by the element x s E X**given in Lemma 2.1.1, i.e.

/

(D)

f = xg

E

x**,

E

where x g ( x * )= JEx*(f) for all x* E X*. By D the set of all Dunford integrable functions was denoted.

Generalizations of Some Integrals

255

Definition 8.2.2. If f : [u,b] t X is Dunford integrable where (D) JE f E X for every measurable E c [u,b](or more precisely ( D )JE f E e ( X ) C X * * , where e is the canonical embedding of X into ,**) then f is called Pettis integrable and

is called the Pettis integral of f over the set E . By P the set of all Pettis integrable functions f : [u,b] t X was denoted. The Lebesgue (= McShane) integral used for the real function x*(f) : I t R is the main tool in this classical cases. The equivalence of the Lebesgue and the McShane integrals for the case of real-valued functions (see Chapter 10 in [G94]) leads to the straightforward observation that in Definition 8.2.1 the Lebesgue integrability of x*(f) : [u,b]+ R can be replaced by McShane integrability. This fact shifts the classical Dunford and Pettis integrals of f : [u,b]t X into the frame of our text concerned mainly with integrations defined by Riemann-type integral sums. In the subsequent subsections we will look for the case when the Denjoy (in the wide sense) and Henstock-Kurzweil integration concepts replace the Lebesgue one in the previous two definitions of the Dunford and the Pettis integrals. We will describe some of the properties of the resulting weak integrals.

8.2.1

Denjoy approach

Here the weak integrals are treated using the (wide) Denjoy integral in the respective definitions in the sense of Definition 8.1.7 which has to be modified to real-valued functions f : [a, b] R. A first attempt to use such an integration belongs to A. Alexiewicz [A50]. The investigation of this sort of "weak" integrals was started again by R. A. Gordon in the paper [G89]. --f

Definition 8.2.3. If f : [u,b] t X is such that the function

256

Banach Space Integration

x*(f) : [a,b] + IR is Denjoy integrable for each x* E X * and if for every interval J c [a,b] there is an element x y E X** such that x;*(x*) = J J x * ( f )for all x* E X * then f is called Denjoy-Dunford integrable on [a, b]. For an interval J c [a,b] we write

/

(DD) f = 2 7

E

x**.

J

Denote by DD the set of all Denjoy-Dunford integrable funct ions.

Definition 8.2.4. If f : [a,b] -+ X is Denjoy-Dunford inteJJ f E X (or more precisely (DD)J J f E grable where (DD) e ( X ) c X * * , where e is the canonical embedding of X into X**) for every interval J c [a, b], then f is called Denjoy-Pettis integrable and

is called the Denjoy-Pettis integral of f over the interval J c [a,bl. We denote by DP the set of all Denjoy-Pettis integrable functions f : [a,b] + X . Comparing Definition 8.2.3 and Definition 8.2.1 of the classical Dunford integral we can see that instead of measurable sets E c [a, b] intervals J C [a, b] are used only. This difference is caused by the fact that the Denjoy integrability of a real function does not imply its integrability over every measurable set while the integrability over subintervals of [a,b] is guaranteed. For the case of real functions of the Denjoy (wide) integral have been studied thoroughly in the book [IS371 of S. Saks and there is a survey on this integration presented in the book [G94] of R. A. Gordon.

Proposition 8.2.5. Assume that f : [a,b] + X is DenjoyDunford integrable on [a, t] f o r all t E [a, b), and for each z* E

Generalizations of Some Integrals

257

s,“

X * the limit limt+b x*(f ) exists. T h e n f i s Denjoy-Dunford integrable o n [a,b],and rb

rt

f o r each x* E X * .

Proof. Since f is Denjoy-Dunford integrable on [ a , t ]for all t E [a,b) and for each x* E X* the limit limtibS,”x*(f) exists, by Theorem 15.12 in [G94], x*(f) is Denjoy integrable on [a,b] for all x* E X * . On the other hand, take any sequence (tn)in [a,b) convergent to b. Define L(x*) = lim n

I”

I” f).

x*(f) = limx*((DD) n

The uniform boundedness principle guarantees that the linear functional L is continuous on X * . Then it is immediate that f is Denjoy-Dunford integrable on [ a ,b]. 0 Proposition 8.2.5 represents a Hake type theorem (cf. Theorem 3.4.5 in the case of Henstock-Kurzweil integrability) for the Denjoy-Dunford integral. Of course a similar result holds also for the case of Denjoy-Dunford integrability on [t,b] for all t E ( a ,b], and the existence of the limit limt,, x*(f) for each

x* E x*.

s:

Let us present a short survey of some of the results of R. A. Gordon given in the paper [G89]. Let us recall that a set E c [a,b] is perfect if it is closed and has no isolated points or equivalently if E equals to the set of its accumulation points.

{fa} be a family of Denjoy integrable real-functions defined on [a,b]. The family {fa} is uniformly Denjoy integrable on [a,b] if for each perfect set E c [a,b]there exists an interval [c,d]c [a,b]with c, d E E and E n ( c , d ) # 0 such that every fa is Lebesgue integrable on E n [ c , d] and for each a the series C , I C n f a l is Definition 8.2.6. (Definition 14 in [G89]) Let

s““

258

Banach Space Integration

convergent where [c,d ] \ E

=

Un(cn,d n ) .

Definition 8.2.7. (Definition 15 in [G89]) Let {Fa} be a family of real-functions defined on [a,b]. The family { F a } is uniformly BVG(ACG) on E if each Fa is BVG(ACG) on E and if each perfect set in E contains a portion on which every Fa is BV(AC). The proof of Theorem 7.2.3 can be adapted to prove the following theorem.

Theorem 8.2.8. (Theorem 16 in [G89]) Let { F a } be a family of functions defined on [a,b]. Suppose that E is a closed subset of [a,b] and that each Fa is continuous on E . Then the family { F a } is uniformly BVG(ACG) on E if and only if E = En where every Fa is B V ( A C ) on each En.

un

The next theorem ties together the concepts of uniform Denjoy integrability of a family of real functions with the property that this family is uniformly ACG.

Theorem 8.2.9. (Theorem 17 in [G89]) Let {fa} be a family of Denjoy integrable functions defined on [a,b] and f o r each a let F , ( t ) = s,” f a , t E [a,b]. Then the family { f a } is uniformly Denjoy integrable on [a,b] i f and only i f the family {Fa} is unaformly ACG on [a,b]. Theorem 8.2.10. (Theorem 31 in [G89]) A function f : [a,b] + X and suppose x*(f ) is Denjoy integrable o n [a,b] for all x* E X * . Then f is Denjoy-Dunford integrable on [a,b] if and only if the family {x*(f ) : x* E X*}is uniformly Denjoy integrable on [a,b]. Theorem 8.2.11. (Corollary 32 in [G89]) Assume that f : [a,b] + X is Denjoy-Dunford integrable on [a,b], and let P be a perfect set in [a,b]. Then there exists a portion Po of P such that f is Dunford integrable on Po, For defining the classical Dunford integral in Chapter 2 the Dunford Lemma 2.1.1 was crucial. This lemma made it possible

Generalizations of Some Integrals

259

to present Definition 8.2.1 based on the Lebesgue integrability of x * ( f ): [a,b]+ IR for each x* E X*. In the case of Definition 8.2.3 we are in the situation that x*(f ) : [a,b] + IR is Denjoy integrable for each x* E X* and the question is if f is Denjoy-Dunford integrable automatically. An affirmative answer to this problem was given by GBmez and Mendoza in their paper [GM98], Theorem 3. The result reads as follows.

Theorem 8.2.12. A function f : [a,b] + X is DenjoyDunford integrable on [a,b] i f and only if x*(f ) is Denjoy integrable o n [a,b]for all x* E X * . Concerning this Theorem it is clear by definition that if f : [u,b]--+ X is Denjoy-Dunford integrable on [u,b]then x*(f) is Denjoy integrable on [a,b] for all x* E X * . If x*(f ) is Denjoy integrable on [a,b] for all x* E X * then for all x* E X*the function x*(f ) is Denjoy integrable on every subinterval J c [a,b] by the properties of the Denjoy integral of a real function (see Theorem 15.5 in [G94]). The following Lemma is the key to prove the remaining implication in Theorem 8.2.12.

Lemma 8.2.13. (Lemma 1 in [GM98]) Let f : [a,b] + X be such that x*(f ) is Denjoy integrable o n [a,b] for all x* E X * . Let P be a closed subset of [a,b] and assume that f is Denjoy-Dunford integrable o n each open interval J disjoint from P . Then there exists a portion Po of P such that if (In) are the open intervals contiguous to Po then the series

is absolutely convergent for every x* E X * . Proof. Let( Jm) be an enumeration of all open intervals in [a,b] with rational endpoints such that Jm n P # 8. Let (Kn) be an enumeration of all open intervals contiguous to P in ( a , b). For each m E N,the sequence(Jm n Kn)n is an enumeration of all

Banach Space Integration

260

open intervals contiguous to the portion J, n P (of course, in this enumeration some intervals may be empty). Therefore, to prove the result it is enough to show that there exists mo E N such that

for all x* E X*. Assume this is not true. For each n E N the function f is Denjoy-Dunford integrable on K,, since K, n P = 8. Therefore

x*

-i

R,Z*

-i

J

x*(f>,

JmnKn

defines a continuous linear functional for each m E N. So we conclude that for each m , j E N,

q m : x*-+ ZI,Z* +

(s,,,, ,I z*(f), ...

...),

Z*(f),O,O,

J , nKj

defines a bounded linear operator. Our assumption means that for each m E N there exists xk E X * such that

Then the Banach-Steinhaus theorem of condensation of singularities (11. 5 in [DS], p. 81) implies that there exists Z; E X * such that

for all m E N. Finally, notice that each portion of P contains a portion of the form J, nP for some m E N,and for each m E N the J,nKn’s are the intervals contiguous to P in J,. Hence, the previous equality and Theorem 15.10 of [G94] show that xE(f) cannot be Denjoy integrable on [a,b], which is a contradiction. 0

Generalizations of Some Integrals

261

If x*(f ) is Denjoy integrable on [a,b] for all x* E X* let us define the singular set S as the set of all points t E [a,b] such that f is Denjoy-Dunford integrable on no neighbourhood oft, i.e. for every subinterval [c,d] c [a,b] with t E (c,d ) there is no element xi:dl E X**for which we would have ~ i , : ~ ( x= * )Jcd x*(f) for all x EX*. It is not difficult to check that the singular set S c [a,b] is closed because its complement is open (relatively to [a,b]). We have the following.

Proposition 8.2.14. Assume that f : [a,b]+ X is such that x*(f) is Denjoy integrable on [a,b] for all x* E X * . Let (c,d ) be an open subinterval of [a,b]. Then f is DenjoyDunford integrable o n [c,d] i f and only if ( c ,d ) n S = 0.

Proof. If f is Denjoy-Dunford integrable on [c,d] then (c,d ) n S = 0 by definition of S . If ( c ,d ) n S = 0 then for every [cl,dl] c ( c , d ) we have f= E X** for which x[c*l,dll(x*) = x*(f) for all x* E X*and the limit

sc:

P D ) SCt1 q&il]

lim

c1+c+,d1--td-

lr

x*(f)

exists for each x* E X*since t t s,” x * (f)is continuous on [a,b] (Theorem 15.8 in [G94]). Proposition 8.2.5 implies the existence 0 of (DD) Jcd f = x;& =€

x**.

The next theorem is the nontrivial implication of Theorem 8.2.12. Theorem 8.2.15. If for f : [a,b] + X the function x*(f) is Denjoy integrable o n [a,b] for all x* E X* then f is DenjoyDunford integrable o n [a,b].

Proof. Let f : [a,b] t X be such that x*(f) is Denjoy integrable for all x* E X*. Let S be the singular set introduced above If S is empty then we are done.

262

Banach Space Integration

Assume that S is nonempty; we will reach a contradiction. Theorem 8.2.11 guarantees that under our hypothesis each closed set in [a,b] has a portion on which f is Dunford integrable. In particular, we get a portion So = S n (co, do) on which f is Dunford integrable. Now it is immediate that the closed set 3 0 satisfies the assumptions of Lemma 8.2.13 on ( c o , d ~ ]So . there exists a portion ~1 = S o n (c1, dl) = n (c1, d,) of S o (of course, S1is also a portion of S)on which f is Dunford integrable, and such that, if ( I n ) is an enumeration of the intervals contiguous to S1 in (c1, dl), then the series JIn x*(f) is absolutely convergent for every x* E X*. To complete the proof it is enough to show that f must be Denjoy-Dunford integrable on (el, d l ) . Since (c1, d l ) meets S, this will contradict the definition of S. Let J be an interval in [cl,dl] and let x* E X*. Since f is Dunford integrable on S1, x*(f) is Lebesgue (and therefore Denjoy) integrable on S1. On the other hand, the sequence (Inn J),, in which we ignore the empty sets, is an enumeration of the intervals contiguous to S n J in J , and notice that with the exception of at most two intervals, for all nonempty I, n J’s we have 1, n J = I,. So omitting at most two terms of the sequence (JImnJ x*(f)),, we can say that C , JInnJ x*(f) is a subseries of C , JIn x*(f),and so it is absolutely convergent. Hence, we can apply Proposition 8.2.5 to x*(f) and n J = S f l J on J to deduce that

s

c,

for each x* E X*. For each m E N define x; by

Since f is Dunford integrable on S n J c S n (c1, d,) and DenjoyDunford integrable on I, f l J C I,, the linear functionals x; are

Generalizations of Some Integrals

continuous on

263

X*.Clearly, (8.2.1) means that

for each x* E X*. Therefore, the uniform boundedness principle guarantees that the linear functional zy defined by

is continuous on X*. Since this happens for all intervals J in [el,d l ] , we conclude that f is Denjoy-Dunford integrable on [ e l , d l ]. 0

For the case of real-valued functions the following two theorems are given in Gordon’s book [G94] (see also [S37]). Theorem 8.2.16. (Theorem 15.10 in [G94]) I f f : [a,b] -+ R is Denjoy integrable o n [a, b] and P i s a closed set in [a, b] t h e n there exists a portion Po of P such that i f f i s Lebesgue integrable o n Po and i f I k = ( a k , b k ) , k E N are the intervals contiguous t o PO then the series x k f i s absolutely convergent and

sIk

Theorem 8.2.17. (Theorem 15.13 in [G94]) Let E be a bounded, closed subset of R with bounds a and b and let I k = ( a k , b k ) be the intervals contiguous t o E an ( a , b ) . A s s u m e that f : [a, b] + R i s Denjoy integrable o n E and o n each [ a k , b k ] . If

sIk

and the series Ck f i s absolutely convergent then f : [a, b] + X i s Denjoy integrable o n [a, b] and

264

Banach Space Integration

Following closely the paper [GM98]we show the next two theorems for the Denjoy-Dunford integrals which generalize Theorems 8.2.16 and 8.2.17.

Theorem 8.2.18. (Theorem 6 in [GM98])I f f : [a,b] -+ X is Denjoy-Dunford integrable o n [a,b] and P is a closed set in [a,b] then there exists a portion PO of P such that i f f is Dunford integrable o n PO and i f I k = ( a k , b k ) , k E w are the intervals contiguous to Po then the series C k ( D D J I kf as weakly unconditionally Cauchy and rt

for each x* E X * .

Proof. By Corollary 32 in [G89] (see Theorem 8.2.11) there exists a portion P, of P such that f is Dunford integrable on

p*. By Theorem 8.2.12 we have x*(f) Denjoy integrable on [u,b] for all x* E X * and f is Denjoy-Dunford integrable on each open interval J disjoint from P by definition. Then by Lemma 8.2.13 there exists a portion POof P, (POis of course also a portion of P ) such that if (In) are the open intervals contiguous to Po then the series

is absolutely convergent for every x* E X * (the integrals JIn x* (f)in the series above are in the Lebesgue sense). Hence

Generalizations of Some Integrals

265

sI,

for all x* E X*. Therefore the sequence C z = l ( D D ) f , n E N is a weak* Cauchy sequehce idi X** and consequently the weak* (DD)JIk f exists by Alaoglu’s Theorem (see limit limn-,m Theorem V.6.2 in [DS]). Assume Itn(DD)JIn f (x*)l < 00 that t = ( t l , t Z , . . . ) E CO. Then and the weak* limit weak* limn+w tk(D;Z>) f exists as well. Define T : cg t X** by

En

cz=l

sIk

T is a bounded linear operator. For any finite set D c N and any choice of signs f we have

By Theorem 6, p. 44 in [D84] the series ck(D2)]Ik f is weakly absolutely convergent in X**. Given x* E X * the real function x*(f) is Denjoy integrable and its indefinite integral is uniformly continuous on [a,b]. Since t limk+m p , ( I k ) = o we conclude that 1imkhm w(Jakx*(f),I k ) = 0. 0

Theorem 8.2.19. (Theorem 7 in [GM98]) Let E be a bounded, closed subset of R with bounds a and b and let I k = ( a k , b k ) be the intervals contiguous to E in ( a ,b). Assume that f : [a,b] 3 X is Denjoy-Dunford integrable on E and on each [ a k , bk].

If rt

s,

for each x* E X * and the series xk(DD f is weakly unconditionally Cauchy then f : [a,b] t X is Denjoy-Dunford integrable

Banach Space Integration

266

o n [a,b]and

Proof. It is easy to see that for each x* E X*the real-valued b] + R matches exactly the assumptions of function z*(f) : [a, Theorem 8.2.17 and therefore x*(f) is Denjoy integrable on [a,b] and

for every x* E X*. Theorem 8.2.12 yields that f : [a,b] + X is Denjoy-Dunford integrable on [a,b]. By the definition of the Denjoy-Dunford integral we have

(’DD)J b

fk*) = (W

I”

03

fXE(X*)

+

lk

f (x*)

k=l

for each x* E X * and this is our statement.

0

In [G89] Gordon presented the following. Theorem 8.2.20. Suppose that X does not contain a subspace isomorphic to co and let f : [a, b] --+ X . Iff is Denjoy-Pettis integrable o n [a, b], then every closed set in [a, b] contains a portion o n which f is Pettis integrable.

Proof. Let E be a closed set in [a,b]. Since f is Denjoy-Pettis integrable on [a, b], then f is also Denjoy-Dunford integrable on [a, b]. By Theorem 8.2.10, {x*(f) : x* E X*}is uniformly Denjoy integrable on [a,b]. Consequently there exists an interval [c,d] in [a, b] with c, d E E and E n (c, d ) # 8 such that each x*(f ) is Lebesgue integrable on E n [c,d] and I:’J x*(f)l < 00 for each z* E X* where ( c , d ) \ E = U,(c,,d,), (cn,d,) are the intervals contiguous to E in ( c , d ) . We will show that f is Pettis integrable on E n [ c ,d]. Since the function f is Dunford

Generalizations of Some Integrals

267

integrable on E n [ c ,d] it is sufficient to prove that the Dunford integral of f xE is X-valued for every interval in [c,d] and apply Theorem 2.3.5. Let t be a point in [c,d] and suppose that t $ E . Choose an integer N such that t E ( C N , dN). Since x d , & t x*(f ) l < 00

sC:

I sc:

for each x* E X*, the series Ed
We then have

Since this is valid for all x* E X* the Dunford integral of f X E on [c,t] is X-valued. The case for t E E is similar and it follows easily that the Dunford integral of f X E is X-valued for every 0 interval [c,d ] . Hence f is Pettis integrable on E n [ c ,d ] . Theorem 8.2.21. (Theorem 8 in [GM98]) Suppose that X

does not contain a subspace isomorphic to co and let f : [a,b] t X. If f is Denjoy-Pettis integrable o n [a,b] and P is a closed set in [u,b] then there exists a portion Po of P such that i f f is Pettis integrable o n PO and i f I k = (ah, b k ) , k E N are the interwals contiguous t o Po then the series c k ( D ? ) JIk f is unconditionally convergent and

for each x* E X * . Proof. By Theorem 8.2.20 f is Pettis integrable on a portion of P and also on any measurable subset of this portion.

Banach Space Integration

268

Applying Theorem 8.2.18 to f on this portion we get a smaller portion Po such that if I k = (ah, bk), k E N are the intervals contiguous to Po then the series C k ( D P )JIk f is weakly absolutely convergent and

for each x* E X * . C k ( D P JIk ) f is a series in X and X does not contain a subspace isomorphic to co the Bessaga-Pelczynski Theorem B.22 from the Appendix implies that this series is unconditionally 0 convergent and also norm convergent. Theorem 8.2.22. (Theorem 9 in [GM98]) Let E be a bounded, closed subset of JR with bounds a and b and let I k = ( a k , bk) be the intervals contiguous t o E in (a,b). A s s u m e that f : [a, b] --+ X as Denjoy-Pettis integrable o n E and o n each [ a k , bk].

If

for each x* E

X*and the series c I , ( D P )JIk f is unconditionally

b] convergent then f : [a, [a,b] and

s” f

(DP)

=

X is Denjoy-Pettis integrable o n

-+

/

(DP)

00

b

fXE

Proof. Using Theorem 8.2.19 we know that f is DenjoyDunford integrable. It is now sufficient to show that (DD) JJ f E X for every closed subinterval J C [a,b]. Assume that J C [a,b] is a closed interval and define Eo =

E n J. Eo is a closed set. Since f

:

[a, b]

-+ X

is Denjoy-Pettis

269

Generalizations of Some Integrals

integrable on E , we have

and f is Denjoy-Pettis integrable on Eo. On the other hand ( ( a k , b k ) n J ) k , k E N forms the intervals contiguous to Eo in J , k E N (the empty intersections omitted). Except possibly two intervals of the system ( ( a k , b k ) n J ) k , k E N (those which meet the endpoints of J ) represent a subsequence of ( ( a k , b k ) ) k , k E N, for them it is clear that ( a k ,b k ) n J is either empty or equals ( a k , b k ) . The two possible exceptional intervals are subintervals of ( a k , b k ) and so f is Denjoy-Pettis integrable on them. Therefore it is clear that C k ( D p )A a k , b k ] n J f is an unconditionally convergent series in X and so w* C,"=l ( D P )h a k , b k ] " J f = CE, (Dp)h a k , b k ] n Jf . Thus if we apply Theorem 8.2.19 to EOin J we get n

n

n

and in particular ( D D )JJ f E X . Hence f is Denjoy-Pettis integrable on [a,b] because J c [a,b] was an arbitrary interval.

8.2.2

Henstocb- Kurzweil approach

In this part we shortly deal with the case of weak integrals where the integration used in the respective definitions is the Henstock-Kurzweil (= restricted Denjoy = Perron) integral of real functions. The definitions read as follows. Definition 8.2.23. If f : [a,b] X is such that the function x*(f) : [a,b] t IR is Henstock-Kurzweil integrable for each x* E X * and if for every interval J C [a,b]there is an element x y E X** such that x;*(x*) = J J x * ( f ) for all x* E X * then f is called Henstock-Kurzweil-Dunford integrable on [a,b]. ---f

270

Banach Space Integration

For an interval J

c [a,b]we write f

= x;* E X**.

Denote by ‘FIICDthe set of all Henstock-Kurzweil-Dunford integrable functions f : [a,b] --f X .

Definition 8.2.24. If f : [a,b] X is Henstock-KurzweilDunford integrable where (3-tlcD)J f E X for every interval J c [a,b] (more precisely ( X K D ) f E e ( X ) c X * * , where e is the canonical embedding of X into X * * ) then f is called Henstock-Kurxweil-Pettisintegrable and --f

j”,

is called the Henstock-Kurxweil-Pettisintegral of f over the interval J c [a,b]. We denote by ‘FIKCP the set of all Henstock-Kurzweil-Pettis integrable functions f : [a, b] -+ X . The function f is Henstock-Kurzweil-Pettis integrable on the set E C [a,b] if the function fXE is Henstock-Kurzweil-Pettis integrable on [a, b] (xE is the characteristic function of E ) . In this case we write (3-tKCP)JE f = (EICP)Jab f X E . By the results of Chapter 11 in the book [G94] we know that the restricted Denjoy, Perron and Henstock-Kurzweil integrals of real-valued functions coincide. This means that in Definition 8.2.23 the Henstock-Kurzweil integrability can be replaced by Perron integrability or restricted Denjoy integrability to obtain formally different but equivalent definitions of the Henstock-Kurzweil-Dunford and HenstockKurzweil-Pettis integral. From the point of view of the aim of our text the Henstock-Kurzweil variant based on Riemann-type integral sums is of interest. Similarly as in the Denjoy case, comparing Definition 8.2.23 and Definition 8.2.1 of the classical Dunford integral we can see that instead of measurable sets E c [a,b] intervals J c [a,b]

Generalizations of S o m e Integrals

271

are used only. This difference is caused by the fact that the Henstock-Kurzweil (= restricted Denjoy = Perron) integrability of a real function does not imply its integrability over every measurable set (cf. the remark after Theorem 4.1.6) while the integrability over subintervals of [a,b] is assured by Theorem 3.3.4. Since every real-valued function which is Henstock-Kurzweil (=Perron=Denjoy in the restricted sense) is Denjoy integrable the results of the previous subsection can be used for deriving some results. For example Theorem 8.2.11 yields immediately the following. Theorem 8.2.25. Assume that f : [a,b] + X is HenstockKurzweil-Dunford integrable on [a,b], and let P be a perfect set in [a,b]. T h e n there exists a portion Po of P such that f is Dunford integrable on Po.

Another specimen is the following. Theorem 8.2.26. A function f : [a,b] + X is HenstockKurzweil-Dunford integrable on [a,b] i f and only i f x * ( f ) is Henstock-Kurzweil integrable o n [a,b] for all x* E X * .

Proof. If f is Henstock-Kurzweil-Dunford integrable on [a,b ] , for every x* E X*, by Definition 8.2.23, x * ( f ) is HenstockKurzweil integrable on [a,b]. Conversely, if x*( f ) is Henstock-Kurzweil integrable on [a,b] for all x* E X * , it follows that x * (f ) is Denjoy integrable on [a,b] and ( D )Jabx*(f)= (‘FIK)Jab.*( f ) . By Theorem 8.2.12, f is Denjoy-Dunford integrable on [a,b]and for every interval I in [a,b] there exists a vector xT* E X** such that x;*(x*) = (73) JI x * (f ) for all x* E X * . Since x*(f ) is Henstock-Kurzweil integrable on I and x;*(x*) = ( D ) J I x * ( f ) = ( H K ) J I x * ( f ) holds for all x* E X * we obtain that f is Henstock-Kurzweil0 Dunford integrable on [a,b].

272

8.2.3

Banach Space Integration

S o m e examples

In this section we present some interesting examples taken from Gordon’s paper [G89] and from the paper [GM98] by Gamez and Mendoza. Example 8.2.27. (Example 41 in [G89]) If X is not sequentially complete then there is a DenjoyDunford integrable function f : [0,1] --+ X that is not DenjoyPettis integrable (and also not Henstock-Kurzweil integrable).

Example 8.2.28. (Example 42 in [G89]) There exists a function g : [0,1] --+ 12 that is measurable, Pettis integrable but not Bochner and therefore also not DenjoyBochner integrable. Note that by Theorem 6.2.1 this function g is McShane integrable on [0, 11. The following example is due to Alexiewicz [A50]. Example 8.2.29. (Example 43 in [G89]) If X is infinite-dimensional then there is a function f : [0,1] -+ X that is Denjoy-Bochner integrable and also Pettis integrable but not Bochner integrable. Example 8.2.30. (Example 44 in [G89]) There exists a function h : [0,1] --+ co that is Denjoy-Pettis and Dunford integrable but not Pettis integrable. In [GM98], p. 128, the following sophisticated example was presented. Example 8.2.31. There exists a measurable function f : [0,1] co that is Dunford integrable, for every subinterval J C [0,1] we have (D) f E co for its Dunford integral over J but f is not Pettis integrable on any subinterval J C [0,1]. Example 8.2.31 leads to the following statement. Proposition 8.2.32. Suppose that X contains a subspace isomorphic to C O . Then there is a measurable function f : [a,b] --+ X such that ---$

Generalizations of Some Integrals

273

(a) f is Dunford integrable, (b) for every subinterval J c [a,b] we have ( D )JJ f E X , (c) f as not Pettis integrable on any subinterval J c [a,b] In [GM98],p. 132, finally the following is presented.

Theorem 8.2.33. The following are equivalent: (a) X does not contain a subspace isomorphic to CO. (b) Each Denjoy-Pettis integrable function f : [a,b] + X is Pettis integrable on a portion of every closed set. (c) Each Dunford integrable function f : [a,b] t X such that ( D )JJ f E X for every subinterval J c [a,b] is Pettis integrable on some subinterval of [a,b]. Proof. (a) implies (b) by Theorem 8.2.20. (b) implies (c) because clearly each Dunford integrable function f : [a,b] t X for which ( D )J J f E X if J C [a,b] is an interval is Denjoy-Pettis integrable. The fact that (c) implies (a) follows from Proposition 8.2.32. 0

8.3

Concluding remarks

This chapter is slightly out of the scope of this text because the Denjoy extension of the Bochner (Lebesgue) integral does not lead to integration based on Riemann type integral sums like the restricted Denjoy integration or the Lebesgue integration itself. Nevertheless this efforts are rightful from the viewpoint of mathematics. Since the paper [A501 of A . Alexiewicz from 1950 it was known that the integration which is able to integrate approximate derivatives can be defined and studied even for Banach space-valued functions. Studies in this direction have been renewed by R. A. Gordon in [G89]and starting with this paper the theory is enriched from time to time. Without going into detail we mention only some results as a specimen.

2 74

Banach Space Integration

In [YLW99] the following convergence results have been derived.

Theorem 8.3.1. (Theorem 5 in [YLW99]) Let a sequence of functions f n : [a,b] + X , n E N be given. Assume that (i) limn+w fn = f almost everywhere on [a,b] where f n E DB for every n E N, (ii) the primitives Fn of f n are ACG uniformly in n E N, (iii) the primitives Fn of f n converge uniformly o n [a,b]. Then f E Di3 and rb

rb

Theorem 8.3.2. (Theorem 16 in [ Y L W 9 9 ] )Let a sequence of functions f n : [a,b] t X , n E N be given. Assume that the following conditions are satisfied: (i) limn+w f n = f weakly almost everywhere o n [a,b] where each f n is Denjoy-Dunford integrable on [a,b], (ii) the primitives Fn o f f n are weakly continuous and weakly ACG uniformly in n E N . Then f E DD and

Theorem 8.3.3. (Theorem 14 in [ Y L W 9 9 ] )Let a sequence of functions f n : [a,b] t X , n E N be given and let the space X be weakly sequentially complete. Assume that the following conditions are satisfied: (i) limn+wfn = f weakly almost everywhere o n [a,b] where each fn is Denjoy-Pettis integrable on [a,b] and f is measurable, (ii) the primitives Fn of f n are weakly continuous and weakly ACG uniformly in n E N. Then f E DP and

Generalizations of Some Integrals

275

The convergence results from Theorems 8.3.1, 8.3.2, 8.3.3 are based on the favorite idea of controlled convergence by Lee PengYee (see his book [LPY89]). In [YeOl] the following result on integration by parts was proved. Theorem 8.3.4. (Theorem 10 in [YeOl]) b] -+ X is Denjoy-Bochner integrable with the If f : [a, primitive F and g : [a,b] X * is of bounded variation and G ( t ) = s,”g, t E [a,b], then G ( f ) : [a, b] + R is Denjoy integrable on [a,b] and -+

Another special form of an integration by parts theorem is the following. Theorem 8.3.5. (Theorem 14 in [YeOl]) Suppose X * has the Radon-Nikodym property. I f f : [a, b] -+ X is Denjoy-Bochner integrable with the primitive F and G : [a,b] --+ X * is continuous and of bounded variation, then G ( f ) : [a,b]+ IR as Denjoy integrable and

On the right hand side of the equality in Theorem 8.3.5 we have the Henstock-Kurzweil-Stieltjes integral which is defined ) Kvia Stieltjes integral sums of the form C i G ( I i ) ( F ( t i )for partitions {(ti,Ii)}. Using the spaces of integrable functions introduced in this chapter as well as in the previous parts of this book we have the following straightforward inclusions:

B c R D B = S7-K c DB, 23 c %ICD c 23.0 and

B = S M c P c RICP c DP

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Appendix A

Classical Banach Spaces

We give a short account and elementary properties of some Banach spaces which are used in some examples presented in the book. For more information we refer to [DS] or the books [LT].

A.l

Spaces of sequences

Assume that x = ( a l ,a2,, . . ) = ( a k ) is a sequence with a k E R, k E N. The family of all such sequences has the usual linear structure. The space 1, defined for 1 5 p < 00 is the linear space of sequences z = ( a k ) for which C klakl, < 00. The norm for x E 1, is given by

1, is a Banach space. The space 1, is the linear space of all bounded sequences x = ( a k ) . The norm for z E 1, is given by

,I is a Banach space. 277

2 78

Banach Space Integration

The space c is the linear space of all convergent sequences x = ( a k ) . The norm for x E c is given by 1)X)1Zm = SUP ) a k l * k

c is a Banach space. The space co is the linear space of all convergent sequences x = ( a k ) with limk', a k = 0. The norm for x E co is given by

11x111, co

= SUP k

1 4

c c is a Banach space.

For the dual spaces we have the following. If 1 < p < 00 and = 1 then the spaces ( I p ) * and I, are isometrically isomorphic. x* E ( I p ) * and y = ( b k ) E (1,) are coupled by the relation

+

k=l We use the evident license (Ip)* = 1, in this case. If 1 < p < 00 then ( I p ) * is reflexive. The spaces (II)* and I, are isometrically isomorphic. x* E (21)* and y = ( b k ) E),2( are coupled by the relation

k=l We use the license (Z1)* = I, in this case. The space Z1 is not reflexive. The spaces (co)* and 11 are isometrically isomorphic. x* E (co)* and y = (bk) E (II) are coupled by the relation

k=l We write (co)* = 11 in this case. The space co is not reflexive.

Classical Banach Spaces

A.2

279

Spaces of functions

A.2.1

The spaces C ( I ) and L p ( I ) , 1 < p

< 00

The space C ( I ) is the linear space of all continuous functions x : I + R. The norm for x E C ( I ) is given by IIXIIC(1) = S UP tEI

Ix(t>l.

C ( I )is a Banach space. The space C ( I ) is not reflexive. The space L p ( I ) , 1 < p < 0;) is the linear space of pmeasurable functions x : I --+ R for which IP is p-integrable. The norm for 12: E Lp(I)= Lp is given by

If

11~1lL, =

If

$+

(JI If

IP)

.

1 then the spaces (Lp)*and L, are isometrically isomorphic. x* E (Lp)*and y E (L,) are coupled by the relation =

x*(x) =

J, x - y, x E Lp.

We write (Lp)*= L, in this case. Lp(I)is a reflexive Banach space.

A.2.2

The spaces

L1

and L ,

Definition A . l . Denote by L1 = L 1 ( I ,p , R) the space of measurable functions f : I t R such that J1 lfldp exists. L1 with the norm

Ilf llLl

=

1 I& I

If

is a normed linear space.

Theorem A.2. A s s u m e that f n E L1,n E N,f : I --+ R. T h e n f E L1 and (1 fn - f ((L1= 0 if and only if (a) fn converges t o f in measure,

Banach Space Integration

280

(ii) limP(E),o Jg I fnldp = 0 uniformly with respect to n E N, (iii) for every E > 0 there as a measurable E, c I such that

l,EE

IfnldP < E , n E

N.

(See [DS]; 111.3.6.) Theorem A.3. The set of simple functions as dense in L1. (See [DS]; 111.3.8.) Theorem A.4. space.

The space L1 is complete, i e .

L1

is a Banach

(See [DS]; 111.6.6.) Theorem A.5. (Vitali convergence theorem) Assume that f n E L1, n E N, f : I + R and f n ( t ) + f ( t ) for almost all t E I. T h e n f E L1 and limn-m I(fn - f ((L1= 0 if and only i f (i) lirnP(E)-+O JE I f n l d p = 0 uniformly with respect to n E N, (iii) for every E > 0 there is a measurable E, c I such that

l\EE

IfnPP < E , n E

N.

(See [DS]; 111.6.15.) Theorem A.6. I f f

2 0 o n I then the function G given by

f o r measurable E C I is countably additive. (See [DS]; 111.6.18.) Definition A.7. Denote by L , = ,?&(I,p,R) the space of essentially bounded functions f : I -+ R, i.e. functions for which there is N c I , p ( N ) = 0 such that f 11,~is bounded. Denote

Classical Banach Spaces

281

(See [DS]; IV.2.19.) Theorem A.8. L , with the n o r m

is a linear Banach space. (See [DS]; 111.6.14.) Theorem A.9. The dual LT to metrically isomorphic where

L1

and the space L , are iso-

F

x*(f) =

JI g ( 4 f W

P

for f E L I , x* E LF and g E L,.

(See [DS]; IV.8.5.) Theorem A.lO.

L1 is weakly complete.

(See [DS]; IV.8.6.) Theorem A.11. A sequence f n E L1, n E N is weakly Cauchy if and only if it is bounded fll fnllL1 < K ) and the limit

lim L f n d P

TI-,

exists for every measurable E c I . A sequence f n E L1, n E N converges weakly to some f E L1 i f and only i f it is bounded fll fnllLl < K ) and lim

n+m

S, 1 fndP

=

E

fdp

for every measurable E C I .

(See [DS]; IV.8.7.) Theorem A.12. A set K c L1 is weakly compact if and only if K is bounded and the countable additivity of f dp is uniform

sE

282

Banach Space Integration

with respect to f E K , i.e. for every increasing sequence En C I of measurable sets having empty intersection the limit lim

n+cc

l,,

fdp =0

is uniform with respect to f E K . (See [DS]; IV.8.9.)

Theorem A.13. If K

c L1 is weakly

compact then

is uniform with respect to f E K Since in our case p ( I ) < 00, this condition is suficient for weak compactness of a bounded set K c L1. (See [DS]; IV.8.11 Corollary.)

Appendix B

Series in Banach Spaces

00

Definition B . l . A series

C x k of elements x k E X , k E N of k=l

a Banach space X is said to be convergent if the sequence of its n

partial sums sn =

z k

converges in the norm of the space

x.

k=l 00

Definition B.2. The series

xk, xk

E

x,k E N is absolutely

k=l M

convergent if

C IIxkIIX

< 00.

k=l M

Proposition B.3. If converges.

00

Cxk

converges absolutely then

k=l

x k k= 1

00

Definition B.4. A series

Cxk

of elements x k E X , k E N of

k=l

a Banach space X is said to be unconditionally convergent if it converges for every rearrangement of its terms, i.e. if the series 03

C x P ( ~ )converges whenever

P is a one-to-one mapping of N

k=l

onto N. KI

Theorem B.5. For a series

xk

of elements x k E

x,k E N

k=l

of a Banach space X the following conditions are equivalent: (a) the series converges unconditionally, (b) all series of the form X n l t X n 2 t X n , . . . where n1 < n2 < n3 < . . . converge, 283

284

Banach Space Integration 00

C Okxk

(c) the series

convergesfor a n y choice of o ! k = %I,

k=l

c 00

the series

(d) for every bounded sequence (ai),ai E

akxk

k=l .~

converges to some element of X , (e) given E > 0 there is an no E

c 00

w such that 11

akxkllx

<E

k=no

for every sequence a

=

(ai ) E 1,

with JJalll, 5 1.

(See [KK97]; Theorem 1.3.2., [Day]; Chap. IV and [BP77]; G. on p.442.)

c 00

Proposition B.6. If X = IW then a series

xk

of elements

k= 1 xk

E

x,k E R is unconditionally convergent if and only if

03

XI, k=l

is absolutely convergent.

The following corollary follows immediately from Proposition B.6. Corollary B.7.

If a Banach space X is finite-dimensional

00

then a series

Cxk

of elements

xk

E X , k E R is uncondi-

k= 1 00

tionally convergent if and only if

xk

is absolutely convergent.

k= 1

For a general Banach space we have

c 00

Theorem B.8. If

00

xk

is absolutely convergent then

k= 1

21,

k=l

is unconditionally convergent.

The next example shows that the converse of Theorem B.8 is not true in general. Example B.9. For k E

N put

xk =

00

w e have

( ( X k ( ( l 2=

and

(0,. . . , O , i , O , . . . ) E 00

( ( x k ( ( 1 2= k= 1

k=l

f

=

00;

12.

the series

Series in Banach Spaces

285

is not absolutely convergent. On the other hand, for any rear00

rangement of

C xk the series converges to

k= 1

1 1 1 s = (1,- - , . . . , -, . . . ) 2’ 3 n

€12

with

i.e. ((slllz< 00 and the series converges unconditionally. 00

Proposition B.lO. If

C xk

is unconditionally convergent k= 1 then all rearrangements have the same sum.

(See [KK97]; Theorem 1.3.1.) The following well-known result shows that the situation described in Example B.9 is not accidental and that it holds for every infinite-dimensional Banach space.

Theorem B.11. (Dvoretzky, Rogers) In an infinite-dimensional Banach space X for every sequence ck > 0, k E N for 03

which

C cz

< 0;) there is an unconditionally convergent series

k= 1

co

Zk, X k E

x,k E N for which l l X k I ( x = C k , k E N.

k= 1

(See [KK97]; Theorem 4.1.1 or [DR50].) If we take e.g. ck = k E N then we get the following consequence of Theorem B . l l .

i,

Corollary B.12. I n every infinite-dimensional Banach space 00

X there exists an unconditionally convergent series

X that is not absolutely convergent.

C X k , xk

k=l

E

Using this and Corollary B.7 we obtain the following interesting characterization of finite-dimensional Banach spaces.

286

Banach Space Integration

Proposition B.13.

The unconditional convergence of a series

00

C xk,

xk

E

X is equivalent to its absolute convergence if and

k= 1

only if the Banach space X is finite-dimensional. Let us now turn our attention to some results involving the weak topology in X .

Definition B.14. A sequence xn E X , n E to x E X if for every x * E X *

N weakly converges

lim x*(xn)= x*(x).

n-cc

05

Definition B.15. A series

C xk, xk

EX,k E

N weakly con-

k=l

verges to a sum s E X if for every x* E X* the limit n

n+co

k=l

n+co

k=l

exists. co

Theorem B.16. (Orlicz, Pettis) Let

C x k , x k E X , k E N be k=l

a series in a Banach space X . I f f o r each set A C N there is x* E X* we have

XA

E

X such that for each

co

then the series

xk

is unconditionally convergent.

k= 1

(See [T84] (2-6), [ID841p. 24, [DU77] Corollary 4, p. 22, [HP57] Theorem 3.2.3.)

Remark B.17. Let us mention that from the conclusion of 00 xk the Orlicz-Pettis Theorem B.16 it follows that the series k=l

converges (in norm).

Series in Banach Spaces

287

Moreover, the assumption of the theorem can be reformuW

lated to read

“

W

C xk is weakly subseries convergent” or C xk “

k=l

k=l

is weakly unconditionally convergent” . The last form says that for every x* E X the series of W

real numbers

C z*(xk) is unconditionally

convergent and by

k=l

Proposition B.6 this means that for every x* E

5 (x*(xk)(<

X we have

00.

k=l

W

Definition B.18. A series

xk

is called weakly absolutely

k=l

convergent if

k=l

for all x* E X*. In [D84], p.44 weakly absolutely convergent series are called weakly unconditionally Cauchy. W

Let us mention that if a series

xk

is unconditionally con-

k=l W

vergent then also the series of real numbers

X*(xk)

is uncon-

k=l

ditionally convergent for every z* E X * and by Proposition B.6 W

this occurs if and only if

Ix*(xk)l < 00 for every x* E

x*.

k=l

In this way we arrive at 03

Proposition B.19. If a series

C xk

is unconditionally con-

k= 1

vergent then at is weakly absolutely convergent. Example B.20. For k E N denote by ek=(O

, . . . , O , l , O, . . . ) E c o

the element of co with 1 on the k-th position in the sequence.

Banach Space Integration

288

Since we have n

C e , k=l

=

(1,1,. . . , l , o, . . . )

for n E N (1 is on the first n positions in the sequence) we can W

see immediately that the series

C ek does not converge in co (in

k=l the norm topology). Assume that x* E c: = 11, i.e. x* W

k

E

N and C la,/

=

=

( a l , a 2 , . .),

ak

E

R,

//x*llc;< 00.

n=l

Then x*(ek) = ak and W

W

k=l

k=l

W

C ek

i.e. the series n

k=l

is weakly absolutely convergent. n

Since x*( C ek) = C a k , n E N we can see that this sequence k=l k=l W

Cak

= x * ( y ) where y = (1,1,. . . ) is a sequence k=l W which does not belong to co. This means that if the series C ek k=l were weakly convergent then its weak sum would be y but co contains no such element.

converges to

An immediate corollary of this example is

Corollary B.21.

If a Banach space X contains an isomorphic W

copy of the Banach space co then there is a series

xk, xk E X , k=l k E N which is weakly absolutely convergent but is not convergent nor weakly convergent to an element of X .

This implies of course that if in a Banach space X every weakly absolutely convergent series converges in the norm topology or in

Series in Banach Spaces

289

the weak topology then X cannot contain subspaces isomorphic to co. An important result in this direction is given by the following theorem.

Theorem B.22. (Bessaga-Pelczyriski) The following assertions are equivalent: (a) The Banach space X does not contain subspaces isomorphic to CO, (b) every weakly absolutely convergent series in X is weakly convergent, (c) every weakly absolutely convergent series in X is unconditionally convergent, (d) e v e y weakly absolutely convergent series in X is (norm) convergent. (See [KK97]; Theorem 6.4.3, [D84], . . . .)

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[YSO2]

[Ye011 [Ye021 [Ye031

Banach Space Integration Swartz, Ch., Introduction to Gauge Integrals, World Scientific, Singapore, 2001. Swartz, Ch., Gauge integrals and series, Mathematica Bohemica 129(2004), 324-332. Talagrand, M., Pettis Integral and Measure Theory, Memoirs of the AMS 307, 1984. Thomson, B. S., Derivates of Interval Functions, Memoirs of the AMS 452, 1991. Thomson, B. S., Differentiation, pp. 179-247 in Handbook of Measure Theory, Ed. E. Pap, Elsevier Science B. V., 2002 Wheeden, R. L., Zygmund A,, Measure and Integral, Marcel Dekker, Inc., New York, 1977. Ye, G., Lee P.-Y., Wu, C. Convergence theorems of the DenjoyBochner, Denjoy-Pettis and Denjoy-Dunford integrals, Southeast Asian Bull. of Math. 23(1999), 135-143. Ye, G., Schwabik, S., The McShane and the weak McShane integrals of Banach space-valued functions defined on R", Mathematical Notes (Miskolc) 2(2001), 127-136. Ye, G., Schwabik, S . , The McShane integral and the Pettis integral of Banach space-valued functions defined on R", Illinois J. of Math. 46(2002), 1125-1144. Ye, G., Wu C., Lee,P.-Y., Integration by Parts for the DenjoyBochner Integral. SEA Bull. Math., 26 (2002),(4) 693-700. Ye, G., An,T., On Henstock-Dunford and Henstock-Pettis Integrals. International J. of Math. and Math. Sci. 25:7(2001), 467-478. Ye, G., Lee,P.-Y., Wu, C., Convergence Theorems of DenjoyBochner, Denjoy-Pettis, Denjoy- Dunford Integrals. SEA Bull. Math. 23,(1999), 145-152. Yosida, K., Functional Analysis, Academic Press, New York, 1965.

Index

absolute continuity, 195 absolutely convergent series, 283 AC, 193, 195 AC*, 193 * A C ,195 AC*, 195 ACv(M), 235 ACG, 194 A C G 8 , 194 *ACG,200 ACG, 200 ACG', 200 additive interval function, 70 approximate derivative, 252 approximately differentiable, 252

continuity, 192 convergent series, 283

B,17

7-close, 234

D ,29 VB,253 b-fine, 46 Denjoy-Bochner integrability, 217 Denjoy-Bochner integrability (restricted), 252 Denjoy-Bochner integral, 253 Denjoy-Dunford integral, 256 Denjoy-Pettis integral, 256 determining sequence, 17 differentiability, 202 DP, 256 Dunford integral, 29

Bochner integrable function, 17, 223, 251 Bochner integral, 17, 223, 251 bounded variation, 194 B V , 193 B V * , 193 * B V ,194 B V , 194 B V * , 194 BVG, 194 BVG*, 194 * B V G ,200 BVG, 200 BVG', 200

figure, 114 F'rechet differentiability, 202 Gbteaux differentiability, 202 gauge, 46 Hake's theorem, 61 Henstock-Kurzweil integrable function, 47, 85 Henstock-Kurzweil integral, 47, 85 Henstock-Kurzweil-Dunford integral, 269 Henstock-Kurzweil-Pettis integral,

297

298

Banach Space Integration

270

‘HK, 47 ‘HK-equi-integrable, 64 ‘HKD, 270 ‘HKP, 270

1,70 indefinite integral, 211

K-partition, 45 K-system, 45 L-Cauchy sequence, 11 L-seminorm, 11 L-zero sequence, 11 Lebesgue integral, 22

M , 47 M-equi-integrable, 64 M-partition, 45 M-system, 45 M*-partition, 113 M*-system, 113 McShane integrable function, 46 McShane integral, 46 McShane’ integral, 113 measurable function. 1 non-overlapping intervals, 45 [c,4, 193 oscillation, 193

W(F,

P , 34 perfect set, 257 Pettis integral, 33 portion, 201 primitive function, 211 property S*YK, 73 property S*‘HK, 73 property S* M , 73 property S * M , 73

RVB, 252

Saks-Henstock lemma, 55, 57, 69 scalarly negligible function, 183 S‘HK, 70 S Y K , 73 simple function, 1 S M , 71

S * M , 73 strong ‘HK-equi-integrability, 83 strong M-equi-integrability, 83 strong absolute continuity, 195, 229 strong continuity, 192 strong differentiability, 202 strong Henstock-Kurzweil integrability, 70 strong McShane integrability, 71 strongly bounded variation, 194 tag, 45 tagged interval, 45 unbounded interval, 85 unconditionally convergent series, 283 uniform absolute continuity, 120 uniformly absolutely continuous family, 120 uniformly ACv ( M ) ,235 variational integrability, 71 variational measure, 227

W A C ,194 wACG, 200 w B V , 194 wBVG, 200 weak absolute continuity, 194 weak continuity, 192 weak differentiability, 202 weakly absolutely convergent series, 287 weakly bounded variation, 194 weakly convergent series, 286 weakly equivalent functions, 183 weakly measurable function, 2 weakly unconditionally Cauchy series, 287