Integration between the Lebesgue integral and the Henstock-Kurzweil integral : its relation to local convex vector spaces

INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces ...

Author: Jaroslav Kurzweil

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INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces

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 &AG Dzvarseisvili translated by P S Bullen

Vol. 4:

Linear Functional Analysis WOrlicz

Vol. 5:

Generalized ODE S Schwabik

Vol. 6:

Uniqueness & Nonuniqueness Criteria in ODE R P Agarwal & V Lakshmikantham

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

Series in Real Analysis - Volume 8 INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces

Jaroslav Kurzweil Mathematical Institute of the Academy of Sciences of the Czech Republic

V f e World Scientific wb

Singapore • Hong Kong Sinqapore • New Jersey • London L

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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INTEGRATION BETWEEN THE LEBESGUE INTEGRAL AND THE HENSTOCK-KURZWEIL INTEGRAL Its Relation to Local Convex Vector Spaces Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

The approach to integration by Riemannian sums was rehabilitated in the fifties of the 20th century by a new interpretation of the concept of a "fine" or "6-fme" partition of the integration interval. It is well known that both the Lebesgue integration and the Henstock-Kurzweil integration can be obtained by the same method, only the integration bases are different. The concept of an integration basis y is very flexible and results in a rich class of ^-integrations. To every integration basis 3^ there corresponds the vector space Py of primitives of 3^-integrable functions (on a fixed interval / = [a, b] C K), a concept of Ey-convergent sequence of functions from Py, and ULci^y) which is the finest locally convex topology on Py such that every Ky-convergent sequence is convergent in (Py:ULc(^y))Lebesgue integration is obtained by a suitable choice of y, y = C. Then Pc is the space of absolutely continuous functions and ULC(^-C) is induced by the norm ||.F||var = vari*1. Hence (Pci^Lci^c)) is a complete space. If y — TiK, then HenstockKurzweil integration is obtained: The topology Uici^wc) is induced by the norm H-FHsup and (PHKMLC^HK)) is not complete. The problem whether (Py ,UL,c(J&y)) is complete is the central problem of this book. A theory is developed which gives an answer for a broad class of J^'s and to an extended problem which includes integrations introduced by Bongiorno and Pfeffer in 1992 and by Bongiorno in 1996.

V

PREFACE

VI

Topics connected with the Riemann approach to integration were reported and discussed in the Seminar on Differential Equations and Real Functions of the Mathematical Institute of the Academy of Sciences of the Czech Republic since the beginning of this approach. I wish to thank the participants of the seminar for their contributions and comments. I express sincere thanks to J. Jarnik and S. Schwabik, who read the manuscript and suggested several improvements. I am grateful to S. Schwabik who encouraged me and transformed the manuscript into the camera ready form. The research which resulted in publishing this book was supported by the grant No. 210/01/1199 of the Grant Agency of the Czech Republic.

Prague, March 2002

Jaroslav Kurzweil

CONTENTS

Preface

v

0. Introduction

1

1. Basic concepts and properties of ^-integration

9

2. Convergence

21

3. Convergence and locally convex spaces

32

4. An auxiliary locally convex space

42

5. £-integration

52

6. .M-integration

69

7. Noncompleteness

76

8. 5-integration

86

9. ^-integration

104

10. An extension of the concept of ^-integration

109

11. Differentiation and integration

116

References

135

List of symbols

137

Index

139

vn

0

INTRODUCTION

The approach to integration which is based on approximation of the integral by Riemannian sums is rather flexible. If the set of partitions which are used in the formation of Riemannian sums is rich then Lebesgue integration is obtained. On the other end of the spectrum a poor set of partitions leads to an integration which is called Henstock-Kurzweil and which is equivalent to Denjoy integration in the restricted sense and to Perron integration. In this book integrations are studied for various sets of partitions. If y is a set of partitions we denote by Py the set of primitives of 3^-integrable functions. For every 3? some sequences Fi G Py are called Ey- convergent IF"

to a limit FQ G Py, Fi —> FQ. Therefore there exists a unique locally convex topology ULc(^y) on Py which is the finest one among locally convex topologies T on Py with the property IF

that Fi —> Fo implies that Fi —> FQ in (Py,T). The problem whether (Py,Uic(^y)) is complete is crucial for this book; the answer depends on y. Let I = [a,b] C R. A set A = {(ti,Ai);i = 1 , 2 , . . . , A;} is called a partition in I if k G N and if (0.1) U G / f o r i = 1,2,...,*;, (0.2) Ai C I is a figure, i.e. a finite union of closed intervals, % =

i , 2 , . . . , K,

(0.3) figures Ai,Aj are nonoverlapping for i ^ j (i.e. \Ai D Aj\ = 0 for i ^ j where \E\ is the Lebesgue measure of 1

2

INTEGRATION BETWEEN L AND H - K

E C R). A is called a partition of / if (in addition) it

|J At = I.

(0.4)

Denote by C the set of partitions in / and by TiK, the subset of C which consists of A such that (0.5) Ai is a closed interval, i = 1, 2 , . . . , k, (0.6) U eAi,i

=

l,2,...,k.

+

Let ( : I -* R - A G £ is called (-fine if (0.7) At C (U - {(U),U + C(tx)) for i =

l,2,...,k.

f : I —» R is called Tf/C-integrable (£-integrable, respectively) if there exists j G R and for every £ G R + there exists ( : / -> R + such that

| 7 -^/(^)l^|| <£ 2=1

whenever A = {(ti,Ai);i = 1,2, . . . , & } G W/C ( A G C respectively) is a C"fine partition of / . 7 is called t h e 7Y/C-integral (^-integral respectively) of / over / . It is well known that / is "WC-integrable if and only if it is Denjoy integrable in t h e restricted sense over i" or equivalently if and only if it is Perron integrable over / , and that the two integrals coincide. On t h e other hand, / is £-integrable over / if and only if it is Lebesgue integrable over / a n d again t h e two integrals coincide. Moreover, two integrations intermediate between reintegration and 7Y/C-integration were introduced in [B-Pf 1992] and [B 1996]. In this book integrations intermediate between £-integration and Tf/C-integration are treated systematically. Let y fulfil

INTRODUCTION

3

TiJC C y C C and some additional conditions which are not important at the moment. Let K C I be a closed interval. / : K —»• R is called yintegrable (over K) if there exists 7 G R and for every e G R + there exists £ : J —> R + such that fe

17-^/(^)1^11 <e 8=1

whenever A = {(£;, Ai); i = 1 , 2 , . . . , k} G ^ is a (-fine partition of J\. If / : I —> M. is 3^-integrable over I and if K C / is a closed interval then the restriction f\x is 3^-integrable over K. Denote its integral by F{K). F maps the set of closed intervals K C I to R and is called the primitive of / . If Fi, i = 1,2 is the primitive of a 3^-integrable / i : J —» R then Fi = F2 if and only if /1 = $2 a.e. In order to avoid the concept of equivalence classes of integrable functions the results will be formulated for the primitives. Denote by Py the set of F such that F is the primitive of some y~integrable / . A sequence Fi G Py, I G N is called Cauchy in Ey if there exists a sequence // : I —> R of ^-integrable functions fulfilling (0.8) there exist (3 : I -> R+ such t h a t k

\Fl(I)-Y,fi(U)\Al\\<2-> i=l

whenever;',/ G N and A = {(ij, Aj);i = 1 , 2 , . . . , fc} G ^ is a (j-fine partition of J, (0.9) the sequence fi(t), I G N is Cauchy in R for every t G J. Of course, F/ is the primitive of / ; . If the sequence F/, I G N is Cauchy in Ey and /j fulfil (0.8) and (0.9) then there exists /o : I —> R such that (0.10) //(<) -»• / 0 ( < ) for / - • 00, < G /

4

INTEGRATION BETWEEN L AND H - K

and it can be proved that for every closed interval K C / there exists FQ{K) e K. and (0.11) Fi{K) -> P 0 ( / i ) for / -> oo, that /o is 3^-integrable and F0 is its primitive. Therefore a convergence Ey can be introduced on Py in a natural way as follows. Let Fm £ Py for m = 0 , 1 , 2 , . . . . The sequence F/, I £ N w is called convergent to Po in E y , F\ —> Po if there exist y~ integrable functions fm : I —> R, m = 0 , 1 , 2 , . . . such that (0.8) and (0.10) hold. Again, Fm is the primitive of / m , m = 0 , 1 , 2 , . . . . Ev

Ey is the set of couples ((P/, I € N), P 0 ) such that Fi -=*+ F0. IF

A topology T on P y is called tolerant to Ey if P/ —>• Po implies that Fi —• P 0 in ( P y , T ) . It follows from general considerations that there exists a locally convex topology ULc{Ey) on Py such that (0.12) ULC{Ey) is tolerant to Ey, (0.13) UiciEy) is finer than any locally convex topology T on Py which is tolerant to Ey. Obviously, Uic(Ey) is unique. It was proved in [K 2000] that ^Lc(E?i/c) is the topology on P-HK induced by the norm || • || s u p where | | P | | s u p = sup{|P(A~)|; K C / i s a closed interval }. Therefore (PnKi^Lci^nK)) is not complete. In this book the answer to the problem of completeness of (Py,Uic(^>y)) is given for various y. The topics of Chapters 1-3 are clear from their headings. In Chapter 2 convergence Qy is introduced in addition to the convergence Ey. A system of sets Qy(6) C Py is defined where 6 is a parameter, 6 e D*.

INTRODUCTION

5

Every set Qy(S) is convex, circled and compact in Py which is endowed with the topology induced by || • || s u p . Moreover,

U Qy(6)-

Py=

6€D*

A sequence F; G Py, / G N is called convergent to Fo in Qy, Ft - A Fo (where F 0 G Py), if there exists 6 G D* such that Fl e Qy{&) for / G N and if ||F, - F 0 | | s u p -> 0 for / - • oo. Qy is the set of couples ((F ( , I G N), F0) such that F/ ^ > F 0 . Elementary relations between convergence and topology are studied in Chapter 3. A topology T on Py is called tolerant to Qy if F/ - A Fo implies that F\ —»• F 0 in ( F y , T ) . Again, there exists a unique locally convex topology ViiciQy) on F y such that (0.14) ULC(Qy) is tolerant to Qy, (0.15) ULc(Qy) is finer than any locally convex topology T on Py which is tolerant to Qy. The convergences Ey and Qy are so close that (0.16) a topology T on P y is tolerant to Ky if and only if it is tolerant to Qy. Consequently, (0.17)

ULC(Ey)=ULC(Qy).

As a consequence of (0.14) and (0.15) we find that (0.18) the topology which is induced on Qy(6) by || • || s u p is finer than the restriction of ULc(Qy) to Qy(S), S G D*; (0.19) if T is a locally convex topology on Py such that for every 6 G D* the topology which is induced on Qy(S) by || • 11sup is finer t h a n the restriction of T to Qy(6), then ULc(Qy) is finer than T .

6

INTEGRATION BETWEEN L AND H - K

Finally, it can be deduced from (0.18) and (0.19) that (0.20)

{conv ( J Qy(S)

D B(-q(6)); rj : D* - •

R+}

S£D*

is a zero-filterbase for VfLciQy)B(a) = {Fe

^-n *be above formula

Py, \\F\\snp < a}, a e R+

and convE is the convex hull of E C Py. (0.20) holds for all y. In Chapter 4 some restrictions on y are introduced and seminorms || • \\ytS, s £ I a r e found in an explicit form such that (0.21) ULc(Qy) is the topology on Py induced by the set of seminorms || • ||y jS , s (E I (cf. Theorem 4.2). In the proof of (0.21) a major part is played by (0.20). In Section 5 the case y = C is treated. Since the restrictions from Section 4 are fulfilled in the case y = C and since for s G I the seminorm || • ||£ i S is equivalent to || • || v a r where H-FUvar = var F, we conclude that ULC(Q.C) is induced on Pc by the norm

ll-llvarFinally, it is deduced that Pc is the set of primitives of Lebesgue integrable functions, ^-integration coincides with Lebesgue integration and (PCMLC(Q.C)) is complete. Observe that F\ —^ FQ implies that \\Fi — Foil var —* 0 for / —> co but not vice versa. The M.-integration is close to ^-integration and {PMMLC{Q.M)) is complete (cf. Chapter 6). In Section 7 some new restrictions on y are introduced which together with the restrictions from Section 4 guarantee that (Py,ULc(Qy)) is not complete. Let A be the set of A : [0, oo) —> [0, oo) nondecreasing, A(0) = 0, A( 0 for a > 0.

INTRODUCTION

7

Let S(\) be the set of partitions A = {(
J]A(dist(
(dist(i, J ) being the distance of t from J ) . Then ( P S ( A ) 5 ^ / L C ( Q S ( A O ) * S n o ^ complete since the restrictions from Sections 4 and 7 are fulfilled by S(X) for A € A (Chapter 8). Let ft be the set of u : R + —• [0, | ] , nondecreasing, u>(a) > 0 for a > 0. Let 7Z(UJ) be the set of partitions A = {(ti,Ai);i — 1, 2 , . . . , k} in J such that K N and

l l ^ l l d i i m l l U A , > " ( d i ^ { M u A 0 , i = l,2,...,fc where d i a m £ is the diameter of E C R and ||A|| is twice the number of the components of the figure A. Then (P-R(LJ)MLC(QTI(^)))

is not complete

(Chapter 9). LetX = (yuy2,y3,...),f:I-+R. / i s called ^-integrable and F is its primitive if / is 34-integrable and F is its primitive for k £ N. Denote by P ^ the set of F such that F is the primitive of some A-integrable / . Let Fm € Px for m = 0 , 1 , 2 , . . . . The space Px is equipped with the topology Ux which is defined as the supremum of the set of topologies ULc(Q>yk), k (=. N. The concept of ^-integration is an extension of the concept of 3^-integration and is studied in Chapter 10. Noncompleteness results are obtained if X = (<S(Ai), «S(A2),<5>(Afc),... ), \ k £ A for k € N and if X = (K(u>i),K(uj2),K(ujk),...), uk € ft for k keN. If \k(a) = 2 a iora>0,X = (<S(A1),<S(A2),«S(Afc), • • •) then ^-integration is the integration which was introduced in [B

8

INTEGRATION BETWEEN L AND H - K

1996], if Lok(a) = 2~k for a > 0, X = ( f t ^ ) , ^ ^ ) , - ^ ^ , . . . ) then ^-integration is the integration which was studied in [B-Pf 1992]. Noncompleteness results in Chapters 9 and 10 are obtained by comparison with <S(A)-integration for a suitable A € A. 3^-differentiation is introduced in Chapter 11 and a general result concerning the relation of ^-differentiation and ^ - i n t e g ration is obtained. Chapter 11 is concluded by a specialization to £-integration. This Chapter will be closed by a list of four parts of the book which are not necessary for understanding the text which follows them: 1. Sections 4.7, 4.8, pp. 49-51 (a necessary and sufficient condition for completeness of (Py,ULc(Qy)) for a subclass of 3>'s); 2. Sections 5.8 - 5.11, pp. 62-68 (relations between convergences in (PC,ULC(QC)), in Qc and in Ec); 3. Chapter 6, pp. 69-75 (A^-integration); 4. Sections 8.9 - 8.12, pp. 91-103 (dependence of 5(A)integration on A).

1 BASIC CONCEPTS PROPERTIES OF

AND

^-INTEGRATION

1.1 N o t a t i o n . By R, R + , N, Z we denote the set of reals, the set of positive reals, the set of positive integers, the set of integers. For E C R, t G R let i n t E , c\E, \E\, d i a m E , d i s t ( t , £ ) be the interior of E, the closure of E, the outer Lebesgue measure of E, the diameter of E, the distance of t from E. Let N be the set of N C R such that |7V| = 0. By intervals we mean compact nondegenerate intervals in R, e.g. L = [c,d]. An open interval is denoted by (c, d) while [c, d), (c, d] are semiopen intervals. A figure is a finite union of intervals. T h e symbol #M stands for the number of elements of a finite set M. Let I — [a, b] be an interval. If K C I , then Iv(A') is the set of intervals L G K and Fig(A') is the set of figures A C K. We shall write Iv, Fig instead of Iv(J), Fig(J). Let K C I. A finite set A = {(i;,Aj); z = l , 2 , . . . , f c } , shortly A = {(i, A)} is called a partition in K if (1.1)
\Aif\Aj\-0

whenever

A is called a partition

(ti,Ai),(tj,Aj)eA,i^j.

of K if in addition

K = \J{Ai; i = 9

l,2,...,k}.

10

INTEGRATION BETWEEN L AND H - K

Let C, : I -> R + , E C I. A partition A = {(i,A)}

is called

(1.3) (-fine, HAc[tC(t), t + ((t)} for (t, A) G A, (1.4) E-tagged, if t G E for (t, A) G A (t is the tag of 4 ) . Let £ be the set of partitions in I and let TiK be the set of partitions A in / such that (t, A) G A G H.IC implies t E

Aelv.

By ^ we denote a set of partitions A in / such that

(i.5) wccycc, (1.6) if 0 C A G ^ , then 0 G y, (1.7) if A = {(ti,Ai); i = 1 , 2 , . . . , A:} G y, 0 = { ( * ; , £ ; ) ; j = 1 , 2 , . . . , / } G WC, | A i n X j | = 0 for i = 1 , 2 , . . . , A;, j = 1 , 2 , . . . , / then A

u © G y.

The set of y s is denoted by Y. Throughout the book it will be assumed that

yeY. Let E C M C I, ( : I ->R+. By y(M,E,C) we denote the set of A = {(£,^4)} G y such that A is a partition in M which is jE?-tagged and (-fine. Occasionally £ is independent of £, i.e. there exists a G K + and ((t) = a for t G / , and we write y(M, E, a) instead of y(M,E,() in that case. Moreover, y(M,E, oo) is the set of A = {(<, A)} G ^ such that t £ E, AcM. The set of 8 : N x I - • R+ such that (1.8)

6(j,t)

> 6(j + l,t) for j£N,t<=

I

is denoted by D*. 8(j,-) : J -* K+ is defined by % , • ) ( * ) = 6(j,t) for jeN, tel. If / : I -f R, M C I , F : Fig -»• R then / | M , ^|Fi g (M) are the restrictions of / to M and of F to Fig(M); F\IV(M) has an analogous meaning. If 8 G D* then <5|M is the restriction of 8 : N x I -> R+ to N x M , i.e. <5| M (j, •) : M -> K + for j G N.

BASIC CONCEPTS

11

1.2 L e m m a (Cousin). Let K G Iv, ( : I -» R+. Then there exists A = {(t, A)} G H)C(I,I,Q, A being a partition of A'. The lemma is well known (cf. e.g. [K 2000], Lemma 1.2). Taking (1.7) into account we have Corollary. If ( : I -> R+, 0 G y(I,I,() then there exists A G ,y(I, / , C) such that A is a partition of I and 0 C A. 1.3 D e f i n i t i o n . Let K G Fig, / y-integrable on K if

: K - • R.

/ is called

(1.9) there exists 7 G R, 6 G £>* such that k

l7-£/(OMI =

fr-£/(*0l*ll<2-' i=l

A

f o r ; G N, A = {(t,A)} = {{U,Ai); 1 = 1,2,...,k} y(K, K, 6(j, •)), A being a partition of K.

G

Notes, (i) The value 7 is unique by Lemma 1.2. It is called the y-integral of f over K and denoted by (y) fK fdt. It follows from (1.5) that every ^-integrable / is "WC-integrable and that

(y) f fdt = (HIC) ( fdt. JK

JK

Therefore we will usually write JK fdt instead of (y) J fdt. (ii) "WC-integrable means Henstock-Kurzweil integrable (cf. [K 2000], Definition 1.4), £-integrable means Lebesgue integrable (see Chapter 5). 1.4 T h e o r e m . Let f : I —> R be y-integrable Then (1.10) the restriction

f\x

is y-integrable

on K.

on I, K G Fig.

12

INTEGRATION BETWEEN L AND H - K

F : Fig —> JR. is defined by

F(K) = (y) f fdt = (y) I f\Kdt JK

and is called the primitive

JK

of f (y-primitive

of f).

Proof. Let 6 G D* correspond to / by Definition 1.3,

K = {J[ck,dk] k=i

where / € N and a < c\ < d\ < C2 < c?2 < ' ' ' < c\ < di < b. Let Lm, m G M be the components of I \ K, (/ — 1 < # M < / + 1). Let A = {(t,A)},A' = {(t',A')} G y(K,K,S(j + 1,-)) be partitions of K. By Lemma 1.2 there exists 0 m G WC(c\Lm,c\Lm,8(j + 1,-)) form G M. P u t

Hj =AU (J 0 m , H 2 =A'U (J 0 m . Both S i and £2 are <5(j + 1, -)-fine partitions of / so that

\F(I)-J2f(t)\A\\<2-*-\i

= l,2

and

\J^f(t)\A\-^f(t)\A\\<2-\ which implies that /|^- is 3^-integrable on K. complete.

The proof is

BASIC C O N C E P T S

13

1.5 D e f i n i t i o n . H : Fig —• R is called additive if H{K1\JK2) for KUK2

= H(K1)

+

H{K2)

£ Fig, {Id n K2\ = 0.

Note. For if : Fig -> R put 5 ( a ) = 0, 5 ( 0 = # ( [ a , <]) for a < < < b. Then # is additive if and only if H([c, d]) = S(d) - S(c) for a < c < d R be ^-i-ntegrable and iet F : Fig —> R be its primitive. Then F is additive. Proof. Let KUK2 £ Fig, 1 ^ n K2\ = 0, j £ N. Let 6 £ D* correspond to / by Definition 1.3. There exist A^ = {(£, A)} £ y(Ki,Ki,6(j, •)), Ai being a partition of A^, z = 1,2. Then

(1.11)

|F(/v 2 )-5]/WI^I<2- J , i = l,2. A;

Moreover, A i U A 2 £ y{Kx U A' 2 , A^ U K2,6(j, is a partition of K\ U A 2 so that

(1.12)

\F{K^K2)-

•)) and A i U A 2

Yl /(*)I^II<2-J'. AiUA2

(1.11) and (1.12) imply that | F ( / w U K2) - FilU)

- F(K2)\

< 3 • 2~].

F is additive since j £ N is arbitrary. The proof is complete.

14

INTEGRATION B E T W E E N L AND H - K

1.7 L e m m a (Saks, Henstock). Let f : / —> E be ^-integrable in I, F : Fig —> R being its primitive. Let 6 G -D* correspond to f by Definition 1.3 and let j G N, A = {(£, A)} G y(I,I,6(j,-)). Then (1.13)

|£(F(A)-/(<)|A|)|<2"'", A

(1.14)

£|F(A)-/(*)L4||<2--'+1. A

Proof. Let L m , m G M be the components of / \ A, p G N. There exist (1.15)

0 m G niC(cl Lm, cl Lm, S(j +p,-))ior

Since A U UmeM ®m

1S a

me

M.

^ 0 ' ')~fine partition of /,

AU |J e ro e;y(i,i,*(j,.)), we get (1.16)

|F(J)-^/(t)|A|- ^ A

^/(0|A||<2--'-.

m€M 0 m

(1.15) implies that (1.17)

|F(clL m ) - ^ / ( i ) | A | | < 2 - J - p for m G M. 0m

By Theorem 1.6 we have F(I) = $ > ( A ) + ] T F ( c l L m ) . A

m€M

BASIC CONCEPTS

15

Therefore (cf. (1.16), (1.17)) | £ ( F ( A ) - f(t)\A\)\

< 2-> + # M • 2 - ' " - '

A

and (1.13) holds since p may be arbitrarily large. Let A1 = {(t,A)eA-F(A)-f(t)\A\>0}, A 2 = A \ Ai. By (1.13) we obtain £

\F(A) - / ( 0 | A | | = | £ 0 F ( A ) - f(t)\A\)\

Ai

< 2~\

Ai

J ] |F(A) - /(0|A|| = | £(F(A) ~ f(*)\A\)\ < 2"J" A2

A2

so that (1.14) holds. The proof is complete 1.8 Definition. Let H : Fig -* R, s € J. if is called J continuous at s if for every e £ R + there exists
for A = {(s,A)}€;y(I,{s},er). JVo*e. Since # A < 2 for A = {(5, J)} e WC(I, {5}, 00) it is clear that H is ?t/C-continuous at s if for every e G R + there exists £ € R + such that |F(J)|<£ if s £ J € Iv, I J\ < £. For brevity if is called continuous at s if it is 7iJC-continuous at s. If if is ^-continuous at s, then it is "H/C-continuous at s.

16

INTEGRATION BETWEEN L AND H - K

1.9 T h e o r e m . Let f : I —s- R be y-integrable and let F be its primitive, s £ I. Then F is y-continuous at s. Proof. Let e £ R + and let 8 correspond to / by Definition 1.3. There exist j G N, o < e , a < % + 2 , 5 ) , 2 a | / ( s ) | < 2"-''- 1 . By (1.14) we have J2 \F(A)\

< •2-^-1 + \f{s)\J2

A

\A\ < 2-i-1

+ \f(s)\2
A

for A = {(5, A)} e y(I, {s},6(j

+ 2, s)). The proof is complete.

1.10 N o t e . Denote by AdC the set of H : Fig -»• R which are both additive and continuous at every s € / . P u t | | # | | s u p = s u p { | i f ( J ) | ; J € Iv} for H G AdC . AdC equipped with the norm || • || s u p is a Banach space. Let Py be the set of primitives of y-integrable functions / : I —> R. Py is a vector space, Py C AdC (cf. Theorem 1.6, Theorem 1.9). 1.11 D e f i n i t i o n . Let H : Fig -> R or H : Iv - • R, s <E / , (3 £ R. H is called differentiable at S and /? is the derivative of H at s if for every e <E R + there exists
<e|J|for

se

J e!v,J

C(s-a,s

+ a).

Note. (3 is unique and it is frequently denoted by H(s). is called differentiable if it is differentiable at every s £ J.

H

17

BASIC CONCEPTS

1.12 T h e o r e m . Let f : I -> R be y-integrable be its primitive. Then (1.18) F(t) exists almost everywhere, erywhere, f is measurable.

F(t) = f(t)

and let F almost

ev-

(1.18) is a consequence of [K 2000], Theorem 1.17 since 7i)C C y and / is 7Y/C-integrable. 1.13 L e m m a . Let h : I - • R, N G M, e G R+. Then exists ( : I —•> R+ such that

there

£>(0PI<e A

forA = {(t,A)}€£(I,JV,C). # m i (cf. [K 2000], Lemma 1.15). For i G N there exist open sets d CR such that N C Gt, \G{\ < e2~2i. Put E0 = 0, Et = {t G iV; |fc(*)| < 21}. L e t ( : I ^ R + fulfil (*-((*),*+((*)) C Gt for t e Ei\ Ei-i, i G N. Then £ does the job. 1.14 T h e o r e m , (i) Let / , # : I —> K be ^-integrable, (1.19)

(3>) / / ( s ) d 5 = (y) Ja

I g{s)ds for < G / . •/ a

Then f = g almost everywhere. (ii) Let / , g : / —> R. Assume £ba£ / is [V-mtegrabie and / = g almost everywhere. Then g is y-integrable and (1.19) holds. Proof, (i) is a consequence of Theorem 1.12 and (ii) follows by Lemma 1.13. 1.15 L e m m a . Let K G Fig, / : K -»• R, F : Fig(A') -> R. Assume that F is additive. Then the following conditions are equivalent: (1.20) / is y-integrable

and F is its

primitive,

18

INTEGRATION B E T W E E N L AND H - K

(1.21) there exists 9 £ D* such that

Y,\F(A)-f(t)\A\\<2-> A

for j £ N, A = {(t, A)} £ y(K, K, 0(j, •)), (1.22) there exist M £ M and r\ £ D* such that (i)

£|F(A)-/(*)|A||<2-'" A

for j £ N, A{(t, A)} £ y(K, K \ M, v(j, •)),

(ii)

J2\F{A)\<2-i A

for j £ N, A{(t,A)}

£

y(K,MMJ,-))-

Proof. Let (1.20) hold. Then (1.21) holds by (1.14). If (1.21) holds then (1.22) is fulfilled for M = 0. Let (1.22) hold and put M, g(t) = 0 for * G M. Then g(t) = f(t) for t£K\

|F(JiQ-5>(0WI< A

<

Yl A,t£K\M

\F(A) - f(t)\A\\ + Y,

\F(A)\<

A,t£M

< 2~ J ~ 1 + 2~J~1 = 2~j for j £ N, A{(t,A)} £ y(K,K,r](j + 1, •)), A being a partition of K. Therefore F is the primitive of / by Theorem 1.14 (ii). The proof is complete.

19

BASIC CONCEPTS

1.16 T h e o r e m . Let S C I be closed and countable. (1.23)

Then

FePy

if and only if (1.24)

F | F i g L € Py(L,L,oo) for Lelv,LnS

= $

and (1.25)

/ is y-continuous

at s

fors^S.

Proof. The "only if" part is a consequence of Theorems 1.4 and 1.9. Assume that (1.24) and (1.25) are fulfilled and that 5 ^ 0 . Let Lm, m G M be t h e components of I \ 5 , where M = N or M = { l , 2 , . . . , p } . Let S = {st; I G L} where L = N or L = { l , 2 , . . . , g } . It is not difficult t o prove (cf. (1.21)) that there exist / : J —> R, £ G -D* such that (1.26)

f(s) = 0 for

(1.27)

seS,

Y,\F(A)-f(t)\A\\<2-* A

for j G N, m € M, A = {(t,A)}

G

y(Lm,Lm,6(j,-)).

Let d(j: t) = 6(j + m + 1, t) for j G N, m G M, * G Lm and let 9(j,si)

be so small t h a t

J2\F(A)\<2-^'-1 A

20

INTEGRATION BETWEEN I AND H - K

for j e N, A e y(I,{si}AJ,*i))y(I,I,0(j,.)). Then

£|F(C)-/(r)|C||< e

£ ©,re/\s

Let j € N, 0 = { ( r , C ) } €

l^(C)-/(r)|C||+ J ] |F(C)|< e,res

< J^ 2_J~m-1 + "^2 2~i~l~1 < 2~3. Hence i*1 is the 3^-primitive of / . The proof is complete.

2

CONVERGENCE

2 . 1 . The following elementary convergence result is an extension to ^-integration of an elementary convergence in WCintegration. The proof is omitted since it is analogous to the proof of Theorem 3.1 in [K 2000]. T h e o r e m . Let 6 G D*, f{ : I -> R, j /o : I —> R. Assume that

(2-1)

z

G R for % 6 N,

\li-Yffi(t)\A\\<2-' A

for i,j G N, A = {(t,A)} of I, (2.2)

G y(I,I,6(j,-)),

fi(t)^fo(t)

A being a

partition

for tG J,z^oo.

Then lim,-_).00 7; = 70 G JR. exists and

(2-3)

|7o-E/oWI^II<2-J A

for j G N, A = {(*, A)} G ? ( J , J", Kh •)), A being a partition of J. iVoie. (2.1) and (2.3) imply that fm is J/'-integrable and 7 m = (3^) Jj fmdt for m = 0 , 1 , 2 , . . . . Let Fm be the primitive of fm, 21

22

INTEGRATION BETWEEN L AND H - K

m = 0 , 1 , 2, We have Fi(I) —> F0(I) for i —> oo. Since /;| A is ^-integrable on K for A' G Fig, i G N, we may conclude t h a t (2.4)

F,-(A") -> F 0 ( J \ ) for i -> oo, Ji G F i g .

Moreover, functions Hi : I —> R, i? : / —> R which are defined by # i ( a ) - 0 , t f ( a ) = 0, #*(*) = F 2 ([a, <]), H(t) = F0([a,t]) for a < t < b, i G N are continuous. Therefore | | i r i - i r o | | s u p - > 0 for

(2.5)

i->oo.

A convergence on Py can be introduced in connection with Theorem 2.1. 2.2 Definit ion. Let Fm G Py for ?n = 0 , 1 , 2 , . . . . Assume that there exist / m : I —>• R for m = 0 , 1 , 2 , . . . and 6 £ D* such

that

(2.6)

£|FTO(A)-/TO(*)|J1||<2->

.

A

for m = 0,1,2,..., (2.7)

jeN,

A = {(t, A)}

ey(1,1,6(j,-)),

/,•(*)-/<>(*) for < e / , i - > o o .

Denote by Ey the set of couples ((F,,z G N ) , F 0 ) fulfilling (2.6) and (2.7). Ey is called a convergence and the sequence i^, i G N F

is said to be convergent

to FQ in Ey, Fi —> FQ.

Remark. The above definition is analogous to Definition 3.5 in [K 2000]. 2.3 L e m m a . Let Fm G Py for m = 0 , 1 , 2 , . . . . The ing conditions are equivalent: (2.8) Ft ^

F0,

follow-

23

CONVERGENCE

(2.9) there exist hm : I -> R, m = 0 , 1 , 2 , . . . , 77 G D* and M e J\f such that

J2\F^A)-hm(t)\AW<2~J

(i) A

form

= 0 , 1 , 2 , . . . , j € N, A = {(t,A)} € 3>(I,I\

M,i7(j,-)),

A

form = 0,l,2,...,jeN,A

(iii)

fcTO(<)

=

{(t,A)}ey(I,M,r,(J,-)),

- • h0(t) for t <E I \ M, i -> 00.

Proof. If ^ - ^ F 0 then (2.9) holds with M - 0, hi = £ , i = 0 , 1 , 2 , . . . , 77 = 6. If (2.9) holds then (2.6) and (2.7) hold with 6(j,t) = rj(j +1, t) and fi(t) = hi(t)fort € I\M, ft(t) = 0 for t e M, i = 0 , 1 , 2 , . . . . 2.4 Lemma. Let Fx ^

F0, G{ - ^ G0, a € R. Tien

IB1

(2.10)

F,(fc) — • Fo for any subsequence

(2.11)

F + G . - ^ F o + Go,

(2.12)

aFi^+aFo.

The lemma follows from Definition 2.2.

i(k), k = 1, 2, 3 . . . ,

24

INTEGRATION BETWEEN L AND H - K

2.5 L e m m a . Let F{ —> i*o(2.13)

Then

Ili^-FolUp^O.

The lemma is a consequence of (2.5). Later on convergence Qy will be introduced; it is more suitable for further treatment. Moreover, both convergences E y , Qy lead to the same locally convex topology on Py (cf. Corollary 3.9 ). 2.6 D e f i n i t i o n . For 6 G D* let Qy{6) Fig —> R. which are additive and fulfil

(2.14)

be the set of F :

pTO^+E^ for jeN,

e =

{(t,A)}ey(i,i,e(J,-))

and (2.15) there exist M <E Af and / : / -> R such that

X) 1^(^-/(01^11 < 2" for j e N, 0 = {(*, A)} € y(I, I \ M , 0(j, •))• 2.7 L e m m a . Let 0 e D*,M sucii that

0

eAf.

Then there exists ( e D *

*0\0

for j € N , 0 = {(t,A)}

€ y(/,M,e(i,-))-

The lemma is a consequence of Lemma 1.13.

25

CONVERGENCE

2.8 L e m m a . Let F : Fig -> R be additive and fulfil (2.14) and (2.15). Then F is the y-primitive of f.

Proof. Let M be defined by (2.15). By Lemma 2.7 there exists £ € D* such that

2^/jr,^ *(M)

z

e

for j € N, 0 = {(t, A)} € y(I,M,Z(J,-))Hence (1.22) holds if r](j,t) < min{0(j + l,t),£(j + 1,*)} and Lemma 2.8 is a consequence of Lemma 1.15. T h e proof is complete. 2.9 T h e o r e m .

U QyW) = py9<=D*

Proof. Lemma 2.8 implies that

U Qy(°) c PyLet F e Py. Then (2.15) is fulfilled with M = 0 by virtue of(1.21). Moreover, (1.21) implies that

Y,\nA)\<2-> + Y,\m\\A\\ A

A

for j G N, A = {(*, A)} G y(1,1,6{j,

l/(*)l < P C M ) ] - 1 for Therefore P y C (J#eD* Qy(^)-

•)) so that (2.14) holds if

jeN,t€l.

^ h e P r o ° f i s complete.

26

INTEGRATION BETWEEN L AND H - K

2.10 T h e o r e m . Let 0 e D*, Gt e Qy(0) for i e N, Gt being the primitive of a y-integrable gt : I —> R7 Go '• Fig —> EL Assume that (2.17)

G 8 (A) -»• G0(A)

for A e Fig,z -> oo.

T i e n there exists go : i" —• R. such that (2.18)

^i —• ^o ha measure for i —> oo.

Note. The theorem has several consequences: (2.19) there exists a subsequence i(k), k £ N and Mo (E TV such that 9i(k)(t) -* #o(0 for k -+ oo a.e., (2.20) G 0 is additive (by (2.17), since G; are additive), (2.21) Go e Qy(0) (since we may write G ,-(£), <7i(£) instead of i71, / in (2.14), (2.15) and pass to the limit for k -> oo by (2.17), (2.19) after having made a suitable choice of M ) , (2.22) Qy(8) is closed in AdC (cf. (2.21), (2.17), (2.19)), (2.23) Gi(fc) —>• Go (since G m e Qy{6) for m = 0 , 1 , 2 , . . . and (2.19) holds, there exist M G Af, rj G D* such that (2.9) (i) and (iii) hold. Moreover, (2.9) (ii) follows from (2.14) by Lemma 1.13 so that (2.23) holds by Lemma 2.3). Proof. In order to prove (2.18) it is sufficient to show that the sequence gi is Cauchy in measure. Let us suppose the opposite, i.e. that there exists a 6 R + such that for every r € N there are p = p(r), q = q(r) such that p,q > r and (2.24)

\{tel;

\gp(t)-gg(t)\>a}\>2a.

CONVERGENCE

27

Let h £ N be chosen such that 2~h+2
(2.25) For H N put Xk = {xGl;

0(h,x)>b-^}.

Then J 5 ^ C Xi forfc< / and (Jj.^Xfc = / so that lim \Xk\ = b — a. k—>oo

Find k eN such that \Xk\ > b — a — a. Denote EVA = {te I ; |flf„(*)-<7,(*)l>*}By Theorem 1.12 <7i is measurable for i £ N, hence -£/p,? is measurable as well. Consequently \EPtq\ > 2<7 by (2.24) and |-X"jfc fl Eptq\ + \Xk \ Epjq\ = |Xfc|, \Xk n Ep,q\ > \Xk\ — \I\

Ep>q\,

so that (2.26)

\Xk 0 Ep>q\ > a for all p = p(r), q = g(r), r £ N.

P u t c/ = a + £(6 - a) for / = 0 , 1 , 2 , . . . , A;, (2.27)

J/ = [c,_i,c,] for / = 1 , 2 , . . . , k.

Let XJPjg be the set of / such that

j,ni t n£ M n(/\M + )/f)

28

INTEGRATION BETWEEN L AND H - K

where M + = | J i e N M i , and let 2/ = ti,p,q be an element of Jl nXkn Ep>q n(I\ M + ) for / e £p,q. Let us set Ap,q — {(tl, Jl)', I € £p,q}Obviously &p,qey(i,i\M+,d(h,-)) and

J]|GJ(J)-^(f)|J||<2-ft

(2.28) A

P,9

for i £ N, p = p(r),q Since evidently

= q(r), r G N.

G , ( J ) - G g ( J ) - (GP(J)

-

9p(t)\J\)

+ Gq(J) -

gq(t)\Jl

(2.28) implies that

E

-

M*)

99(*)\\J\ <

2 h+1

~

+ E \°^J) A

"P,!

G J

i( )\-

P,9

If (2, J) £ A P ; g, then J is one of the intervals (2.27) and by (2.17) there exists r G N so large that E A

\Gp(r)(J)-Gq(r){J)\<2~h-

P(r),«(r)

This inequality together with (2.28) implies that

(2.29)

E

l^(r)(0-^(r)(<)PI<2- f c + 2 .

CONVERGENCE

29

On the other hand, if (t, J ) G A p ( r ) ) g ( r ) then £ G Ep(r)q(r) and IfipO)^) ~~ <7?(r)(0l — <7- Moreover, A P j g was constructed in such a way that Xk n Ep,q n ( J \ M + ) C | J { J ; (t, J ) G A P ) , } so that (cf. (2.26)) a<\XknEPtqn(I\M+)\<

V | J A

?,«

Therefore

A„ ,

M*)-^(*)im>^'EiJi^ff2'

which contradicts (2.29) and (2.25). Hence (2.18) holds. T h e proof is complete. 2.11 D e f i n i t i o n . Let X be a vector space over R, S G X. S is called (i) convex if a x + (1 — a ) y (E S for x, y G 5 , a G [0,1], (ii) circled (balanced) if x G 5 , a G [— 1,1] implies ax G 5 , (iii) radial (absorbing) if for every x G -X" there exists Ao G such that x G AS for A > AQ. 2.12 T h e o r e m . Let 9 G D*. Then Qy(0) is a convex, circled and compact subset in the Banach space (AdC, || • ||SUp)Proof. Let 6 £ D*. It follows directly from Definition 2.6 that Qy(9) is convex and circled. The compactness of Qy(9) is a consequence of Theorem 2.10, (2.21), the Arzela-Ascoli theorem and t h e following observation: For j G N, t G I put a = 2~j-19(j,t). If J G Iv, t G J C (t -
IGWIS^ + j ^ S Z - ^ . The proof is complete.

INTEGRATION BETWEEN L AND H - K

30

2 . 1 3 L e m m a . Let 771,772 G D*. Then there exists 773 G D* such that

(2.30)

Qy(m) + Qy(m)cQy(Vs).

Hint. It can be verified that (2.30) holds for V3(j,t) = -min{»7iO' + l,<),J72(i + l , * ) } , j C N, t G J since 1 rii(j,t)

n2(j,t)

2 <

;

:

;

1 :

r^ <

min{77i(j,<),r/ 2 (i,<)}

»?3(i,*)"

2.14 D e f i n i t i o n . Let F t G P y for i G N and F 0 € PyAssume that (2.31) there exists 6 e D* such that F m G Qy(<5) for m = 0,1,2,..., (2.32) ||P, - Pollsup -> 0 for i -> 00. Denote by Qy the set of couples ((F 8 , i G N ) , F 0 ) fulfilling (2.31) and (2.32). Qy is called a convergence and the sequence Pi, z G N is said to be convergent

to FQ in Qy, Ft —> Fo.

2.15 L e m m a . Let F{ ^U F0, Gt ^ Then (2.33)

(2.34)

(2.35)

G0, a G M.

Pi(jfe) — • Po for any subsequence

i^ + G ^ F o

+ Go,

a f ^ a F o .

i(k), k = 1, 2, 3 . . . ,

CONVERGENCE

31

Proof. By Definition 2.14 there exist 771,772 £ D* such that FleQy(r]l),GleQy(r]2)

for i = 0 , 1 , 2 , . . .

and 11-ft - -Po| I sup —> 0, ||Gi - Go||sup —>• 0 for z —• 00. Obviously, (2.33) is valid. (2.34) holds since there exists 773 £ D* (cf. Lemma 2.13) such that Ft + Gi GQyM

for i = 0 , 1 , 2 , . . .

and H^ + G i - F o - G o l l s u p - O . Let a £ R, k £ N, | a | < 2*. Making use of Lemma 2.13 repeatedly we conclude that there exists 0 £ D* such that 2*ft GQy(O)

for i = 0 , 1 , 2 , . . . .

Hence a f t £ # y ( 0 ) for z = 0 , 1 , 2 , . . . and (2.35) holds since \\aFi — a-Fo||sup —• 0. T h e proof is complete. 2.16 T h e o r e m , (i) If F{ ^

F0, then ft ^

(ii) If Fi — • Fo then there exists a subsequence such that Fi(k) ^

F0. i(k), k £ N

FQ.

Proof. Let Fi —^» -Fo- By Definition 2.2 there exist / m , m = 0 , 1 , 2 , . . . and 6 £ D* such that (2.6) and (2.7) hold. Moreover, there exists 9 £ D* such that

0(j,t)<8(j,t),

|/m(t)|<^J_j

for m = 0 , 1 , 2 , . . . , j £ N, < £ I. Therefore (2.14) holds for F = 0, F = Fm, f = fm so that Fm Note 2.1 implies that \\Fi — F0\\snp (ii) holds by (2.23). T h e proof

Fm and (2.15) holds for M = £ Qy{6) for m = 0 , 1 , 2 , . . . . —* 0 for i —> 00 and (i) holds. is complete.

3 CONVERGENCE LOCALLY C O N V E X

AND SPACES

3.1 P r e l i m i n a r i e s . Let Y be a set. Let T be a system of subsets of Y such that (3.1)

(3.2)

(3.3)

®eT,YeT,

if UUU2&

if V C T,

T,

then

then tfi n U2 G T ,

\J{U;

U G V} G T .

Then T is called a topology on Y, (F, T ) is called a topological space and the sets £/ from T are called open (T-open). T is called Hausdorffif for x,y G F , x 7^ y there exist Ui,U2 G T such that x G Z7i, y G ?72, U\ H f72 = 0. If T is a topology on F and Z C Y then T | z = {U 0 Z; U G T } is a topology on Z (the restriction of T to Z). Let #i G Y for i = 0 , 1 , 2 , . . . . The sequence x,, i G N is said to be convergent to XQ in (F, T ) , if for every U G T with xo G £7 there exists A: G N such that x^ G U for i > fc. Let T\,T2 be topologies on F . 7i is called ./mer than T2 if 7i D T 2 . From now on let Y be a vector space over R. Let T be a topology on F . If y G F , t/ G T implies that y + U C T, then T is called invariant with respect to shifts. 32

CONVERGENCE AND LCS

33

Let 53 C 2 y fulfil (3.4)

(3.5)

(3.6)

03 ^ 0, 0 i 03,

every V from 03 is radial, circled and convex,

for Vi, V2 G 03 there exists V3 G 03

such that v3 c Vi n v2, (3.7)

(3.8)

for V G 03 there exists Vi G 03 such that 2Vi C V,

for x G F , x 7^ 0 there exists V G 03 such that x ^ V.

Let V(03) be the set of U C Y such that x G U if and only if x + V C U for some V G 03. V(03) is a topology on F which is invariant with respect to shifts and Hausdorff. V(03) is called the topology induced by 03. (Y,T) is called a locally convex space if there exists 03 such that T = V(03). Let r : Y -+ [0, oo). If (3.9)

(3.10)

r(x + y) < r(x) + r(y)

for

i,t/6F,

r ( ^ x ) = |)u|r(x) for x G F, /u G R

then r is called a seminorm. Let i J b e a set of seminomas. Assume that for every x G F there exists r G -R such that r(x) > 0. The set (3.11)

{x G F ; r,(x) < £ for i = 1 , 2 , . . . , k}

INTEGRATION BETWEEN L AND H - K

34

where k € N, r; G R, e G R + is convex, circled and radial. Let 2G be the set of sets (3.11). 2U fulfils (3.4) - (3.8) and V(2U) is called the topology induced by R. Let P* C AdC be a vector space. Let A be a set the elements of which are couples ((Fi,i G N),Po) where Fm G P* for m = 0 12 V,

-L, - : , . . . •

We shall write (3.12) Ft - ^ F0 instead of ((Fi,i G N ) , P 0 ) G A and assume (3.13) i f P , - ^ P 0 , G 4 - ^ G 0 , a , / ? G E t h e n ( a ^ + / ? G O ^ (aF0+/3G0),

aFi -^

aF0,

(3.14) if Ft - ^ P 0 then ||P, - P 0 | | s u p - • 0. A is called a convergence on P* and the sequence Pj-,i G N is said to be convergent

to F0 in A, Fi

A

> P0.

3.2 L e m m a . Let P* C P2* C AdC, Pi*,.P2* being spaces. Let A* be a convergence on P ? , i = 1,2. T i e n (3.15)

Ax C A 2 iff P, - ^ P 0 implies Ft-^

vector

F0.

This is just a reformulation of (3.12). 3 . 3 D e f i n i t i o n . Let P * C AdC be a vector space, let A be a convergence on P * and let T be a topology on P * . T is called tolerant to A if Ft - ^ F0 implies Ft -> P 0 in

{P*,T).

3.4 N o t a t i o n . Denote by 7^ up the topology on AdC which is induced by the norm || • ||Sup- Let P * C AdC be a vector space and let A be a convergence on P * . By U L C ( A ) let us denote the set of topologies T on P * such that (3.16) (3.17)

(P*,T)

is a locally convex space, y

is tolerant to A.

Note. U L C ( A ) / 0 since { 0 , P * } G U L C ( A ) .

CONVERGENCE AND LCS

35

3.5 L e m m a . Let P* C AdC be a vector space and let A be a convergence on P*. Then there exists ULC(A) G U L C ( A ) such that (3.18)

W L C (A) is finer than any T G U L c ( A ) .

3.6 N o t e . Uic{A) is unique. Moreover, ULC(A) dorff since it is finer than T SU p|p*.

is Haus-

Proof. If T G U L C ( A ) then (P*,T) is a locally convex space and there exists 03 G 2 P * fulfilling (3.4) - (3.8) such that T = V(03). For a finite set 7i = V(03i), T2 = V(03 2 ), . . . , Tk = V(03fc) G U L C ( A ) and Vi G
y = Vi n y2 n • • • n v* and let 03 be the set of such V. Then 03 fulfils (3.4) - (3.8) and V(03) is finer than any T G U L C ( A ) and tolerant to A. Hence ULC(A) = V(03). T h e proof is complete. 3.7 T h e o r e m . Let P* C P2* C AdC, P^P^ spaces. Let Ai be a convergence on P*, i = 1,2. Assume

being

vector

that

(3.19)

A]CA2.

Then ULC{A\)

is finer than

ULC(A2)\P*.

Proof. Since Ft - ^ F0 implies Fi - ^ F0 (cf. (3.16)) we conclude that if T is a topology on P2* which is tolerant t o A 2 then Tip* is tolerant to Ai

INTEGRATION BETWEEN L AND H - K

36

so that T e U L C ( A 2 ) implies T\P* G U L C ( A I ) . Therefore WLc(A2)|P. G U L C ( A I ) and WLC(AI)

is finer than W L C ( A 2 ) | P 1 * .

T h e proof is complete. 3 . 8 L e m m a . Let 03 fulfil (3.4) - (3.8), V € 23,

Fo = |J (1 - -)V. T i e n y 0 is open in (Y, V(Q3)). Proof. Let F G Fo. There exists m G N, m > 3 such that F G ( 1 - £ ) F SO that F + — F C ( 1 - — )V. m m Let / € N, 2' > m. Making use of (3.7) repeatedly we conclude that there exists V £ 53 such that 2lV C V. Hence

i>c2 _ 'v c — v m and F + VC(1-—)V

m

CV0

so that Vo is open. The proof is complete. 3.9 T h e o r e m . Let T be a topology on Py. tolerant to Qy if and only if it is tolerant to Ey.

Then T is

CONVERGENCE AND LCS

Corollary. ULC{Ey)

= VLC(Qy),

ULC(Ey)

37

=

ULC(Qy).

Proof. If T is tolerant to Qy then it is tolerant to Ey by Theorem 2.16 (i). Assume that T is tolerant to Ey and that there exist Fm G Py, m = 0 , 1 , 2 , . . . such that Fi —> FQ but the sequence Ft, i G N is not convergent to F0 in ( P y , T ) . Hence there exist U G T and a subsequence i(k), k G N such that FQ £ U and (3.20) However, F^)

F i ( f e ) £ U for fc G N. ~~^ -Fo by (2.33) and Theorem 2.16 (ii) implies

that there exists a subsequence &(/), / G N such that Fj(fc(;)) —> FQ. Therefore there exists IQ G N such that F;(j.(/)) £ U for I > IQ, which contradicts (3.20). This contradiction proves that T is tolerant to Q y . T h e proof is complete. 3 . 1 0 L e m m a . For a G R + p u t 5 ( a ) = {F G AdC; | | F | | s u p < a}. Let 0 G 17 G ^ L c ( Q y ) , 0 £ D*. Then there exists a G R + such that Qy(0) D £( < 2~\ Hence Fi > 0 which is impossible since ULci'Q'y ) is tolerant to Q y . This contradiction makes t h e proof complete. 3.11 L e m m a . For E C AdC denote by c o n v F the convex hull of E. LetOeUe ULC(Qy)Then there exists ( : D* -> + R such that conv | J Q y ( 6 ) n B(C(6)) C U. seD*

INTEGRATION BETWEEN L AND H - K

38

Proof. Since 0 G U € WLC(Q;>;) and WLC(Q3>) = V(2J) for some R + such that | J Qy(6) n B(((6))

C Vo

and conv \J Qy{8) n # ( ( ( £ ) ) C Vo C V C f/. seD* The proof is complete. 3 . 1 2 L e m m a . For ( e D* define V>(C) : N x J -+ R+ by

(3.21)

^(C)0',0 = |C0" + 1,0-

Then (3.22)

il>:D*^

(3.23) V>(C) = * iff <M,t) = 26(j-l,t) and ((l,t) (3.24)

2Qy(C)

D*,

for j =2,3,4,...,t

> ((2,t)

C Qy(H0)

el

for t e I, for

C € #*•

Proof. (3.22) and (3.23) are immediate consequences of (3.21). Let F e Qy(0- Then

CONVERGENCE AND LCS

39

for j G N, A = {(t, A)} G y{I, I, CO', •)) and there exist M G AT and / : I —> R such that ^|F(A)-/(t)|A||<2-

for j G N, A = {(*, A)} G y(I, I \ M, CO', •))• Moreover, (3.21) implies (3.25)

y(i, I, v(C)0\ •)) c y(i, / , CO" + 1 , •)), j € N .

Let j G N, A = {(t,A)} we get (3.26)

£

e y(/,I,V(C)0',-))- Since F G Qy(C)

|2F(A)| < 2[2->" 1 + £

<2"i + E A

^ L _ _ ] <

^

IKOO\O

If j G N, A = {(*, A)} G y ( 7 , 1 \ M, tf(C)(j, 0) we get (3.27)

Y^ \2F(A)

~ 2/W|A|| < 2 • 2 - ' - 1 = 2 - J '.

Now (3.24) is a consequence of (3.26) and (3.27). The proof is complete. 3.13 L e m m a . For A : D* -> R+ put V(A) = conv ( J Qy(C) n J3(A(C)), Ceo* ^

= {V(A); A : D* -

40

INTEGRATION BETWEEN L AND H - K

Then V3y fuffls

(3.4) - (3.8).

Proof. It follows immediately from the definition of 03j; that (3.4), (3.5), (3.6) and (3.8) hold. For ( e D* put 6 = V>(C), rtO = IHHO)Then 2Qy(C) C

Qy(H0),

2 |J Qy(C) n 5(^(0) C |J ^ ( ^ ( 0 ) 0 5 ( ^ ( 0 ) ) . Ceo*

C€D*

Since ?/>(£>*) = D* by (3.23), we have

U QyMC))n5(A(iKC)))= U W ) n f l ( ¥ ) ) C€D*

0€D*

and finally 2V(/,) = 2conv | J Q y ( ( ) n % ( C ) ) C <e£>* C c o n v \J

Qy(O)nB(X(0)).

eeD* Hence (3.7) holds and the proof is complete. 3 . 1 4 T h e o r e m . ULC(Qy)

= VpUy).

Proof. Since QJy fulfils (3.4) - (3.8) by Lemma 3.13, V(3Jy) is a topology on Py and (Py, V(Q3y)) is a locally convex space. Let Ft ^ F0. Then Fm e Qy(tf) for some £ G D*, m = 0 , 1 , 2 , . . . and \\Fi — .Fo||sup —> 0. Lemma 3.12 implies that Fi-F0e

Qy{i>{6))

for i = 1 , 2 , 3 , . . . .

If A : D* ->• R+ then there exists i e N such that \\Fi - ^o||sup < A(V>(£)) for « > A;.

CONVERGENCE AND LCS

41

Hence Ft-F0e

Qy(il>(6)) H B(X{^(6)))

C V(X)

for i > k

so that V(V3y) is tolerant to Qy. Therefore V(93y) G U ( Q y ) and consequently £ / L C ( Q ; V ) is finer than V(%3y). On the other hand, V(^Oy) is finer than ULc(Q.y) by Lemma 3.11. Hence ^Lc(Qy) = V(2J;y) and the proof is complete. 3.15 L e m m a . Let y, Z e Y, y C 2 , ( : I -»• R + , ^ ( 1 , ^ 0 C 2 ( 1 , 1 , C), Tien (3.28)

-PW,J,C)

= ^ y . %(/,/,c) =

(3.29)

(3.32)

PzcPy,EzcEy,

Qz(6)cQy(6)

for 6 e D*,

Q s C Q y , ^ ^ F 0 implies F 2 ^

ULC(Ey)\Pz (3.33)

^'

Qy(i,i,o=®y>

(3.30)

(3.31)

E

ULC{EZ)

is£nerthan

F0,

i.e. ULC(Ey)\Pz.

Proof. (3.28) - (3.32) are obtained immediately from Lemma 1.15, Definitions 2.2, 2.6 and 2.14. (3.33) follows from (3.30) by Theorem 3.7.

4 A N AUXILIARY LOCALLY C O N V E X

SPACE

4.1 P r e l i m i n a r i e s . For F £ AdC, s £ I put \\F\\y,s = s u p { ] T \F(A)\;

A = {(3, A) £ y(I, {s}, 00)},

A

Ty = {F £ AdC; ||F||y, a < 00 for s £ I}. Obviously, Ty is a vector space and || • H ^ is a seminorm on Ty. Since WC C y we have (4.1) \\F\\wc,s = sup{|F([c, s])\ + \F([s,d\)\; < \\F\\y,s

for

a
F£Ty,

SEI,

\\F\\sup<2\\F\\nKtS,\\F\\nK,s<2\\F\\sup for F £ Ty, s £ I and || • \\yiS is a norm on For k £ N, s1,s2,...,3k

£ I, £ £ K + put

Wy(s!,s2,... = {F£TyWy = {Wy(s1,s2,...,sk;e);

Ty.

,sk;e)

=

\\F\\y>a. < e for i = 1, 2 , . . . , k}, k £ N , s i , s 2 , • •., sk £ I,e £ 42

R+}.

AN AUXILIARY LCS

43

Wy(s\,S2, • • •,sjt;e) is a convex, circled and radial set in Ty and Wy fulfils (3.4) - (3.8). Define Ty =

V(Wy).

Ty is a topology on Ty and (Ty,Ty) is a locally convex space. Ty is induced by the set of seminorms {|| • \\yj3; s £ / } . (4.1) implies (4.2)

Ty is finer than

Tsup\Ty

For K e Iv , F e AdC define TKF

TKF(A)

: Fig -»• R+ by

J P(cl(A n int K)) if cl(A fl int K) ^ 0, = j 0 if cl(A n int K) = 0.

T^-F is well defined, since either cl(A fl int K) £ Fig or cl(A fl int K) = 0 for A £ Fig . It is not difficult to prove that TKF £ AdC . Observe that TKF : Fig - • R while F\K : Fig (A") -> E. 4.2 T h e o r e m . Assume that (4.3)

P y C Ty,

(4.4)

Ty\py

(4.5)

TKF

(4.6)

for A' e Iv

is tolerant to

Qy,

£ Py for K € Iv ,F £ P y , there exists

I I F / c P b , , < K\\F\\y,„

K£ R

such that

FePy.se

Then (4.7)

ULC{Qy)

= Ty\Py.

T h e proof will be performed in several steps.

I.

INTEGRATION BETWEEN L AND H - K

44

4 . 3 L e m m a . Let K(i) = [ci,dt] G Iv, K(i + 1) C K(i) zeN, s e l , f]ienK(i) = {s}. Let Ft G Py fulfil

for

(4.8) Fi(A) = 0 if A G Fig , A C [a,ct] U [d,-, 6], « G N, (4.9) for e G R+ there exists a G R + such that

E l*K^)l ^ £ A

fori€N,A = {(s,A)}6y(I,{s},
for % G N.

Remark. Fi is J^-continuous at s since i^ G Py for i G N (cf. Definition 1.8). a in (4.9) is independent of i which may be interpreted as the uniform ^-continuity at s of functions Fi, i G N. Proof. By assumption there exist Si G D*, /,• : I —> R and Mt e JV for z G N such that

(4.10)

JX^I^+^J^ A

A

u

'

y

for i , ; G N, A = {(*,4)} G ? ( / , J , * ( j , •)),

(4.11)

^lF^)-/«(t)lAH^2"i A

for i,JGN, A = { ( t , A ) } G ^ J \ M , , ^ 0 V ) ) . Without loss of generality we may assume that (4.12) fi(t) = 0 for a < t < a, di < t i,tel.

AN AUXILIARY LCS

Put (4.14) ui(t) = dist(«, K(i)) for i e N, t e I and define 9 as follows: (4.15) 0U,t)=u1{t)

for j G N , t G / \ / i ( l ) ,

9(j,t) =min{u)t+1(t),6t(j foriJtN,

+ i + l,t)}

f € A ^ ) \ A ' ( « + l),

9(j,s) is so small that ^|Fi(A)|<2--'--1 A

for i,j € N, A = {( 5 , A)} G ? ( ! , { * } , 0 O » ) , 0(j + l,s) <
M = { S }UUM,. We have to prove that

(4.16)

X>(A)|<2-' + £JiL for i,j € N, A = {(<, A)} £ 3>(/,J,0OV))

and (4.17)

^|FJ(A)-/,(0I^||<2^ A

for i,j e N,A = {(t,A)}

Let j e N, A = {(t,A)} e

e

y{I,I\M,0(j,-)).

y(i,i,e(j,-)).

INTEGRATION B E T W E E N L AND H - K

46

Put A0 = { ( M ) e A ;

tei\K(i)},

A, = {(*, A) 6 A; t € A'(/) \ K(l + 1)}, / € N, Aoo = {(<,A) G A; t = s}. Systems A;, / = 0 , 1 , 2 , . . . , / = oo are pairwise disjoint and their union is A. Let i G N. It follows from (4.8) and (4.15) that (4.18)

^2\Fi(A)\=0ioT

/ = 0,l,...,i-l.

A,

Let / G N, I > i. Then (cf. (4.10), (4.15))

(4.i9)

s>wi*2-'"'- + E Aj

s,u!f+u)-

Aj

< 2-J-1-1 + V

^

Moreover (cf. (4.15)),

(4.20)

X]I^( A )I < 2 _ i _ 1 -

(4.18), (4.19) and (4.20) imply that (4.16) holds. (4.17) is valid by an analogous argument. The proof of Lemma 4.3 is complete. 4.4 L e m m a . Let ( : D* -> R, s G I. (4.21) there exists rj G R

+

Then

such that

{F G Py, F(A) = 0 for A G Fig , AcI\(s-r,,s

+ T,),\\F\\y,a
[) 6£D*

Qy(8) H

B(((6)).

47

AN AUXILIARY LCS

Proof. Let (4.21) be false. Then there exists a sequence Fi, i G N such that (4.22)

Ft G Py \ | J Qy(6) n £(((<$)), : € N,

(4.23)

P^ly,, < 2 " \ zGN,

(4.24) F 2 (A) = 0 for A G Fig , A C I \ (s - 2 ~ \ s + 2 " J ) , z G N. Now Lemma 4.3 may be applied. P u t Ci = max{a,s—2~l},di

= min{s+2~*, &}, -ftT(i) = [c^,
Then f | l € N ^ ( i ) = {s} and (4.8) is fulfilled. Let e > 0. There exists k G N such that 2~fc < £. Hence (cf. (4.23))

A

for 2 > k, A = {(5, A)} G ? ( / , {s},oo) and (4.9) is fulfilled. By Lemma 4.3 there exists 9 G D* such that Fi G Qy{9) for z € N and by (4.23) and (4.1) ||-Fi||SUp -> 0, which contradicts (4.22). This contradiction makes the proof complete. 4.5 L e m m a . Assume that (4.4) holds. Let ( : D* —• R + . T i e n there exist r G N, c*i G i" for i = 0 , 1 , . . . , r, Si G I for i = 1 , 2 , . . . , r and £ G M + such that (4.25)

a = a0 < ai < • • • < ar — b, S{ G [ a i - i , ai] for i = 1, 2 , . . . , r,

(4.26)

{ F G P y ; F ( A ) = 0 for A G Fig ,

^C/\(oi-i,ai),||F||y,a<e}C

|J

Qy(5)n5(C(«))

INTEGRATION B E T W E E N L AND H - K

48

for ii = 1 , 2 , . . . , r. Proof. Lemma 4.4 implies that for every s £ I there exists rj — r](s) e R+ such that (4.21) holds. By Lemma 1.2 there exists

A = which is a form A = min{r)(sl); holds. The

{(s,J)}eWC(I,I,ri)

partition of I. Therefore A can be written in the {(si, [ci!i_i,Q!,-])} where ai,Si fulfil (4.25). P u t £ = i = 1,2, . . . , r } . Then (4.21) implies that (4.26) proof is complete.

Proof of Theorem 4-2. (Py,Ty\py) is a locally convex space by (4.3). Moreover, (4.4) implies that Ty\Py <E VLC(Qy) so that (4.27)

ULc(Qy)

is finer than

Ty\Py.

It remains to prove that (4.28)

Ty\py

is finer than

ULc(Qy)-

Let ( : D* -> R+ and let r e N, at € / for i = 0 , 1 , . . . , r, st e I for i = 1 , 2 , . . . , r, £ € R + correspond to £ by Lemma 4.5. Let (4.29)

FeWy(s1,s2,...,sr--^-

)f\Py rK

(cf. (4.6)). We have r

r

(4.30)

T

F = Y, i°i-i,*i]

F

1

= E ; I Wx.".-r i r -

(4.6) and (4.29) imply that | | r [ a , - _ l j 0 t ] ^ b , S i < K||rF||y ) S . < £, i = l , 2 , . . . , r ,

AN AUXILIARY LCS

49

hence by Lemma 4.5 T[ai_uai]rF

e

| J Qy(6) n B(C(6)),i

=

1,2,...,r.

S£D*

By (4.30) F is a convex combination of r r a t . _ i a . i r F , i = 1 , . . . , r, so that F€conv

| J Qy(6)nB(((6)) 6eD*

=

Vy(0

and (4.31)

Wy{sus2,.

..,sr;^)r\PyC rp

Vy(Q.

Since ULC(Qy) = V(Wy) (cf. Theorem 3.14) and Ty = V(Wy) we conclude by (4.31) t h a t (4.28) holds. T h e proof is complete. 4.6 N o t a t i o n . For A 6 Fig let us denote by C o m p A the set of components of A. 4.7 L e m m a . (Ty,Ty)

is

complete.

Proof. For basic concepts of a filter etc. see e.g. [K 2000] Appendix. Let Z = {Z} be a Cauchy filter in (Ty,Ty) so that (4.32) for every s 6 I , ( 6 1 \\G-F\\y,s
+

there exists Z e Z such that for

G,FeZ.

Since Z is Cauchy in (Ty,Ty), Ty C AdC, Tsup\xy C Ty we get that Z is Cauchy in (AdC,7^ u p ). Since (AdC,7^ u p ) is complete there exists H £ AdC such that Z^H

in ( A d C , T s u p ) ,

i.e. (4.33) for every 5 € I, e € R + there exists Z £ Z, Z C B(H,e)

= {Ge

AdC; ||G - # | | s u p < e}.

INTEGRATION B E T W E E N L AND H - K

50

Let s £ I. By (4.32) there exists Z\ G Z such that \\G-F\\ytS

< 1 for F,G e ZL

(4.33) implies that for e G R+ there exists Z(e) G Z, Z(e) C Z i O B ( i f , £ ) . Let G(e) G Z(e), A = {(s,A)} G }>(!, {s},oo). We have

(4.34) Y, \H(A)\ < E lG(e)(A)l + E l^(^) - GWA)\ ^ A

A

A

< \\G(e)\\y>a + \\H - G(e)|| 8 u p E

# Comp A.

A

Let F e Zx.

Then

l|G(e)||y )S < ||F||j, > a + \\G(e) - F\\y,a < \\F\\y,3 + 1 and by (4.34) we get

E \H(A)\ < \\F\\y,a + l + ^ # C o m P 4 , A

A

which implies that

E)|ff(A)|<||F|| 3 ,,. + l, A

||-H"||y,« < ll-^lly.s + 1 and H G Ty. Assume that (4.35)

Z^H

in (Ty,Ty)

is false, i.e. (4.36) there exist s G I , £ G E + such that for every Z G Z there exists K € Z,

\\K-H\\y,s>t

AN AUXILIARY LCS

51

By (4.32) there exists Z2 G Z such that (4.37)

\\G-F\\y,s<\i

for F, G G Z2

and there exists K2 G Z2, \\K2-H\\y,a>Z. Therefore there exists A = {(s, A)} G y(I, {s},oo)

such that

£|A'2(A)-Jf(A)|>^. A +

Let e G R , ££#Comp.4<|. A

By (4.33) there exists Z3 G 2 , Z 3 C B(H,e) Let F G Z 3 . Then

D Z2.

A

> 53 |7^2(A) - lf(A)| - J2 \HiA) - F(A)\ > 1$ - \C = \t, A

A

which contradicts (4.37). Therefore (4.36) is true and the proof is complete. 4.8 T h e o r e m . Assume (4.38)

that (4.3) - (4.6) hold.

(Py,ULC(Ey))

is

Then

complete

if and only if (4.39)

Py is closed in

(Ty,Ty).

Proof. ULC(Ey) = Ty\Py by Theorem 4.2 and Corollary 3.9. Theorem 4.8 holds since a subset of a complete space is complete if and only if it is closed. The proof is complete.

5

£-INTEGRATION

5.1 P r e l i m i n a r i e s . For F e AdC, K G Iv put k

varA-F = s u p { J ] | F ( J , ) | }

where k G N, Ju J2,..., Jk G Iv(iJST), | J,- D «/> | = 0 for i / I. Write var F instead of var/ F and put llFUva^varF. By AdCBV let us denote the set of F G AdC such that ||F||var < OO.

AdCBV with the norm || • ||var is a Banach space. TVSLI is the topology induced on AdCBV by || • || v a r . By AdAC we denote the set of F G AdC such that for every e G IR+ there exists ( T 6 R + such that k

ElWOI<e if k G N, J i , J 2 , . . . , J f c a

s weu

^2i=i \Ji\ — - It i space in (AdCBV, Tva,T).

G br{K),

\Ji n Ji\ = 0 for i ^

/,

known that AdAC is a closed sub-

52

^-INTEGRATION

53

If A G Fig and / : I —• R is Lebesgue integrable then (Leb) fA fdt is the Lebesgue integral of / over A. f : I —> R. is called a stepfunction if there exist ceo, « i , . .. , otr G I-i ii i £2, • • •, ir £ IK. such that a = ao < a i < • • • < aT = b, f(t) = & for t G ( a j _ i , a j ) , z = 1,2, . . . , r . St denotes the set of primitives to stepfunctions. The main result are the next two theorems. In the first, procedures from Chapters 3 and 4 are applied to £-integration while in the second it is made clear that /^-integration is Lebesgue integration. The rest of this chapter is devoted to the relations between E ^ , Qc and the convergence in (PCMLC{QC))5.2 T h e o r e m . (5.1) (5.2) (5.3) (5.4) (5.5)

Tc = AdCBV, Tc is the topology on Tc induced by \\ • 11var, Pc C AdAC, 1~c\pc is tolerant to ULC(®C)

=

Qc,

TC\PC.

Remark 1. (5.3), (5.5) and (5.2) imply that ULC(Qc) topology on Pc induced by || • || va r-

is the

Remark 2. Let £* be the set of partitions A = {(t, J ) } G £ such that J G Iv for (t, J ) G A. Since £* C £ , every £-integrable / : I —» R is £*-integrable. On the other hand, if A = {(i,A)} £ £ then replace every couple (t,A) G A by couples (*,Ji),(i,J2),...,(*,Jr) where { J i , J 2 , . . . , J r } is the set of components of A. The resulting partition A* belongs to £*. If ( : i" —> R + and if A is £-fine then A* is £-fine. Therefore every £*-integrable / is £-integrable. Moreover, £-integrals and £*-integrals coincide.

INTEGRATION BETWEEN L AND H - K

54

5.3 T h e o r e m . (5.6) Pc = AdAC, (5.7) if F e AdAC, / : I -> R, F ( A ) = (Leb) J A /eft for A G Fig, then / is C-integrable and F(A) = ( £ ) /

/dt.

iVoie. (5.6), (5.5) and (5.2) imply that (PCMLC(QC)) complete space. Theorems 5.2 and 5.3 will be proved in several steps. 5.4 L e m m a . Let a < c < b, F <E Pc- Then T^F

is a

£ PC-

Proof. There is nothing to prove if c = a. Let a < c < b. Since F £ Pc there exist / : / — * • R and S <E D* such that (cf. Lemma 1.15)

Y,\F{A)-f{t)\A\\<2-i A

f o r j e N , A = {(t, A)} eC(I,I,6(j,-)). Without loss of generality we may assume that /(c) = 0 , 6 ( j , t ) < \ t Let j eN, A = {(t,A)} Observe that

e

c\ for

jeN,t?c.

C(I,I,6(j,-)).

A C [a,c) if (t,A)

€ A,t

e [a,c)

Ac(c,b]

€ A,<€(c,6].

and if (t,A)

Put A i = {(t,A)

6 A; t 6 [a,c)}U

{(c, cl(A n [a, c)); (c, A ) e A , A n [a, c) ^ 0},

^-INTEGRATION

A 2 = {(t,A)

55

£ A ; t e (c,6]}U

{(c,cl(An(c,6]); ( c , A ) € A , A n ( c , 6 ] ^ 0 } 5 and write A x = { ( s , £ ) } , A 2 = {(s,E)}. Obviously A a U A 2 G C(I,I,6(j, •)) so that

£ |F(£) - f(s)\E\\ + J2 \F(E) - f(s)\E\\ < 2">. Ai

A2

Observe that

J2\H(A)-h(t)\A\\< A

Ai

A2

if i l G AdC, h:I^R, Put

h(c) = 0.

{

0 for t G [a,c],

f(t)

for t € ( c , 6 ] .

We have A

< E |r[c,6]F(^) - g(s)\E\\ + E |r[c,6]F(E) -

9(S)\E\\

Ai

= AElF(^)-/(5)^H^2" 2

Hence r ^ j F 6 Pc- T h e proof is complete. The next lemma is analogous. 5.5 L e m m a . Let a < d < b, F G Pc- Then T^^F

G Pc-

56

INTEGRATION BETWEEN L AND H - K

5.6 L e m m a . Let a < c < d < b, F G Pc-

Then

r M ] F e Pc. Lemma 5.6 is a consequence of Lemmas 5.4 and 5.5 since F[c,d]F = T[a:d](T[Ctb]F). Proof of Theorem 5.2. As a consequence of the definitions of C and || • ||£ i S we have (5.8)

\\F\\c,s = l l ^ l k r for F e AdC,s

e I.

Hence (5.1) and (5.2) are valid. Let F e Pc, £ € K + , p e N, 2~P+1 < e. By Lemma 1.15 there exists 8 (E D* such that

YJ\F{A)-f{t)\A\\<2-> A

i f j e N , A = {(t, A)} e £ ( / , / , %,-))• Let A = {(t, A)} e WC{I, I , 6(j, •)) be a partition of I. Then A can be written in the form (5.9)

A={(Ti,[ai-1,ai]);

i = l,2,...,r}

where r G N, a = a0 < a\ < • • • < ar — fe, r,- € [ a t _ i , a j ] for i = 1,2,...,r. Let L\, L2,..., Lk € Iv, |L^ D Lm \ = 0 for i ^ m. Put (5.10)

0 = { ( r j , [ a i _ i , a i ] n £ T O ) ; |[aj_i,a,-] n L m | > 0, i = 1 , 2 , . . . , r , m = 1 , 2 , . . . , A;}

and write 0 = {(5, A")}. Since 0 £ £ ( / , i", 6(p, •)) we get

Y,\F(K)-f{s)\K\\<2->, 0

^-INTEGRATION

(5.11)

57

Y,\F(K)\<2-' + '£l\f{8)\\K\< © e k

<2-

p

+ max{|/(r0|; i = l,2,...,r}

] £ \Lm\. m= l

A is a partition of I. Therefore Lm = \J{K; (s,K)eQ,KcLm},

m = l,2,...,fc

and | A'i n K2 \ = 0 if ( s i , Iu), (s 2 , A"2) € 0 , ( s i , A'i) ^ ( s 2 , A~2). Hence k

(5.12)

£|F(L

m

)|<£|fWI<

m=l

© it

<2-

p

+ max{|/(r 8 )|; i = 1,2,... ,r} J ] | L m | m=l

and (5.3) holds. In order to prove that Tc\pc is tolerant to Q£ it is sufficient to prove that Tc\pc is tolerant to E^ (cf. Corollary 3.9). Let Ft ^

F0.

There exist gm : I —> R for m = 0 , 1 , 2 , . . . and 8 € I?* such that (5.13)

J]|Fm(A)-<,m(*)|A||<2-' A

for j e N, m = 0 , 1 , 2 , . . . , A = {(*, A)} e C(I, I,8(j, •)) and (5.14)

gi(t)

-»• flr0(*)» < G I , * - • oo.

58

INTEGRATION B E T W E E N L AND H - K

By (5.13) we have (5.15)

Y. I ^ A ) - F »04)l < 2 - J + 1 + Y. 1^(0 " ^ ( < ) i m A

A

for i,j e N, A = { ( M ) } G C(I,I,6(j,-)). Let j £ N, A = {(*,^)} € WC(I,I,8(j, •)), A being a partition of / . Then A can be written in the form (5.9). Let L1:L2,

• • • ,Lk G Iv, \LiC\ Lm\ = 0 for i ^ m.

Define 0 by (5.10) and write 0 = {(s,K)}. Since 0 G C(1,1,6(j, •)) we have (cf. (5.15)) J2 \Fi(K)

(5.16)

~ F^K)\

^ 2-J+1 +

0

+ max{|^(r m ) - g0(Tm)\; m = 1,2,... ,r}(b-

a), i = 1,2,

Moreover, k

(5.17)

^

|F,(L m ) - F0(Lm)\

m=l

WW

~ F0(K)\

0

in analogy with (5.12). (5.16) and (5.17) imply that there exists n(j) e N such that k

J2 \Ft{Lm) - F0(Lm)\

< 2 - ' + 2 for i > n(j).

m=l

Therefore \\F, — -Fo||var —• 0 for i —> oo and (5.4) holds. Now Theorem 4.2 will be applied for y = £. We have AdAC C AdCBV so that (4.3) holds by (5.1) and (5.3). (4.4) holds by (5.4). By Lemma 5.6 we obtain (4.5). Moreover, by (5.8) we have (since F € AdCBV) WKFWCS

= v a r r A - F < v a r F - \\F\\C,,

and (4.6) holds as well. Therefore (5.5) holds by Theorem 4.2. The proof is complete.

£-INTEGRATION

59

5.7 L e m m a . Let h : I —> R+, (Leb) fjhdt there exists (:!—>• R + such that (5.18)

< oo.

Then

V fc(*)|A| < 4.(Leb) / hdt • "

A

forA =

{(t,A)}eC(I,I,0-

Proof. For / G Z put C| = {*€/; 2 ' R + fulfil (<-C(«).< + C ( < ) ) n / C V / for * G C / , / € Z . For A = {(*,il)}€£(/,J,C) put A, = {(<,A)€ A; t e C , } , l e Z ,

A* = { ( M ) e A ; * G l \ | J C j } . fez Then />(*) = 0 for (t,A)

A

£ A*,

/€Z A,

ZGZ

60

INTEGRATION BETWEEN L AND H - K

<J22l+1\Ci\ <4.(Leb) I hdt and (5.18) holds. The proof is complete. Proof of Theorem 5.3. Our first goal is to prove (5.19)

AdAC C Pc-

Let j e N, F e AdAC. There exists / : / -> R such that F(A) = (Leb) / fdt

for A <= F i g .

Since the set of stepfunctions is dense in the space of Lebesgue integrable functions there exists a stepfunction g : I —» R such that (Leb)

f\f-g\dt<2-J-4.

By (? let us denote the primitive of g. Then ||i71-G||var<2-J-4.

(5.20)

By Lemma 5.7 there exists (^i : / —• R such that (5.21)

^|/W-5(t)||A|<2-^

2

A

for A = {(t,A)} € £ ( / , / , 0,i )• Obviously G £ Pc- Therefore there exists £j,2 : ^ -> ^+ that

(5.22)

J]|G(A)-^)||>1|<2-^ 1 A

for A = {(t,A)}

€ £ ( / , / , 0,2).

such

^-INTEGRATION

61

Put (j(t) = min{Cj,i,Cj,2}, * € I-

Let A = {(*,A)} € £ ( / , / , 0). Then £ |F(A) - /(t)|A|| < £ \HA) ~ G{A)\+ A

A

+ Y,\G{A)-git)\A\\ + Y,\9(t)-f(t)U\A

A

The first term on the right hand side does not exceed 2 _ J - 2 (cf. (5.20)), the second does not exceed 2~}~1 (cf. (5.22)) and the third does not exceed 2 ~ J - 2 (cf. (5.21)). Hence

Y,\F{A)-f(t)\A\\<2-\ A

Putting S(j,t) = min{Ci(t), (2(t),..., obtain that 6 £ D* and that

£(*)} for j G N, t£ I we

£|F(A)-/(*)LA||<2-'" A

f a r i G N , A = {(*,A)}€£(/,I,«(i,-))Therefore (5.19) and (5.7) hold. (5.6) is a consequence of (5.3) and (5.19). The proof is complete. If y G Y then ULc(Qy)

is tolerant to Qy (cf. Lemma 3.5)

so that F, ^ » F 0 implies that Fj -»• F 0 in ( P y , Z 4 c ( Q y ) ) . Therefore Ft ^ F0 (cf. (5.5), (5.2)) implies ||F, - F 0 | | v a r -> 0. The next example and proposition demonstrate that condition Fi — • Fo is stronger than condition \\Fi — Fo||Var —>• 0.

62

INTEGRATION BETWEEN L AND H - K

5.8 E x a m p l e . For m € N put

JT = [ — , ~ ] , fc = l,2,...,m, m

m

/ - 1 ifc-1 / , , , m r]fc- 1 V, , = [ + —T, + - r , fc, 7 = 1 , 2 , . . . , m m mz m ml so that m

Am = U V T" * = l,2,...,m. Put

^r,...,/m = U Ki

k

k-1

where U 6 { 1 , 2 , . . . , m } for z = 1, 2 , . . . , m and let xT l '• m I —> R be the characteristic function of W, , . Functions Y?" , are called functions of order m. P u t m

ai(m) = ^ n " , n € N. n=l

(5.23) Let / i = x{ a n d for i = ui{m — 1) + 1 , . . . ,cu(m) let / ; be all functions xT I °^ o r d e r m, m — 2, 3 , . .. . Let Fi be the primitive of / „ i £ N. Observe that (5.24)

ft(t)e

{0,1} for

ieN,teI,

l i m s u p / i ( i ) = l,liminf/,•(<) = 0, t £ I,

(5.25)

(Leb) / Jo

fidt = m

if ft = xru...,im,me

N,

^-INTEGRATION

(5.26)

63

H^llsup -> 0, ||Fi|| v a r -> 0 for i -> oo.

Finally (since WC C y), (5.27)

G, ^

Go implies that G> ^

G0

and (cf. Definition 2.14 and (5.26)) (5.28)

if there exists FQ 6 P-HK such that F, ^

F0 then F0 = 0.

P r o p o s i t i o n . The sequence F{, i £ N is not convergent 0 in QnK. Proof. Assume that (5.29)

Ft ^

0.

Then there exist 8 € D* and M; G jV, i G N such that (5.30)

^|Fl(K)-/l(t)|K||<2^ A

for i,j G N , A = { ( * , # ) } € For e € R+ put

WC(I,I\Mi,6(j,-)).

E ( e ) = { < € / ; 6 ( M ) > £}• Since 6(1, •) : I -»• R+ there exists e+ € R+ such that (5.31)

\E(e+)\

> \.

Let m € N, m > 4, ^ < £+. P u t Z ( m ) = {fc; 1 < Jfc < m, \E(e+) n J f | > 0}.

to

INTEGRATION BETWEEN L AND H - K

64

Then (5.31) implies that (5.32)

#Z(m) >

For every k € Z(m) that

^ .

there exists l{k) G N, 1 < /(A;) < m such

inmwn^(m)|>o. Let T

k € Ffcm,(fc) n £ ( m ) \ M for Ar G Z ( m ) ,

where M = | J - e N M t . Then (5.33)

A = {(r fc , J™); k G Z ( m ) } G W C ( / , I \ M, 6(1, •))

since Tk G £ ( m ) , 5(1, n) > e+ > ^ , rfc G Jfcm, | J f | = £ for jfc G Z ( m ) . P u t l(k) = 1 for & £ Z ( m ) . There exists p G N, u;(m — 1) 0 for J G lv([0,l]). Therefore (5.32) and (5.34) imply that

(5.35)

Fp([0,l})>

J2

F

P(JH>

keZ(m)

> 3m ~~ 4

1 m

1 > _1 2 _ 4

On the other hand, Fp([0,1]) = (Leb) / fp(t)dt Jo

= z] T \Vkmvl{k)\ = m— = —' ' ' m* m

which contradicts (5.35) since m > 4. The proposition is valid.

^-INTEGRATION

65

5.9 Theorem. Let Ft G Pc for i G N, M G J\f. Assume that (5.36)

(5.37)

H.Fillvar-0,

Fi(t)-*0

for

t£l\M.

Then (5.38)

Ft - ^ 0.

Proof. Let r G N, |J| < 2 r , £(/, fc) = { t e / \ M ; |F,(i)| < 2~'-r-3

for i > A;}, k, I G N.

Then E(l,k + 1) D E(l,k) \jE(l,k)

for Jfe,/eN,

=

I\M.

For / e N there exists k(l) G N such that k(l)>l,

\I\E(l,k(l))\

<2~\

\\Fi\\v„ < 2 - ' - 3 for i > k(l). Put 5(/) = P | E(m,k(m)),

I G N, 5(0) = 0.

Then 5(/ + 1 ) D 5 ( 0 ,

leN,

|J\5(/)|<2-' + 1 , |J\1J 5(01 = 0.

INTEGRATION BETWEEN L AND H - K

66

Moreover, S(l)cE(l,k(l)) so that (5.39)

\Fi(t)\ < 2~'-r-3

for

t G S(l),i > k(l),l G N

and (5.40)

HFiUvar < 2 " ' ~ 3 for i>Jfe(/),/eN.

Define

(5.41) V

;

{ Fl(t) for * e I \ M 0 , /,(*)=< W [ 0 for teM0,i G N,

where Mo = M U ( J \ U J G N 5 ( 0 ) Since i^ G Pc there exists Si G F)* such that (5.42)

£|fl(A)-/,-(<)|A||<2-'" A

for i, j G N,_A = {(*, A)} G C(IJMh-))There exists 9 G D* such that (5.43) (5.44)

0(j,*) < mm{Si(j,t); 6(j -2,t)< l,jeN,l>j,

i < k(j)}

mm{6i(j,t);

if t G I, j G N,

i < k(l)} if teS(l)\S(l-l).

Let j € N , A = {(t, A)} € £ ( / , / , tf(i,-))If z < &(j) then (5.45)

J]|FJ(A)-/l(t)|A||<2^

£-INTEGRATION

67

by (5.42) and (5.43). If k(l — 1) j , then

(5.46) J^\Ft(A)-Mt)\A\\< A

+

£

( l ^ ) l + l/*WPI)+

A,teS(l)

£

\Fi(A) - Mi)\A\\ + £

A,tel\S(l)uM0

1^(^)1-

A,t€M 0

The first term on the right hand side does not exceed 2 ~ ' _ 1 (cf. (5.39) and (5.40)), the second does not exceed 2 ~ J _ 2 (cf. (5.44) and (5.42)) and the third does not exceed 2 ~ ' - 2 (cf. (5.40) and (5.41)). Hence

Yt\Fi(A)-Mt)\A\\<2-' for i,j e N, A = {(t,A)} (5.46)).

€ C(1, 1,8(j, •)) (cf.

(5.45) and

Moreover, fi(t) —> 0 for t € / , zi —> oo so that F 2 - = ^ 0. The proof is complete. 5.10 R e m a r k . Let Fi e Pc, Yl'Zi ll^llvar < oo. Then there exists a sequence /% £ R + , i G N such that fli —> 0 and oo

..

E^HF'l|var<00. Since Pi

it can be deduced in a standard way that fi(t) —> 0 a.e. Therefore Ft - ^ 0.

INTEGRATION BETWEEN L AND H - K

68

5.11 R e m a r k . Let F,-, t ' e N b e defined by (5.23), F0{J) = 0 for J G Iv, y G Y. Then (5.47)

Fi-Jb

in (P;y,WLc(Qy))

but (5.48)

Fj is not convergent to FQ in

Qy.

(5.47) holds by (5.26) and (3.33) since y C C while (5.48) is a consequence of (5.26), the proposition and (3.32) since y C TCIC. 5.12 T h e o r e m . Let Ft G PC for i G N. (5.49)

Then

i^-^Fo

holds if and only if (5.50)

||^-Fo||var^0

and (5.51) t i e r e exists M G M such that Fi(t) -> F0(t) for t G I\M, i -»• oo. Proo/. Let (5.49) hold. Then Z 4 c ( Q £ ) is tolerant to Ec (cf. Corollary 3.9 and Lemma 3.5) and is induced by || • ||£ (cf. Theorem 5.2), which implies (5.50). Moreover, (5.51) holds by Lemma 2.3. On the other hand, (5.50) and (5.51) imply (5.49) (cf. Theorem 5.9). The proof is complete.

6

M-INTEGRATION

6.1 N o t a t i o n . M.-integration is ^-integration flavoured by 7Y/C-integration. In this section S C I is countable, closed and nonempty, {Lk; k G K} where K = {1,2, . . . , & * } or K = N is the set of components of I \ S. For t £ / \ S there exists a unique k(t) G K such that t G Lk{t). For k G K let n(k) G N fulfil -n(*)+i < diamL . 2 fc Put 7y(0=-dist(i,5), < € / , Jk,n(k)-l

= 0' & € K,

Jfc;J = { 2 _ i } , i £ K , i >

n(k),

\\F\\°k)i = varF| Jjkil . for F G AdC, fc G K,i > n(k). 6.2 D e f i n i t i o n . M is the set of A = {(*, A)} G £ fulfilling (6.1) (6.2)

if t G 5 then t G A G Iv, if * G I\S

then t G A G Iv or A C

Lk(t).

Since S is countable and nonempty we may write S = {sf, I G L} where L = { 1 , 2 , . . . , / * } or L = N. Obviously M. G Y. The next theorem is the main result of this chapter. 69

70

INTEGRATION BETWEEN L AND H - K

6.3 T h e o r e m . (6.3) PM is the set of F G AdC such that F\Jki

G AdAC( Jk,i)

for keK^>

(6.4) ULC(Q.M) is the topology the set of seminomas

on PM

II • llsup, || • ||°,i, kGK,i> (6.5) (PMMLC(QM))

is

n(k), which is induced

by

n(k),

complete.

The proof will be performed in a series of steps. An obstacle we meet is the fact that the results of Chapter 4 cannot be applied directly since ||.F||./v<,s = °° for some F G PM a n d s G / (cf. (6.2)). Therefore M* G Y is introduced such that M*integration is A^-integration and the results of Chapter 4 can be applied in the case of A4 ""-integration. However, rj plays an auxiliary role in Definition 6.4 and the concept of M.*-integration does not change if it is replaced e.g. by rj2. On the other hand, the dependence of A'f-integration on S is essential. 6.4 D e f i n i t i o n . M* is the set of A = {(t, A)} G C fulfilling (6.1) and (6.6)

if t e I \ S

then t G A G Iv or

A e Fig, A C[t-ri(t),t Obviously M* eY,M*

+ r,(t)].

CM.

6.5 L e m m a . (6.7)

QM{6)

C QM-(6)

for 6 G D* there exists

for

6 G

D*,

0 G D* such that

A1-INTEGRATION (6.8)

71

QM*(6)CQM(0),

(6.9)

PM=PM-,

(6.10)

®M=QM*,

(6.11)

ULC(QM)

=

ULC(QM-)-

Proof. (6.7) holds since M* C M. Define 0 : N x I - • R+ by 0( j , i) = min{77(i), S(j, t)} 0(j,t)

= 6(j,t)

for < G I \ S,

for < e 5 .

Then M{I,E,eU,-))cM'{I,E,6{j,-))

for

j€N,Ed

and (6.8) holds. (6.9) and (6.10) follow from (6.7), (6.8) and Theorem 2.9. (6.11) is a consequence of (6.9), (6.10) . The proof is complete. 6.6. Observe that (cf. Section 4.1) (6.12) H-Fllwc.t = sup{|F([c,*])| + \F([t,d})\;

a
and

(6.i3)

\\F\\
Specializations of concepts from Section 4.1 are the object of the next two lemmas.

72

INTEGRATION BETWEEN L AND H - K

6.7 L e m m a . (6.14) \\F\\M*,3 = \\F\\wc,s forseS,Fe (6.15) \\F\\M*,t = max{\\F\\nK,t,\\F\\°ki;

AdC, k G K,i > n(k)} for

teI\S. 6.8 L e m m a . (6.16) TM* is the set of F G AdC such that \\F\\0ktt < oo for k G K, i > n(k). (6.17) TM* is the topology on TM* induced by the set of seminorms || • || s u p , || • \\°ki, k G K, i > n(k). 6.9 L e m m a . PM* is the set of F G AdC fulhlling

(6.18)

F\Jki € AdAC(Jfc)J) for keK,i>

n(k).

Proof. Let F G PM*- Then F G AdC since UK C M* so that PM* C P-HK C AdC. Let / G I \ S. By Definition 6.4 (cf. also Definition 1.3 and Theorem 1.4) we have -F|i(t) <= Pc(L(t),L(t),oo) where L{t) = [t - &(*),*+ h(')]Moreover, Pc C AdAC (cf. (5.3)) and (6.18) holds since Jkti is covered by a finite set of intervals [t — \r]{t), t + \f}{t)}Let now F G AdC and let (6.18) hold. Then (cf. (5.6)) F\jk,i £ £(Jk,i, Jk,i, oo) for i e K , ! >

n(k).

There exist / : I —> R and 6fc,i £ -D* such that f(s) s G 5 and

= 0 for

Xy(A)-/(t)wi<2-'A

for j G N, A = {(t,A)}

G C(Jkji,Jkti,6k,i(j,-)),

k G K, i > ra(fc).

73

^-INTEGRATION

Let j e N and let Cj : I -> K+ fulfil (6.19) if / £ L then Cj(si)

1S s o

small that

|F([c,5,])| + | J F ( [ 5 ^ ] ) | < 2 - ^ 1 - ' for si - Cj(si) i+1(j + 1 + z + k, t) and [< - (j(t),i + 0 ( 0 ] C J M + i , keK,i> n(k). Now let (6.21)

e =

{(T,M)}eM*(I,I,(i).

Denote 0', = { ( T , M ) } £ 0 ; r =

S,},

0'/fc = {(r,M)} £ 0 ; r £ J M \ Jk,i-i},

/ £ L,

k£K,i>

n(k).

Then (6.22)

Yl \F(M) - f(r)\M\\ < £ ] T \F(M) - / ( r ) | M | | + e ieh e;

+ ££|F(M)-/(r)|M||. (6.19) implies that £

|F(M) - / ( r ) | M | | < 2 - ^ 1 - ' , / G L.

©I (6.20) implies that

J2 \F(M) - /(r)|M|| < 2 --'- i -*- 1 , A: G K,i > n(fc).

74

INTEGRATION BETWEEN L AND H - K

Define 6 : N x i" -> R+ by 0(j\t)==rnin{Ci(t),...,(,(*)}• Then ^ f l * and ]T|F(M)-/(r)|M||<2-' 0 for j € N, 0 = {(T,M)} T h e proof is complete.

G A T ( I , J, #(.?,•)) so that F € P ^ . .

6.10 L e m m a . TM* \PM* is tolerant to QM* • Proof. Let Fi -^-> FQ. Then there exists 8 G D* such that Fm € QM*(S) for m = 0 , 1 , 2 , . . . and \\Fi - F0\\sup -* 0. Let s 6 J \ S, Vs — [s — ^rj(s),s + ^rj(s)]. Then i^lv, is convergent to F0\ys in Qc(vs,v.,oo) a n 0 for fc € K, i —»• oo. Moreover, \\F{ — i*o||sup —* 0. Lemma 6.10 is a consequence of (6.17). T h e proof is complete. 6.11 L e m m a . ULC(®M*)

= 1~M*\PM* •

Proof. Theorem 4.2 will be applied. (4.3) holds by Lemmas 6.8 and 6.9, (4.4) holds by Lemma 6.10, (4.5) and (4.6) are consequences of Lemmas 6.9, 6.7 and of (6.13). Therefore Lemma 6.11 holds by Theorem 4.2. T h e proof is complete. 6.12 L e m m a . (PM* ,MLC(Q.M*)

iS

complete.

Proof. Since the set of seminorms (6.17) is countable and Lemma 6.11 is valid it is sufficient to prove t h a t for every sequence Fj € PM* J j € N which is Cauchy with respect to each of the seminorms (6.17) there exists Fo G PM* such t h a t ||*> - ^ollsup - 0, | | F , - Fo\\0kti "> °

for

3 - oo.

A4-INTEGRATION

75

Since Fj is Cauchy with respect to || • || s u p and PM* C AdC there exists Fo £ AdC such that (6.23)

H^-.follsup-O.

Fj\jki is Cauchy with respect to || • \\\ i, which together with (6.23) implies that F0\jkii C AdAC(Jk,i), k € K and i > n{k) and v a r ( F j | J t i i - F0\jki) -»• 0 for j -> oo. Hence FQ E PM* (cf- Lemma 6.9) and

Fj^FQ

in

(PM-,ULC(QM*)).

T h e proof is complete. Proof of Theorem 6.3. Theorem 6.3 follows by (6.9), Lemma 6.9, (6.11), (6.17) and Lemmas 6.11 and 6.12.

7

NONCOMPLETENESS

7.1 A restriction o n y. Denote by \\A\\ the number of boundary points of a figure A so that | | | A | | is the number of components of A. For s el, a £ R+, A = {(s, A)} £ y(I, {s}, oo) put (7.1)

A ^ a ) = {(s,A)

£ A; |A D (7 \ [s - 0},

A 2 (s,
and such that

IML||
Ai(«,ir)

for 5 G 7, a G R+, A = {(s, A)} G y(I, {s}, oo). (7.3) Py C Ty, (7.4) 7j; is tolerant to Qy. Proof. Let F £ Py. Qy(0). Therefore (7.5)

There exists 9 £ D* such that 7 1 G

£|F(A)|<2-> + £

| A |

forj€N,e = {(<,A)}ey(/,/,»(i,-))76

NONCOMPLETENESS

Let s G I, A = {(s, A)} G y(I, {s}, oo). By (7.5) we have (7.6)

£

| ^ ) | <

2

-

+

^_

E M

<

2

-

1

+

Moreover (cf. (7.1)),

E

I*WI*

A^a.^M))

E

2IMiiimU<

^(.,((1,.)) "

<^(0(l,s))\\F\\supi which together with (7.6) implies that my s

'

~ 2_1 + »(M)

+

^W' 3 ))ll F ll"p-

(7.3) holds. Let F - ^ F 0 . Then there exists 9 G D* such that (7.7)

Fi - F 0 € Qy(0) for i = 1,2,...

(cf. Lemma 2.13) and | | F ~ F0\\SUp -* 0 for i -> oo. Let 5 G F For j G N put

(7.8)

^ =2-''- 1 6>(.7», f ^i for t = 3,

INTEGRATION BETWEEN L AND H - K

78

Obviously 6 G D*. Let

i,i€N, A =

{(s,A)}ey(iAs},s(j,-)).

Then

Y,

\Fi(A) - F0(A)\ < ^(a.OII^-FolU

Al(s,(Tj)

and (cf. (7.5))

J2

\Fi(A) - F0(A)\ < J2 \F*(A)\ + E

A2(s,(Tj)

A2(s,(Tj)

< 2- 2 ^ + 2 — ^ — ~ 0(;,s)

V ^^

l^(A)|<

A2(s,(Tj)

|A| < 2 - J + 1 + 2 - J + 1 = 2 " J + 2 .

Hence

£|*KA)-F0(A)| = A

= J2 \Fi(A) - F0(A)\ + £ Al(s,(Tj)

MA) - F0(A)\ <

A2(s,(Ti)

< 2 - J + 2 + ^(aJ)||Fi-JFo||sup, which implies that \\Fi - F0\\y!S

-»• 0 for i - • oo.

Therefore

F 2 ^ F 0 in (r^Ty) and (7.4) holds. The proof is complete. The main result of this chapter is

NONCOMPLETENESS

7.3 T h e o r e m . Assume

that (7.2) holds.

79

Then

(7.9) there exist Hj G Pc for j G N, H G Ty \ P-HK such that

J2H>-+H in ( T y , ^ ) for i —> oo. The proof will be performed in several steps. 7.4 L e m m a . Let (7.2) hold, F G AdC, s G J, a G R + . Then (7.10)

||F||y, s < v a r / n ^ - * , ^ ] ^ + ^ ( 0 1 1 * 1 1 sup.

Proo/. Let A = {(s,A)}

G ^ ( 7 , {s}, oo). Then

E i F ( A ) i = E iF(A)i+ E A

A2(S,CT)

i F ( A )i^

AI(S,
< vaxIn[a_a+0.]F+

2K(a)\\F\Uup-

(7.10) holds and the proof is complete. 7.5 N o t a t i o n a n d s o m e o b s e r v a t i o n s . Let S : R —> [0,1] be periodic with period 2, S(t) = \t\ for te

[-1,1],

let e2 € R + , ft G N for i G N, (7.11)

^e,
Put Gi(t) = eiS(qit)

for t G R, i G N,

80

INTEGRATION BETWEEN L AND H - K oo

G(t) = Y^Gt{t) for

teR.

G is well defined. There exist additive functions Hi : Fig —> R, H : Fig such that Hi([c,d\)

= Gi(d) - G,-(c) for [c,rf] e I v , i € N,

#([c,<*]) = G ( d ) - G ( c ) for

M]elv

(cf. Note 1.5). Hi and i ? are unique. Obviously (7.12)

H% G Pc, \\Ht\\sup < et for i G N, # G AdC .

Since gj£i is the Lipschitz constant of Gi we have (7.13)

Hi([c, d}) < etqi(d - c) for i G N, [c, d] G Iv .

7.6 L e m m a . Let K : R + —»• R + be nonincreasing. there exist e; G R + , & G N, a,- G M + for i G N such that holds and - . 2 — 1

(7.14)

£l

- -^qjSj

OO

1

- J2 £j > ^i,

Qi

l e N

s-i

(7.15)

qid —>• oo for i —> oo,

(7.16)

e,-(2<Tigi + « ( a 0 ) < 4.9\
(7.17)

£V-<7.-
'

NONCOMPLETENESS

81

Proof. There exist <7; (E R+ and qi £ N such that (7.17) holds and (7.18)

ql+1 > 4.9.g 2 , qx > - ^ - for i G N.

Put 9* (7.19)

et = - .

Now (7.15) is obvious and (cf. (7.18)) (7.20)

^

= 9-^- < -

£i

qi+i

4

so that (7.11) holds. Moreover (cf. (7.19), (7.20)), 1

i _ 1

°°

3=1

9*

1

J=i+1

y_

&_i

9^ i

- ^ ~ ^ 8 ~ ^

(

19!

4

i _ 1

°°

3=1

]=i+l

_i_ _i_ +

1

4^ + 4^ .

+

"-)-

M

and (7.14) is fulfilled. Finally, Si(2aiqi + K.((Ji)) < 3cneiqi = 3 • 9Vi and (7.16) holds. The proof is complete.

£

INTEGRATION BETWEEN L AND H - K

82

7.7 L e m m a . Let sequences (7.14) and (7.15). Then (7.21) the derivative

Si and qi, ! 6 N fulfil (7.11),

H(t) does not exist at any t G inti".

Proof. Let t G i n t / . Since qi —>• oo for i —> oo (cf. (7.11) and (7.15)) there exists m G N such that 1

1

,t + - C / for i > m

< Qi

Qi

and there exist k{ G Z such that t 6

—, ——— .Qi

= Ki C I for i > m.

Q

Let i > m. Then \KA — —,

(7.22) -^±

= qiYiHi(Ki)

+ qiRi{Ki) + qi J ] ^(J^)-

By the definition of Hi and by (7.12) and (7.13) we get \Hi(Ki)\

= £,, | # , ( / i t ) | < —

^

J < i,

Qi

\Hj(Ki)\

< Sj for j > i.

(7.22) and (7.14) imply that Ki)\ l#TOI

^

V^

V^

i=i

j=»+i

Therefore (cf. (7.15)) \H(K, \Ki

oo for i —> oo.

(7.21) is valid and the proof is complete.

v, 1

NONCOMPLETENESS

7.8 L e m m a . Let (7.2), (7.11), (7.16) and (7.17) hold. (7.23)

83

Then

H € Ty

and i

J2H^H

(7.24)

Proof. Let s£l,jeN.

in

(Ty>Ty)-

If

A =

{(s,A)}ey(i,{s},^)

then (cf. (7.10), (7.13), (7.12), (7.16)) E

\HJ(A)\

^

var

/n[ s - < 7 j , s +
A

< ejqj2<jj + K(aj)ej

< 4.9-Vj,

^^(A)|=^|J]^(A)|< A

A j€N

< E E i^(^)i <4^>v jGN A

j€N

Therefore (cf. (7.17))

ii-^iiy,* < ^ E ^ 7 " 7 - ? < ° ° and (7.23) holds. In an analogous manner we conclude that i

oo

||]T^-tfb, s <4 ]T 9V„ which implies that (7.24) holds. T h e proof is complete. Proof of Theorem 7.S. Theorem 7.3 is a direct consequence of Lemmas 7.6, 7.7 and 7.8.

INTEGRATION BETWEEN L AND H - K

84

7.9 L e m m a . Assume (7.25)

that (7.2), (4.5) and (4.6) hold.

ULC(Qy)

=

Then

Ty\Py

Proof. (4.3) and (4.4) hold by Lemma 7.2 so that (7.25) is true by Theorem 4.2. The proof is complete. 7.10 T h e o r e m . Assume that (7.2), (4.5) and (4.6) hold. Let P* be a vector space equipped with a topology U* such that (7.26)

(P*,U*)

is a locally convex

(7.27)

space,

Py C P* C P-HK,

(7.28)

W\Py

CULC(Qy),

(7.29)

ULC(.QHK)\P-CU*.

Then (7.30)

{P*,W)

is not

Corollary. (Py,ULc(Qy))

complete.

is not

complete.

Proof. By Theorem 7.3 there exist a sequence Hj £ Pc, j € N and H G Ty \ PnK such that 2

(7.31)

J2HJ^H

in

(Ty'Ty)

for

i^oc.

NONCOMPLETENESS

85

By (7.25), (7.27) and (7.28) the sequence £ J = 1 Hj is Cauchy in (P*,W*). Assume that (7.32)

(P*,U*)

is complete.

Then there exists G £ -P* such that i

(7.33)

^Hj^G i=i

in (P\W)

for i - > oo.

(7.29) implies that i

|| y

Hj — G|| s u p —> 0 for i —> oo,

|| \

Hj — H||sup

but —> 0 for z —> oo

by (7.31) and (7.29) so that G = H € AdC \P-HK which contradicts (7.33). Therefore (7.30) holds and the proof is complete.

8

^-INTEGRATION

8.1 P r e l i m i n a r i e s . A summation condition is crucial in the definition of ^-integration. Let A be the set of A : [0, oo) —> [0, oo), nondecreasing, A(0) = 0, A(cr) > 0 for a > 0. D e f i n i t i o n . For A G A let <S(A) be the set of A = {(f, A)} G C such that (8.1)

(8.2)

if (t,A)

G A then A G Iv,

J^A(dist(*,A)) < 1. A

Notes, (i) Obviously 5(A) G Y for A G A. (ii) If A(CT) > 1 for a > 0 then <S(A) = HK. 8.2 T h e o r e m . Let A G A. T i e n (8.3)

(-PS(A),WLC(Q5(A)))

Corollary. {PUKMLC{Q.HK))

J S

'

not

complete.

is not complete (cf. Note 8.1

8.3 T h e o r e m . Let A G A, y = S(\). (4.6) hold.

Then (7.2), (4.5) and

Theorems 8.2 and 8.3 will be proved in several steps. 86

S-INTEGRATION

8.4 L e m m a . Let A G A, s G / , a G R+. (8.4)

A =

87

For

{(s,J)}6S(\)(I,{s},<x>)

put (8.5)

A a ( ^ a ) = {(s, J) G A; \J n (I \ [s - a,s + a])\ > 0}.

Then

(8.6)

^

l|J||<4 +

2|/| A(<J)

Proof. There are at most 2 couples (s, J i ) , (5, J2) such t h a t I Ji D [s~a,s

+ a]\ > 0,\Jif](Ji

\ [s -<7,s + cr])| > 0,i = 1,2

while for the rest we have dist(s, J ) > A(
#A 1 (,,a)<2+-Sand (8.6) holds since \\J\\ = 2 for J G Iv. The proof is complete. 8.5 L e m m a . Let A G A, F G -P 5 (A), C G [a, 6), A' = [c,b], s e l . Then (8.7) (8.8)

TKF

G P5(A),

| | r ^ F | | 5 ( A ) > s < 3||F|| 5 ( A ) > a .

Proof. There is nothing to prove if c = a. Assume t h a t a < c R and 6 G D* such that (8.9)

£|F(J)-/(i)|J||<2-'

88

INTEGRATION BETWEEN L AND H - K

for j G N, A = {(*, J)} G 5(A)(J, J, *(j, •))• Without loss of generality we may assume that (8.10)

/(c) = 0, 6(j,t) <\t-c\

for t ^ c,j G N.

Put 9

^~

=

/

m

\

f

°r

C < t

~

b

'

0 for a < t < c

and observe that

TKF

W

= [

o if | * n J |

= o.

Let j G N, 6 = {(i, J)} G <S(A)(J, I, % , • ) ) . There exists at most one (c, J ) G A such that \[a, c] fl J\ > 0 and there exists at most one (c, L) G A such that |[c, 6] f\L\ > 0. Put Si = {(c, [a,c] fl J)} if |[a,c] fl J | > 0 for some (c, J ) G A, Si = 0 otherwise, S 2 = {(c, [c, 6] n L)} if |[c, 6] fl L\ > 0 for some (c, I ) G A, S2 = 0 otherwise, 0 i = { ( * , J ) G 0 ; *G [a,c)}USi, 0 2 = { ( i , J ) G 0 ; < € ( c , 6]} U S 2 . Then e1ue2G5(A)(/,J,5(j,-)). Therefore ^|rjrF(J)-^(t)|J||< 0

J] ©iue 2

|rK-F(J)-^)|J||<

5-INTEGRATION

89

< Y \TKF(J) - g(t)\J\\ + E FKF(J) - g(t)\J\\ = 0!

02

Y\F(J)-f(t)\J\\<^3i 02

which implies that g is <S(A)-integrable and T^-F is its primitive, i.e. (8.7) holds. Let s € / , A = {(s,J)} € S(X)(I, {s}, oo). If s e A' put A i = {(s, X); (s, J ) G A, A = K n J, |X| > 0}. Then A

Ai

^A(i:)<J]A(|j|)
A

so that Ax = {(s, A)} e S(\)(I, (8.11)

{s}, oo) and

^|rA-F(J)|<||F||5(A)),. A

If 5 £ [a, c) put A 2 = { ( * , J ) € A;

JCK}.

If there exists (5, [ai,&i]) £ A such that put

a < a i < c < 6 i < 6

A 3 = {(a,L); A = [s,&i]}, A 4 = {(s,L);

L=[s,c]}

and put

A 3 = 0, A 4 = 0 otherwise. Then

(8.12) Y FKF(J)\ < Y \F(J)\ + E 1^(^)1 + E 1^(^)1 ^ A

A2

A3

< 3||i r ||5(A),s and (8.8) holds. The proof is complete. In analogy with Lemma 8.5 we have

A4

90

INTEGRATION BETWEEN L AND H - K

8.6 L e m m a . Let A G A, F G Ps(\)> d £ (Gi&]; -ft" = [a;<^]? 3 G -ft Then (8-13)

rA-FGP5(A),

(8.14)

| | r A - F | | 5 ( A ) i S < 3||F|| 5 ( A ) ) S .

8.7 L e m m a . Let A G A, ft1 G PS(A)> -ft' = [ c ^] C I, s <E I. Then (8-15)

r A - F G P,s(A),

(8.16)

||rAF||5(A)>s < 9||F||5(A)ii.

Lemma 8.7 is a consequence of Lemmas 8.5 and 8.6 since T/cft1 = r[ a ; d ](r[ C j 6 ]i r ). Proof of Theorem 8.3. Theorem 8.3 follows from Lemmas 8.4 and 8.7. Proof of Theorem 8.2. Theorem 8.2 is a consequence of theorems 8.3 and Corollary 7.10. 8.8 R e m a r k . For a G R+ put £(a) = {A = {(t, J)} G £; J G Iv, dist(*, J ) < a\ J\}. The problem which was put at the end of Section 9.2, [K 2000] can be given in the following form: Is (P£(a)MLc(Q.£(a)))

complete?

Put X(a) = aa for a G R+. Then obviously £(a) C

S(\).

5-INTEGRATION

91

By Lemma 3.15 we have Ps(\) C Pc(a)i ULc(®£ia))\ps(x) ULc(®n>c)\pe{a)

CULC(QS(x)), cULC(Q£(a))

so that Theorem 7.9 may be applied with 3^ = S(\). P* = P£(a), W = ULC(Q£(a)) (cf. Theorem 8.3). Therefore (Ps(a),^Lc{Q.e(a))) is n ° t complete. Theorem 8.2 plays a fundamental role in the proofs of various noncompleteness results. The dependence of Ps(\) a n d on ULC(^S(\)) ^ is described in the next three theorems (cf. Corollary 3.9). 8.9 T h e o r e m . Let A , f i G A , (8.17)

0 < l i m i n f - ^ f , l i m s u p - ^ r < °°-

Then (8-18)

PS(\) = Ps(n)i

(8-19)

E5(A) = E 5 W ,

(8.20)

ULC(Es(x})=ULc(Es(tl)).

8.10 T h e o r e m . Let G0 G AdC, G0(J) A,// e A, ak,Pk elforkeN, (8.21)

= 0 for J € Iv,

b > /?i > oil > fi2 > a2 > • • • > a, lim f3k = a,

92

(8.22)

INTEGRATION BETWEEN L AND H - K

^ - > 2

k

for ^ N , ^

[ak,pk],

$a) -> 0 for a - • 0. Then there exist F m € AdC for m = 0 , 1 , 2 , . . . such that

(8.23) F ( t ) exists for t <= I, i e N, F 0 ( t ) exists for t £ (a, 6]

(8.24)

JPO is continuous

on (a, 6],

(8.25)

Fi is continuous

for i £ N,

(8.26)

Fo£P5(A)\^s(,o,

IF

(8.27)

(8.28)

^

^

Go,

| | F | | 5 ( / i ) j 0 ->• oo for i -> oo,

(8.29) ^ does not converge to G in (Ps(ll),ULc(^s(fi))) f°r any G £ PsM, (8.30) Fi does not converge to G in E s ^ ) for any G £ Ps(/i)Note. Fi does not converge to G in Qs(ii)) f ° r a n y G £ Psbt) since ULC(QS(II)) = ULC{^S(H)) °y Corollary 3.9 and 1S ULC(QS(»)) tolerant to (^(^-convergence.

S-INTEGRATION

93

8.11 T h e o r e m . Let A,/j £ A, <Xk,[h G I for k G N, p G (0, |J|). Let (8.21) and (8.22) hold and assume (8.31)

\(a)

> p(a)

Let us denote by Kg(llj\ps.x)

for a G [0,p\. the set of couples

((Gi,G2,Gs,.. such that Gm £ Ps($ f°r Then (8-32) (8.33)

. ) , G o ) G E5(/x)

m

= 0,1,2,... .

^S(M)

C

P 5 ( A ) \ Ps(ll)

p

s(\), + 0,

(8-34)

E%)CE5(A))

(8-35)

E5(A)|P5(/i)\E5(,)/0,

(8.36) (8.37)

ULC(ESM)

that

D

ULC{Es{x))\pSM,

ULC(Es(ll))\ULc(Esw)\pSM

^ 0.

Corollary. Let A,/i G A,

(8.38)

lim ^ 4 = °°-

Then (8.32) - (8.37) hold (since there exist am,/3m G I for m = 0 , 1 , 2 , . . . and p G (0, \I\) such that (8.21), (8.22) and (8.31) are fulfilled). Theorems 8.9, 8.10 and 8.11 will be proved in a series of steps.

94

INTEGRATION BETWEEN L AND H - K

8.12 L e m m a . Let A,// G A, n G N, p G R+. Assume that (8.39)

^ ( a ) < 2 n A(a), /i(a) < -

for a G [0,/o].

Then (8-40)

P5(/i) C P5(A),

(8.41)

E5(,t) C E 5 ( A ) .

PTOO/.

Let F G P 5 ( ^ ) . There exist / : I -»• R and 6 G D*

such that

£W)-/(0MI<2-'" A

for j G N, A = {(*, J ) } G S(fi)(I,I,6(j,-)) Let 0 : N x I -»• R+ be defined by (8.42)

(cf. Lemma 1.15).

^(j,<) = m i n { 5 ( ; + n + 1 , <),/»} •

Obviously 0 e D*. Let J G N , 0 = {(i,J)}G5(A)(I,7^(j,-))We have

J2 A(dist(<, J)) < 1 e and (cf. (8.39))

£>(dist(*,J))<2 B . © Since dist(£, J ) ) < /? for (i, J ) 6 0 we conclude (cf. (8.39)) that 0 can be divided into sets © i , ©25 • • • , ©g such that

5-INTEGRATION

f o r p = 1,2, ...,q-

95

1,

]T>(dist(<,J))
and g < 2 n + 1 (i.e. 0 = \Jqp=1 Qp, 0 p n 0 r = 0 for p ^ r). Therefore 0 P G
E l*V) - f(t)\J\\ < 2-i-n~\ p = 1,2,... ,q, ®P

E \F(J) - /wiJii < E E i*v) - /wii ^2_J' e

P

= i ©p

^ G Ps(\) and (8.40) holds. F

Let now Ft ^ that

F0. There exist fm

: I -»• E, 6 € D* such

El^m(J)-/mWI^II<2^' A

for j e N, A = {(*, J ) } 6 5 ( ^ ) ( / , J, 6(j, •)), m = 0 , 1 , 2 , . . . , (8.43)

fi(t)^fo(t)

for

i->oo,<eI.

Let j € N, let 9 be defined by (8.42) and e = {(*,J)}eS(A)(/,/,0(j,.)). As above, we can write 0 = (J»=i ®p where 0 p e 5 ( A i ) ( 7 , 7 , t f O ' + n + l,-)) for

p=l,2,...,q

96

INTEGRATION BETWEEN L AND H - K

and a < 2n+1.

We conclude that

Y, \Fm(J) ~ fm(t)\J\\
0

l F -( J ) - fm(t)\J\\ < ^

0„

for m = 0 , 1 , 2 , . . . which together with (8.43) implies that Fi

> FQ. (8.41) holds and the proof is complete.

Proof of Theorem 8.9. (8.17) implies that there exist p G IR"1" and n G N such that p(a) < 2nX(a),

\(a)

< 2n^(a),

\(a),p(a)

< 1

for a G [0,p]. Therefore (8.40), (8.41) and symmetrically -Ps(A) C Ps(ii), E 5 ( A ) C E 5 ( / i ) hold by Lemma 8.12, which implies that (8.18) and (8.19) are fulfilled. (8.20) is a consequence of (8.18) and (8.19) (cf. Theorem 3.7). The proof is complete. Proof of Theorem 8.10. There exist r(h) G N for h G N and CP G / for p G N such that

(8.44)

b>6 >6 >6 >

> a, lim £p = a, p—>oo

(8.45)

£r(2fc-l) = Pk, £r(2k) = ajt for k G N,

r(2k)

(8.46)

Pi-a<

Y, r(2Jfc-l)

Kit

a) < 2(ft - a)

5-INTEGRATION

97

for k G N. Denote by K the set of couples (k,p) G N 2 such that r ( 2 f c - l ) + l
r(2k).

There exists H : / —> K such that (8.47) H(t) = 0 for t G {a} U {£ p ;p G N} U U i t e N ^ + i , " * ]

U

F ( | ( e p + £ P - i ) ) = | A ( e p - a) for (fc,p) G K, H is increasing on [£p, ^(£ p + Cp-i)] a n d decreasing on [|(£ P + C p - i ) , £ P - i ] for (fc,p) € K, if is continuous on (a, 6]. Let us define (8.48)

Hi =

HXi

where Xi(t) = 1 for t G [ a 2 ; , A ] , Xi(0 = 0 for t G I \ [a 2 .,/?i], i G N, (8.49)

(8.50)

F 0 ( M ] ) = i 7 ( d ) - # ( c ) for

Fj([c, d]) = Hi(d) - Hi(c)

[c,d]elv,

for [c, d] G Iv, i G N.

(8.23), (8.24) and (8.25) are direct consequences of (8.47) (8.50). Moreover, (8.47) - (8.50) imply that there exists 9'e D* such that

(8.51)

J2 l F -( J ) " ^(01^11 < 2 _i ,

forjeN, Sl = {(t,J)}£C{I,{a,b],6(j,-)), Let /* G N, L = { 1 , 2 , . . . , / * } , q G N,

m = 0,1,2,....

0 = {(a,[c,,d,]; ZGL}G<S(A)(I,{a},/? g )

98

INTEGRATION BETWEEN L AND H - K

where (8.52)

a < ci* < di* < c/._i < d/»_i < • • • < cx < dx < Pq.

Since 0 G S(X) we have ^ A ( d i s t ( a , [ Q , ^ ] ) ) < 1, 0 i.e.

J ] A(c/-a) < 1. Let us introduce Lo - {/ G L; H(ci) = H(d{) = 0}, Li = {7 <E L; # ( d / ) > 0}, L2 = { / G L ;

fr(rf,)=0,fT(c,)>0}.

Since i?(i) > 0 for t G / we have (8.53)

L = L 0 U Li U L 2 , L, n L,- for i, j = 0,1,2, i ^ j .

For (k,p) G K let us put (8.54)

M M , p = {le

L i ; d, G ( e P , e P - i ) } ,

M 2)fc>p = {/ G L 2 ; c, G

(ZP,£P-i)}.

If / G Li then there exists a single couple (k,p) / £ M^jt.p so that

(8.55)

|J (k,p)eK

M1)k>p = U.

G K such that

S-INTEGRATION

99

Of course, the sets M j ^ p are pairwise disjoint for (k,p) G K. Let Co = flq; obviously Co > d\. If (k,p) G K and Mi;jt,p / 0, let us p u t i>{k,p) = minMi j f c ) P - 1 so that £ p < c^kiP) and A(f p ) < A(c^( fc)P )). Moreover, if / G Mljk,p then k > q (cf. (8.52) and (8.45)), di G (£p,£p_i) and there exists at most one / + G M i ^ p such that C/+ < £p. If no / + exists, then ]T

|F m ([c,,d,])| < v a r t f ^ ^ j = | A ( £ P - a)

so that (8.56)

\Fm({ci,dl})\
Yl

in every case. U(k*,p*),(k,p) 0 then

G K, (k*,p*) ? (k,p), tp(k*,p*)

MhktP ^ 0, M 1 ) f c ., p . ^

^ip(k,p).

By (8.55), (8.52) and (8.56) we get (8-57)

£|F

X

m

(h,^])|<

E

I^([Q,^)|<

(fc,p)enc,M1,fc,p^0 /eMi, fc , p

9

(fc,p)€K,M litiP ^0

< - 5 ^ A ( c , - a ) + A ( c 0 - a ) < - [ 1 + A(/?, - a ) ] , m = 0 , l , 2 , . . . . 9

i€L

9

INTEGRATION B E T W E E N L AND H - K

100

If / G L/2 then there exists a couple (k,p) G K such that M.2^,p = {/}. Hence

|J

M 2>fc , p =L 2 .

(k,p)€K

Moreover, if {/} = M 2)fc)P then H(dt) \\{ci - a) < ^ A ( Q - a), m = 0 , 1 , 2, Therefore (8.58) V

= 0 and |F m ([c/,d/])| <

|F m ([c,,d,])| < - V A ( c , - a ) < - , m = 0 , 1 , 2 , . . . .

JGL2

^ /€L

*

Since .F m ([c/,dj]) = 0 for / G Lo we conclude by (8.56) and (8.57) that (8.59)

X;|FTO([c/,d,])| = E IM[c/,*])l + E lF-([c'^'])l ^ 0

i€Li

/€L 2

< - ( 1 + \(/3q - a)), m = 0 , 1 , 2 , . . . . If j G N let q(j) G N be so large that (8.60)

^ ( i

+ AG^-a))^-'"-

1

and let 8 G D* satisfy (8.61) Kh a) < PqU) - «, &(j, t) < K3 + M ) for j G N, t G [a, 6]. Let j G N, A = {(*, J ) } G «S(A)( J, I , * ( j , •))• Let us put 0 = {(a, J ) G A } , O = {(*, J ) 6 A ; f e (a, 6]}.

101

5-INTEGRATION

P u t fm(a) = 0, fm{t) = Fm(t) for t € (a, 6], m = 0 , 1 , 2 , . . . . We conclude by (8.59), (8.60), (8.61) and (8.51) that (8.62)

£ |Fm(J)-/m(*)|J|| < J ] |Fm(J)| + J ] |Fm(J)-/ro(<)|J|| < A

0

<2-J~1

+2-3-1

Q

=2-\

m = 0,1,2....

(8.62) implies that (8.63)

Fm e Ps{x)

for m=

0,1,2,... .

Since (8.64)

Fit)

^ 0 for i -+ 00, i G I

we conclude from (8.63) and (8.64) that (8.27) holds. For i € N let K.(i) be the set of couples (k,p) £ K such that i < k < 2\ A," = {(a, [£„, ^ P + £ , - i ) ] ) ; (fc,p) €

K(i)}.

Since K(P

- a) ~

if ak < ip < pk (cf. (8.22)), we conclude from (8.45) and (8.46) that r(2k)

(8.65)

Me P -a)<2- fc+1 (/?!-a).

Yl p=r(2k-l)

+l

Therefore

J]/i(dist(a,[e p ,^(^ + ^-i)])) = A,-

102

INTEGRATION BETWEEN L AND H - K 2*'

r(2k)

=E

v^p ~a)-

E

fc=z p = r ( 2 f c - l ) + l 2'

< E 2 _ f c + V i <2-' + 2 /?i fc=t

so that A 8 G 5(/i) if 2 - l + 2 / ? i < 1. If i G N then # , ( t ) = H(t) for t G [a 2 .-,ft] by (8.48). If (fc,p) G K(i) then i < k < 2 \ r(2k) '+i) = <*2>, £r(2i-i) = A- Therefore

^ ( f o , ^(£P + tp-i)]) = Fi([£p, \{iv + &_!)]) = ^A(^ - a) for (k,p) G K(i), «' e N. Hence (cf. (8.47)) (8.66)

E lF°([^ ^ A,-

+ k-i)DI = E l F ^ ' ^

"

+ ^-01)1 =

A,-

=E

E

fc=j p = r ( 2 f c - l ) + l

^&-

2'

> E TJfe' A > /?i(iln2-lni), * £ N. (8.66) implies that (8.28) holds and

hmEiM^,;k£ P + £p-i)Di^^-

2—fOO * — '

'

"

2

A;

Hence F0 £ Ts(ll) and F 0 g P 5 ( / l ) since P 5 ( / t ) C T 5 ( / i ) .

5-INTEGRATION

103

Taking (8.63) into account we conclude that (8.26) is true. Observe that (cf. Corollary 3.9, Theorem 8.3 and Lemma 7.9) ULC{^S{»))

=ULC{QS(»))

=

1~S(H)\PSM-

(8.29) holds since \\Fi - G\\s(p),a

- • °o for i - * ° °

and Ts(p) is induced by the set of seminorms {|| • ||$( M ) )S ; s G / } . IS Moreover, ULC(^S(II)) tolerant to E s ^ - c o n v e r g e n c e , hence (8.30) holds. The proof is complete. Proof of Theorem 8.11. (8.31) implies that S(\)(I,I,p) C S(fi)(I,I,p) so that (8.32) and (8.34) hold by Lemma 3.15. (8.36) is a consequence of (8.32) and (8.34) by Theorem 3.7. Moreover, by assumption there exist a*, (5k G I such that (8.21) and (8.22) are fulfilled. (8.33), (8.35), are consequences of (8.26), (8.27) together with (8.29). Since ULC{^s{\)) is tolerant to Es^)-convergence (8.27) and (8.29) imply (8.37). The proof is complete.

9

^-INTEGRATION

9.1 P r e l i m i n a r i e s . A regularity condition is crucial in the definition of ^-integration. The regularity of a couple (t,A) £ I x Fig is defined by

91

f''

r

IAI ^ - > Mdi«m(WuA)<

A

=

Since ||A|| > 2 for A £ Fig, we have (9.2)

0 < reg(i, A) < -

for t £ I, A £ F i g .

Let £1 be the set of u : E + —• (0, | ] , nondecreasing. D e f i n i t i o n . For ui £ £1 let 72.(a;) be the set of partitions A £ £ such that (9.3)

reg(t, A) > w(diam({t} U A)).

Note. Obviously H{u) £ Y. 9.2 T h e o r e m . Let ua £ Q,. Then (9.4)

(P7J.(W),WLC('^-(W)))

is not

complete.

Theorem 9.2 is the main result of this chapter; it is a consequence of Lemma 9.4 and Theorem 7.10. 104

71-INTEGRATION

9.3 N o t a t i o n . For A = {(t,A)} (9.5)

eA

= {(t,J);

(t,A)e_

105

G C put A, J eComp

A}.

Obviously 0 A G C. 9.4 L e m m a . Let ( : I -> R+ 7 u G ft and iet A G A fulfil (9.6)

A(
j

^ for
Then (9.7)

0Ae<S(A)(/,£,C) for £ C I , A = {(*, A)} G ft(u,)(I,£,C),

(9-8)

Q5(A)Wcg%)(^)

(9-9)

P

(9.10)

Ft ^

(9.11)

5 (

A)CP

for

K M

tfeZT,

,

F 0 impiies that F,

^LC(Q^(U,))|^5(A)

C

Q

- ^ } F0,

ULC(QS(X))-

Note. It is obvious that for u G O, there exists A G A such that (9.6) is fulfilled. Proof. Let ( : f ^ (t,A) G A . Then

K+, A = {(t,A)}

G

K(u)(I,E,0,

" * < ' • A ) = pndJ^' W u^) - u(diam({(> u •4))-

106

INTEGRATION BETWEEN L AND

H-K

\A\ 2 # C o m p A = IIAII < 77-T— t . , ,. , , , v ^ " " _ diam({*} U A)tu(diam({£}U A)) Moreover, dist(t, J ) < diam({i} U A) for (t, A) G A, J G Comp A. Hence (9.12)

Y^

A(dist(<, J)) < # Comp A • A(diam({i} U A)) <

Comp /I

<

!4

x

~ 2 diam({t} U A)w(diam({t} U A)) 2diam({i} U A)u;(diam({t} U A)) _ |A| x

\i\

~W

Therefore (9.13)

^A(dist(*,J))= ©A

£

J ]

A(dist(t,J))<

((^JeACompA

- v w_ and (9.7) holds by (9.13) since ©A is E'-tagged and (-fine. Let 6 G £>*, F €
(,i4)

I'l Eiwi^-'+E^o

for j£N,$ = {(t, J)} G 5(A)(J, J, S(j, •)) and (9.15) there exist M <E Af and / : I -> R such that

J2\F(J)-f(t)\J\\<2-> for j G N, $ = {(t, J)} G S(A)( J, J \ M, % , •))•

^-INTEGRATION

If j € N, A = {(t,A)}

107

e n(u){I,I,8{j,-))

then

£ \F(A)\ < £ \F(J)\ < 2-* + £ J£L = 2- + £ J£L. A

0A

©A

If j € N, A = {(t,A)}

V

'

A

€ K(u>)(I,I\M,6(j,-))

u

'

;

then

£ IF(A) - f(t)\A\\ < 53 w ) - J W I I < 2^'A

0a

Therefore (9.8) holds. (9.9) is a consequence of (9.8) (cf. Theorem 2.9). (9.10) is a consequence of (9.9), (9.8) and Definition 2.14. (9.11) follows from (9.9) and (9.10) by Theorem 3.7. The proof is complete. Proof of Theorem 9.2. Let A fulfil (9.6). Theorem 7.10 can be applied with y = 5(A), P* = Pn(u), W = W L C ( Q * ( U , ) ) . We have to verify that (7.26) - (7.29) are fulfilled. It is obvious that (7.26) holds and Ti.fC C 7£(w). Therefore P-R,(U)) C P-HK which together with (9.9) implies (7.27). Moreover, (7.28) is a consequence of (9.11). (7.29) follows from HK C 7£(k->) (cf. Lemma 3.15 and Corollary 3.9). The proof is complete. 9.5 N o t e . For u> £ Q, denote K(LJ,l)

= {{t,A)eK(u,);

teA},

n(u},2) = {(t,A)eH(u); Aeiv}. Obviously K(u,k)

CK(w),

k = 1,2,

which implies that (9.16)

P-n(u) C Pnu,k),

k = l,2

108

INTEGRATION BETWEEN L AND H - K

and

(9.17)

Q R H C Q R W J . ^ U

so that (9.18)

KLC(Qn^k))\PnM

C W L C ( Q K ( W ) ) , * = 1,2.

Putting together (9.9) with (9.17) and (9.11) with (9.18) we may conclude by Theorem 7.10 that (9.19) Theorem 9.2 remains valid if R(u>) is replaced by R(u>, fc), k = 1,2.

10 A N E X T E N S I O N OF THE C O N C E P T OF ^-INTEGRATION

10.1 Introduction. Two integrations have appeared which can be defined by means of integral sums in a way which differs slightly from ^-integration. In this chapter a new concept of integration is introduced together with a concept of topology. Noncompleteness results are proved which apply in particular to the two integrations mentioned above. Let X = {yi,y2,...} where y{ € Y for i e N. 10.2 Definition. Let / : I -> E, F : Fig -»• R. / is called A'-integrable and F is its primitive if / is J V integrable and F is its primitive for k — 1,2,3,.... The set of primitives of A'-integrable functions is denoted by PxNote. Definition 10.2 implies that (10.1)

PX=f]Pyk. km

10.3 Definition. TAx is a topology on Px such that (10.2)

(10.3)

{PxMx)

is a locally convex space,

Ux D ULc(®yk)\px 109

for k € N,

INTEGRATION BETWEEN L AND H - K

110

(10.4) if V is a topology on Px such that (Px,V) is a locally convex space, V D l^Lc(^yk )\px for k £ N, then VDUX.

Note. Ux exists. The proof is analogous to the proof of Lemma 3.5. Obviously, Ux is unique. 10.4 ; f ( S * ) - i n t e g r a t i o n . Let \ k e A for k <E N. T h e o r e m . Let X(S*) Then (10.5) {Px(s-)Mx(s-))

= {<S(A 1 ),«S(A 2 ),... }. is not

complete.

Proof. Since there exist A € A and (k € K.+ for k £ N such that 0 < A(
C ^(A*)*

WLC(Q5(AO)IP5(A)CWLC(Q5(A))

for

i6N.

By (10.1) and Definition 10.3 we get (10-6)

(10.7)

Ps(x)

C

Px(s-),

W*(S*)|P,(;0CWLC(QS(A)).

Now Theorem 7.9 may be applied with y = <S(A), P* = Px(s*), U* = Ux(S')- It is obvious that (7.26) holds. (7.27) and (7.28) are consequences of (10.6) and (10.7). From WC C

EXTENSION OF ^-INTEGRATION

111

S(\k) for k G N we deduce (cf. Lemma 3.15, Corollary 3.9 and Definitions 10.2 and 10.3)

ULc(®n>c)\ps{Xk) cULC(Qs(xk))

for

ULC{.QHK)\pX{s.)CULc(QS(xh))\pxls.) ^Lc{Q.nK)\px(s*)

kN, for

keN,

cUx(S')-

Hence (7.29) holds. The proof is complete. 10.5 R e m a r k . The concept of "C-integral" was introduced in [B 1996]. / : 7 -> R is called C-integrable on 7 = [a,b] if there exists 7 G R such that for each e > 0 there is ( : 7 —> R + such that p

for each C - n n e partition {(xi, J\),...,

x,G7, JiGlv

(x p , Jp)} of 7 satisfying

for i = 1 , 2 , . . . , p , y ^ d i s t ( a j j , Jj) < - . t=l

7 is called the C-integral of / over 7 and denoted by (C) jj fdt. Moreover, if / : 7 —> R is C-integrable on 7 and J G Iv then / | j is C-integrable on J and F : Iv —> R defined by 7^(7) = (C) J j / | j£ft = (C) J j /c?t is called the primitive of / . Let / : I -* R and F : Iv -»• R. Then / is C-integrable and F is its primitive if for every e > 0 there exists rj : I —• R + such that

J]|F(J0-/(^)l^ll<£ i=l

112

INTEGRATION BETWEEN L AND H - K

provided { ( x i , J i ) , . . . , (xq, Jq)} is an 77-fine partition in / fulfilling Ji G Iv for i = 1 , 2 , . . . , q and 9

1 2 ^ d i s t ( x j , Ji) < i=l

(cf. Lemma 1.7). Therefore it is not difficult to prove that F is the primitive of a C-integrable / : I —> R if and only if F € -P;tr(s*) where # ( S * ) = { A i , A 2 , . . . } with At(cr) = 2-J(T for < T > 0 , I ' G N . It was proved in [B-Pi-Pr 2000] that / : / —> R is C-integrable if and only if there exists a derivative h such that / — h is Lebesgue integrable. Moreover, Theorem 8.9 implies that F € AdC is the primitive of a C-integrable / : I —• R if and only if for every e > 0 there exists r\ : I —» R + such that

X)|*W)-/MI*II<£ i=l

provided {(a?i, J i ) , . . . , (ccg, J g ) } is an 77-fme partition in I fulfilling Ji 6 Iv for i — 1 , 2 , . . . ,q and 2 j d i s t ( x i , J i ) < 1. 2=1

10.6 ^ ( ^ * ) - i n t e g r a t i o n . Let uk € ft for jfc G N.

Theorem. Let #(ft*) = { f t ^ ) , ^ ^ ) , . . . }. Then (10.8) (.P;t(-K*),£/*(•£.)) is not

complete.

EXTENSION OF ^-INTEGRATION

113

Proof. There exist Aj 6 A such that (10.9)

Ajfc(
2auJk

^

for

and there exist A G A and ^ G R (10.10)

a £ R+, k G N

such that

0 < A(
Put < V(5*)

= {5(A1),5(A2),...}.

Lemma 3.15 implies that ^S(A) C P 5 (A.) for fc G N, ULc(Qs(xk))\psw

CULC(Qs(x))

for

^ N

and by Lemma 9.4 Ps(\k)CPn{uk),keN, ULc(®m»k))\psvk)cULc(Qs(xk)),ke

N

so that Ps(X) C Pn(Uh), ULc(Qx{u,k))\PsW

keN,

C WLC(Q5(A)), * € N

and finally (10.11)

(10.12)

PS(\)CPX{K.),

W* ( tt.)|p 5 ( x ) CW L c(Q5
INTEGRATION B E T W E E N L AND H - K

114

Now Theorem 7.10 may be applied with y = £(A), P* = Px{W)-> U* — UX(K*)^ is obvious that (7.26) holds. Since HK C K(uk) for k e N we have P-R(wk) C P-HKi ULc(®wc)\pn{Wk)

C

ULC(Qn(Uk))

so that (10.13)

(10.14)

Px(1i*)

C PHK,

ULciQwcyPx^cUxw

Therefore (7.27) is a consequence of (10.11) and (10.13) while (7.28) and (7.29) follow from (10.12) and (10.14). The proof is complete. Remark 1. Let u>k G fi for k E N. Put A'(7e*(/)) = ( 7 e ( a ; 1 , / ) , ^ ( c u 2 , / ) , . . . ) , 1 = 1,2. By Note 9.5 we have P-R(^) C ^ , ) ) ,

Z = l,2,fceN,

WLC(Q«(W4)O)IPW(^) C«LC(QR(UI))»

l=

Hence ^W(tt*) C PAT(-K*(0)» ^ = l i 2 ^ ^(•K*(/))|P^(7I.) ^ ^ ( r ) i

/ = 1,2.

Making use of (10.11) and (10.12) we get -Ps(A) C -P^(7J*(i)), / — 1,2,

l,2,keK

EXTENSION OF ^-INTEGRATION Ux(K*(J.))\pS(x)

C£4c(Qs(A))>

115

1=1,2

for a suitable A G A. We conclude by Theorem 7.10 that Theorem 10.6 remains valid if X(K*) is replaced by X(TZ*(l)), 1 = 1,2. Remark 2. In [B-Pf 1992] the properties of a general n-dimensional integration are described both from the point of view of the descriptive theory of integration and from the point of view of Riemannian sums. In Section 3 the integration is specialized to one dimension. Considering the integral from [B-Pf 1992] in one dimension let us define (10.16) / : I -» R is i? 0 -integrable if there exists 7 G R and if for every e > 0 there exists £ : I —• R + such that

| 7 -£/(*)|A||< £ A

whenever A = {(t,A)}

fc

for (t,A) P u t uk(a)

G C(I,I,()

and

' H^lldiamA -

G A.

= 2~k for k G N, X{K°)

=

{11(^,1171(^,1),...}.

It is not difficult to prove that / is -R°-integrable if and only if its primitive F fulfils F G PX{TI<>)- The space (Px(K°),Mx(n°)) is not complete.

11 D I F F E R E N T I A T I O N A N D

INTEGRATION

11.1 D e f i n i t i o n . Let y C Y, H : Fig -> R, s G J, /? € R. if is called y-differentiable at s to f3, DyH(s) = f3, if for every e > 0 there exists 77 > 0 such that |ff(A)-/?|4|j<e|A| whenever {(s,A)}

e y, A C [s-rj,s

+ r)], \A\ >

edi&m({s}{jA).

Notes, (i) H is called y-differentiable at s if there exists (3 such that H is differentiable at 5 to (3. (ii) W/C-differentiable at s means differentiable at s (cf. Definition 1.11), D-uKH(s) = H(s). (iii) If DyH(s) exists then H(s) exists and DyH(s) = # ( s ) (since y D W/C). 11.2 D e f i n i t i o n . Let if : Fig -> R, 5 € I , /? G R, A" = { ^ i , J 2 , • • • } where ^ € Y for k G N. if is called X-differentiable at s to fl, DxH(s) = (3, if DykH(s) =/3 for Jfc G N. Note, if is called X-differentiable at s if there exists f3 G R such that if is /f-differentiable at 5 to /?. 1 1 . 3 T h e o r e m . Assume

that y

satisfies

(11.1) if A = {(*, A)} £y,Q = {(s, B)} G y, \A fl 5 | = 0 for (*, A) G A, (5, B ) G 0 tLen A U 0 G 3>. 116

DIFFERENTIATION AND INTEGRATION

Let f : I —> R be y-integrable there exists M C. I such that (11.2)

(11.3)

and let F be its primitive.

117

Then

M e TV,

DyF(s)

= f(s)

for

seI\M.

Proof. For j G N let Mj be the set of s G I such that there exists a sequence A(s, I) € Fig, / G Kf fulfilling

{(s,A{s,i))}ey, \F(A(s,l))

-

f(s)\A(s,l)\\>2->\A(s,l)\,

\A(s, l)\ > 2-J diam({s} U A(s, I)) for / G N, diam({.s} U A(s,l))

-> 0 for I -> oo.

lt follows that P u t M = \JjmMJ(n-3) In order to prove (11.2) assume that

(11.4)

holds

-

\Mj\ > 0 for some j G N.

Let us denote by J(.s, /) the smallest interval containing {s} U A(s,l). Let (: I->R+. The set { J ( * , 0 ; « G M i5 Z € N, J ( 5 , / ) C [5 - C ( * ) , 5 + C(s)]} is a covering of Mj in the Vitali sense. Therefore there exists a finite set (s(i),/(i)), i = 1,2,...,772 such that the intervals J ( s ( i ) , l(i)) are mutually disjoint and

X)|JW0,K0)l>2lMil-

118

INTEGRATION BETWEEN L AND H - K

Since (cf. (11.1)) {(s(i),A{s(i),l(i)));

i =

1,2,...,m}ey

and > 2 _ J 'diam({s(i)} U A(s(i),l(i)))

\A(s(i),l(i))\

= 2-*\J(s(i),l(i))\,

i =

=

l,2,...,m

we have m

J ] |F(A(3(i),/(,•)))-/( B (0|AW0,/(«))ll> i=l m

>2->X;IAKO,/(O)I> 771

i=l

so that F is not the 3^-primitive of / (cf. Lemma 1.15). Hence (11.4) is false, i.e. \Mj\ = 0 for j 6 N and (11.2) holds. The proof is complete. 11.4 T h e o r e m . Let 0 < a < 1 and iet J? satisfy (11.1) and (11.5)

|A| > o - d i a m ( { s } U . 4 ) for

Let H : Fig -> R be Then (11.6)

{(s,A)}ey.

additive.

ffGPy

if and oniy if tiiere exist M C I and 6 £ D* such that (11.7)

MeAA,

DIFFERENTIATION AND INTEGRATION

(11.8)

DyH(s)

(11.9)

exists at

119

s£l\M,

£|2T(A)|<2-' A

for j G N, A = {(*, A)} G y(I, M, S(j, •))• Proof, (i) Let (11.6) hold. By Theorem 11.3 there exists M Cl such that (11.7) and (11.8) hold. Since H G Py there exist a ^-integrable h : I —* R. and 8 £D* such that

Y,\H(A)-h(t)\A\\<2-i A

for j G N, A = {{t,A)} that

G y(I,M,6(j,-))

(cf. Lemma 1.15) so

X>(A)|<2-'" + 5 > ( 0 p | A

A

and (11.9) can be proved by Lemma 1.13. The "only if" part holds. (ii) Assume that there exist M C I and S G D* such that (11.7), (11.8) and (11.9) hold. P u t (DyH(t) W

for

teI\M,

~ j 0 for t G M.

Let j , r G N, \I\ < 2 r . Definition 11.1, (11.5) and (11.8) imply that for every j G N there exists (j : I —> R + such t h a t \H(A)-h(t)\A\\<2-*-r\A\

iortel\M,Ae

Fig, Ac(t-

G(<),* + 0(*)), {(M)} G 3>-

P u t (cf. (11.9)) | min{Cjk(<);* = l 5 2 , . . . , j } for j e N,* € / \ M ,

Then (1.22) is fulfilled and (11.6) holds by Lemma 1.15. The "if" part holds. T h e proof is complete.

120

INTEGRATION BETWEEN L AND H - K

1 1 . 5 R e m a r k s , (i) We have H

ePnK

if and only if there exist M C I and 8 G D* such that

M eAf, H(t)

exists for t G

I\M

and ^|^(J)|<2^ A

for j G N , A = {(<, A)} G W ^ ( J, M , <5(j, •))• This is a consequence of Theorems 11.3 and 11.4 since \J diam({i} U J ) for {(<, J ) } G UK. (ii) Let a G (0, §], w(<) = a for < G J. Then H G P*(w) if and only if there exist M G J\f and <$ G D* such that Dn{u)H{t)

exists for

teI\M

and £

|2T(J)| < 2~>

A

for ; G N, A = {(*, A)} G ft(w)(I, M, 8(j,.)). This is a consequence of Theorems 11.3 and 11.4 since \A\ > a\\A\\ diam({*} U A) > <J2 diam({*} U A)

for{(t,A)}eR(u).

DIFFERENTIATION AND INTEGRATION

121

11.6 N o t a t i o n . Let a € [0,1] and let us define Z(a)y

= {Aey;

\A\ > a d i a m ( { t } U A ) for ( t , A ) e A } , 1 2 3

x* = x\y) = {z(2- )y, z(2- )y, z(2~ xy,...}. Obviously Z{a)y

€ Y and Z{a)y

C y.

(11-10)

PZ{„)y => Py,

(11.11)

Ez(cr)y

Therefore

DEy,

(11.12) if 5 € J, H : Fig -> E and DyH(s) Dz((r)yH(s) exists.

exists then

By (11.10), (11.11) and Theorem 3.7 we have (11.13)

ULC(EZ{fT)y)\Py

C

ULC(Ey).

Moreover, (11.14) if y fulfils (11.1) then Z(a)y

fulfils (11.1).

11.7 T h e o r e m . Let y fulfil (11.1), H : Fig - • R, 5 G I ,

/? e R. Tien D*. ( 3 ;)jy(s) = /? if and only if DyH(s)

= /3.

Proof. The "if" part follows by (11.12) and Note 11.1 (iii). Let Dx*(y^H(s) = /?. For every i £ N w e have Dz{2-i)yH(s)

= (3

so that there exists r)i > 0 such that \H(A)-^\A\\<2-i\A\ whenever {(s, A)} E y, A C [s-rn,s + rn], \A\ > 2 _ J diam({.s} U A). Therefore DyH(s) = (3 and the "only if" part is true. The proof is complete.

122

INTEGRATION BETWEEN L AND H - K

11.8 T h e o r e m . Let y fulfil (11.1) and let H : Fig -> R be additive. Then (11.15)

# G P^(^)

if and only if there exist M C /', St G D* for i G N such that (11.16)

(11.17)

Me AT,

DyH(s)

(11.18)

exists for

seI\M,

£|lf(A)|<2-> A

for z,j G N, A = {(t,A)} G 2 ( 2 - ^ ( 1 , M , ^ , - ) ) . Proof, (i) Let i ? G P^*(;y). There exists h : I —> E which is Z(2 _ t )[V-integrable and i f is its i?(2~ J ^ - p r i m i t i v e for i G N. By Theorem 11.3 there exist Mi G M such that Dz{2-i)yH(t) P u t M = \Ji&NMl.

= h(t) for t G I \ Ml: i G N. Since

Dx*{y)H(t)

= h(t) for

t<El\M,

(11.17) holds and (11.16) holds as well. Let z G N. Since h is Z(2 _ J ),y-integrable, r / is its primitive and M G Af, there exists <5; G .D* such that (cf. Lemmas 1.15 and 1.13)

£|tf(A)-/*(t)|A||<2-'-\ A

DIFFERENTIATION AND INTEGRATION

123

J2\Kt)\\A\<2-^ A

for j G N, A = {(t,A)} G 2(2-^(1, MMJ,-)) so that (11.18) is valid and the "only if" part holds. (ii) Let there exist M C I, Si for i G N such that (11.16), (11.17) and (11.18) hold. Then (cf. (11.12)) (11.19)

Dz(2-i)yH(t)

= DyH(t)

for

teI\M,ieN

and (11.18), (11.19) and Theorem 11.4 imply that H G PZ(2-')y for i G N, i.e. H G Px*(y)- The "if" part is valid and the proof is complete. 11.9 T h e o r e m . Assume

that y fulfils (11.1) and

(11.20)

Px.{y)=Py.

Let H : Fig -+ R be additive.

Then

(11.21)

£Py

H

if and only if there exist M C I, 6 G D* such that (11.22) (11.23)

(11.24)

MeAf, DyH(s)

exists for

seI\M,

J ] \H(A)\ < 2-' A

for j G N, A = {(t, A)} G ^ ( 1 , M , 6(j, •))• Proo/. (i) If .ff G P y then (11.22) and (11.23) hold by Theorem 11.3. (11.24) holds by Lemmas 1.15 and 1.13. T h e "only if" part is valid. (ii) Let there exist M C 1,8 G D* such that (11.22), (11.23) and (11.24) hold. Then (11.16), (11.17) hold and (11.18) holds with Si = 8 for i G N, so that H G Px*(y) by Theorem 11.8. Therefore (11.21) holds by (11.20) and the "if" part is valid. The proof is complete. T h e rest of this chapter is devoted to t h e case y = C. It will be shown that ^(
124

INTEGRATION BETWEEN L AND H - K

1 1 . 1 0 T h e o r e m . Let 0 < a < \. (11.25)

Pz(a)c

Then

= Pc

1 1 . 1 1 L e m m a . Let K = [c,d] elv,

Ke

R+,

c = a0 < ai < • • • < ap = d, T\ <E Ji = [ a / - i , ai]

fori Eh = { l , 2 , . . . , p } , h : {ri;leh} Assume that (11.26)

Y,\H(A) e

- h(t)\A\\

-> R.

whenever 0 = {(rj,j4j);7 € L } , 4 / C J/, |Aj| > | ( a / o / _ i ) for / £ L, (11.27) M C N is Gnite, Lm € Iv(if), | 7 m n I „ | = 0 for m,n (= M, m ^ n. Then (11.28)

J ] |ff(Im)|<8/c + max{|/i(r,)|;/eL}- J ] |7m|. m€M

mGM

Proof. Let us put

(11.29)

G = {(/,m)eLxM; | j , n 7 m | >o,H(JinLm)-h(Ti)\JinLm\

> o},

H= {(/,m) G L xM;

|J/ni ro | >o,^(J/n7 m )-/i(r,)|j/ni: m | < o}. Then

(11.30)

(J i m = m€M

|J (J,m)€GuH

j,nim,

DIFFERENTIATION AND INTEGRATION

125

|J,nL m n j;nL'J = o for (Z,m),(Z',m') e G u H , ( ! , m ) / ( ! > ' ) and, since H is additive, (11.31)

Y,\H(Lm)\< mgi

Y,

\H(JinLm)\+

(l,m)£G

Y,

\H(JinLm)\.

(/,m)€H

We will estimate the first summand on the right hand side. Let us put (11.32)

G(Z) = {m € M; (Z,m) G G},

4, =

|J

j,nx r o , ^ = c i ( j , \ A , ) , / G L .

m€G(l)

Obviously (11.33)

G=|jG(Z).

Moreover, H(Ai)-h(rt)\Ai\=

Y

(H(JinLm)-h(n)\JinLm\)

meG(l)

for I € L. The summands on the right hand side are nonnegative so that (11.34)

\H(Al)~h(rl)\Al\\

=

= Yl \H(JinLm)~h(Ti)\JtnLm\\ for ZeL. mGG(J)

126

INTEGRATION BETWEEN L AND H - K

Put (11.35)

L+ = {/ G L; |A/| > \\Ji\),

L - = L \ L+.

Now (11.26) implies that

J2 |^T(^/)-/l(T-,)|^|||
\H(A!)-h(T,)\Al\\
let-

J2 \H(Jl) - h(T,)\J,\\ < K. Since

H{Ai) - h(n)\A,\ = H(Jt) - HrOlJtl - (H(AJ) - M ^ M I ) for / e L, we obtain (cf. (11.35)) (11.36)

^|if(A0-M^)l^ll<4«, let

which together with (11.33) and (11.34) gives

J2

J2 1-^(^1 n £ m ) - /I(T-,)|J, n £ m || < 4K,

l£h m€G(l)

(11.37)

J2 \H(Ji Ci Lm) - hWlJ, n Lm\\ < 4K.

It can be proved in an analogous way that

(11.38)

J2 \H(Ji men

n L

™) ~ Kn)\Ji n Lm\\ < 4«.

DIFFERENTIATION AND INTEGRATION

127

Therefore

Yl \H(Jin

L

8K + m a x { | ^ ( r / ) | ; / € L } (

^

(11.39)

-)i + E \H{<Jtn L -)i ^ \JlnLm\+

(l,m)eG

J^

l^nlm|).

(l,m)£W

(11.28) is a consequence of (11.31) and (11.39). The proof is complete. Proof of Theorem 11.10. In order to prove (11.25) it is sufficient to prove that (11.40)

PZ((T)C

C AdAC

if a e [ 0 , | ] (cf. (11.10) and Theorem 5.3). Let H G PZ(a)CThen H : Fig —> R. is additive and there exist h : I —> R and 6 G D* such that (11.41)

£|2r(A)-fc(t)|A||<2-'" e

for ; G N, 0 = {(t, A)} G Z(
\H(Lm)\<8-2-i+max{\h(Ti)\;leh}-

J^

\Lm\

m€M

if M, Lm fulfil (11.27). Therefore H G AdAC. The proof is complete.

INTEGRATION BETWEEN L AND H - K

128

1 1 . 1 2 T h e o r e m . Let H : Fig -» R be additive. (11.42)

H E Pc

if and only if there exist M G I, 8 £ D* such (11.43)

(11.44)

(11.45)

Then

that

MeAf,

DcH(s)

exists for s e l \

£

M,

|17(A)| < 2">

A

for j 6 N , A = {(t,A)}

e

£(I,M,8(j,-)).

Theorem 11.12 is a consequence of Theorems 11.9 and 11.10. 1 1 . 1 3 T h e o r e m . Let y fulfil (11.1). Let G : Fig -> R be additive, /3 € R, s € J,
(11.46)

(11-47)

DcG(s)

= P,

i ^ ( . ) c G ( s ) = /?,

(11.48) for every e > 0 there exists 7/ > 0 such that vaxH\K

< e|if |

for if e Iv, s e if c [^ - 77,5 + 77] n /.

DIFFERENTIATION AND INTEGRATION

129

Proof, (i) (11.46) implies (11.47) (cf. (11.12)). (ii) Let (11.47) hold, e > 0. Then Dzia)cH(s) = 0 and there exists r? > 0 such that (11.49)

£

\H(A)\ <

-

for A e Fig, A C [s-?7,s + T?]n J, |A| > a d i a m ( { s } U A). Let A' G Iv, s <E A' C [s - 77, 5 + 77] n J. Then Lemma 11.12 may be applied with p = 1, TJ = s, J\ — K, h(T\) = 0, K = (cf. (11.49)). We get

J2 where M,Lm

\H(Lm)\<e\K\

fulfil (11.27) so that (11.48) holds.

(iii) Let (11.48) hold, e > 0. There exists 77 > 0 such that V&TH\K

<

e2\K\

for K G Iv, s G K C [s — 77, s + »7] H I . Let A <E Fig, A = UmeM Lm, be the smallest diam({5} U A).

A C [s - 77,5 + 77] n J, |A| > ediam({5} U A), Lm being a component of A for m G M. Let K interval containing {s} U A. Obviously \K\ = Then

\G(A)~f3\A\\

= \H(A)\<

< varH\K

<£2\K\

Y,

\H(Lm)\<

<e\A\.

Therfore (11.46) holds and the proof is complete.

130

INTEGRATION BETWEEN L AND H - K

1 1 . 1 4 R e m a r k . Let / : / —> R be £-integrable, F being its primitive. Then s 6 I is called a Lebesgue point of f if for e > 0 there exists r\ > 0 such t h a t

(Leb)y |/(*)-/(s)|d* < e |J| for J <E Iv, 5 e J C [^ - r), s + 77] D I. Let us put # ( 4 ) = F(A) - / ( s ) | 4 | for A G Fig. Since v a r # | j = ( L e b ) / j | / ( « ) - f(s)\dt for J € Iv, Theorems 11.13 and 11.12 imply that almost every s £ I is a Lebesgue point of 1 1 . 1 5 R e m a r k . For # : Fig -» R, M C I let us put vary ( j y , M ) = = i n f { s u p { ] [ ] \H(A)\;

A = {(t,A)}

G y(I, M, ()}; ( : I - R + } .

A

(Obviously v a r i J = var-^/c(if, / ) . ) The above concept can be used at many places of this book, e.g. instead of (1.22) (ii) we can write vaxy(F, M ) = 0 and Theorem 11.12 can be written in the following form: P r o p o s i t i o n . Let H : Fig —>• R be additive. (11.50)

Then

ff£P£

if and only if there exists M € A/" such that (11.51) (11.52)

£>£#(<) exists at

teI\M,

var£(#,M)=0.

Observe that Proposition also holds if £ is fulfilling (11.1) and (11.5) (cf. Theorem 11.4), y = Z{o)C where 0 < a < 1, y = R(LO) where u{i) = /3foTt€l,y = £{a) where a € R+ (cf.

replaced by y e.g. y = WC, /3 6 (0, \] and Note 8.8).

DIFFERENTIATION AND INTEGRATION

11.16 T h e o r e m . Let 0 < a < \. (11-53)

Ez((T)c

(11.54)

ULc(Ez{a)c)

131

Then

= Ec,

=

ULc^c).

Proof. We have Ec C E2((T)£ since Z(a)C C £, and we will prove that (11-55)

Ez{a)c

C Ec.

Let Ez(«r)£

i*i —• i*o tor z —> oo. By Lemma 2.3 there exist hm : I —> R, 8 € D*, M € jV such that (11.56)

|F m (A) - M < ) M | < 2 - J

^ A

for m = 0 , 1 , 2 , . . . , j G N, A = {(t, A)} e Z ( a ) £ ( / , / \ M, 6(j, •)),

(11.57)

^|Fm(A)|<2^ A

for m = 0 , 1 , 2 , . . . , j € N, A = {(t,A)}e£(a)£(I,M,%,-)), and (11.58)

hi(t) -> /i 0 (i) for t € / \ M, i - • oo.

132

INTEGRATION BETWEEN L AND H - K

Without loss of generality we may assume that hm(t)

= 0 for m = 0 , 1 , 2 , . . . , t G M,

which together with (11.56), (11.57) and (11.58) implies (11.59)

J ] \Fi(A*) - F0(A*) - (hi(t) - h0(t))\A*\\ < 2"'+ 2 A

for i,j G N, A = {(t,A*)}

(11.60)

G

Z(v)£(I,I,6(j,-)),

hi(t) - h0(t) -> 0 for t G I, i -> oo.

Let j € N. There exists 5 = { ( T / , J / ) ; / = 1,2, . . . , p } G 7{IC(I, I, S(j, •)), H being a partition of / . Without loss of generality it can be assumed that there exist oto, « i , . . . , a p € i" such that a = a0 < a\ < a2 < • • • < ap = 6, J/ = [ a / - i , a;] for / G L = { 1 , 2 , . . . ,p}. Since E G l-UC(I,I,$(j, TI

•)) we have

E Ji C [ri - £(j, 77), r; + 6(j, n)]

for / G L and (11.59) implies that (11.61)

J2 MA) ~ MA) - (hi(r) - h0(r))\A\\ < 2~>+2 A

whenever 0 = {(r/,A;); / G L} where Ai e Fig, Ai C J;, I Ai \ > - ( a , - a / _ i ) for / G L.

DIFFERENTIATION AND INTEGRATION

Lemma 11.11 may be applied with K — I, K = 2~J+2, Fi — F0, h = hi — h0, i e N. We obtain

(11.62)

133

H —

J2 \F(Lm) - F0(Lm)\ < < 2 " J + 5 + max{|/i,(r,) - /i 0 (r,)|; / € L}(b - a) provided M C N is finite Lm £ Iv, \Lm fi Ln\ = 0 for m , n G

M,m/n.

Hence (cf. (11.60)) (11.63)

\\Fi - Fo||var -» 0 for i - • oo.

Moreover, there exists M + G Af', M+ Fi(t) = /ii(<),^b(*) = M * )

for

D M such that * e / \ M + , i e N.

By (11.60) we have Fi(t) - F0(t) - • 0 for i e J \ M + , i -> oo. Theorem 5.12 implies

Fi^Fo so that (11.55) holds and (11.53) holds as well. (11.54) is a consequence of (11.53). The proof is complete.

REFERENCES

[B 1996] Bongiorno B. Un nuovo integrale per il problema delle tive. Le Matematiche 51 (1996), 299 - 313

primi-

[ B - P f 1992] Bongiorno B. and Pfeffer W. F. A concept of absolute continuity and a Riemann type integral. Comment. Math. Univ. Carolinae 33, 2 (1992), 189 - 196 [ B - P i - P r 2000] Bongiorno B., Di Piazza L. and Preiss D. A constructive minimal integral which includes Lebesgue integrable functions and derivatives. J. London Math. Soc. (2) 62 (2000), 117 - 126 [K 2000] Kurzweil J. Henstock-Kurzweil pological Vector Spaces.World

Integration: Its Relation to ToScientific (2000), Singapore

135

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LIST O F S Y M B O L S A 34 \\A\\ 76 AdAC 52 AdC 16 AdCBV 52 B(a) 37 [c,d\,(c,d) 9 [c,d),(c,d\ 9 c\E 9 Comp A 49 conv E 37 Z>* 10 diam E 9 dist(*,£) 9 |£| 9 Ey 22 F ^ F o

34

Iv 9 Iv(K) 9 ( L e b ^ / d t 53 C 10 £* 53 M 69 X * 70 N 9 M 9 PA: 109 Py 16 Qy 30 Qy(£) 24 R 9 R+ 9 reg(i,A) 104 ft(w) 104

F^Fo

22

ft(u,l)

F . ^ F o 30 Fig 9 Fig(tf) 9

107

n{u,2) 107 5(A) 86 St 53

1

-P |Fig(M) 10 -Pllv(M) 10

10 ft/C 10 H(s) 16 7 = [a, b] 9 int£ 9 /|M

TVi 52 Ty 42 T* 43 T\z 32 U L C (A) 34 WLc(A) 35 137

Ux 109 v a r F 52 varA- F 52 var(#, M) 130 V(9J) 33 9J y 39 2rjy 42 AT 109 #(ft*) 112

W E , ( ) 10 .y(M,£;,oo) 10

x{n*{i)) 114 A"* 121

x*(y)

121

A , (^°) 115 A-(5*) 110 y 10 Y 10 y(M,E,a) 10

(y)JKfdt Z

9

Z(a)y 121 #M 9 52 52 ||var Fl|0 * lljfc.i 69 ' lljfc.i 69 F\ y,s 42

n

138

\\-\\y,s 42 | | # | | s u p 16 II • Hsup 16 8(j,-) 10

INTEGRATION BETWEEN L AND H - K

S\M 10 Tk(F) 43 A 86 A!( S ,a) 76

ft

A2(s,a) 76 6 A 105 104

INDEX

absorbing additive balanced C-integrable C-integral continuous at s convergence on P* convergent to FQ in A convergent to Fo in Ey convergent to -Fo in Qy convergent to xo in (F, T ) convex circled derivative of H at s differentiable differentiable at s F-tagged figure finer topology Hausdorff topology interval invariant with respect to shifts Lebesgue point of / locally convex space open (T-open) 139

29 13 29 111 111 15 34 34 22 30 32 29 29 16 16 16 10 9 32 32 9 32 130 33 32

140

INTEGRATION BETWEEN L AND H - K

partition in K partition of K primitive of / radial regularity of (t, A) restriction of T to Z 7£°-mtegrable seminorm stepfunction

tag topological space topology induced by R topology induced by V topology invariant with respect to shifts topology on Y topology tolerant to A -Y-integrable A'-differentiable at s A'-differentiable at s to p1 J'-continuous at s 3^-differentiable at s ^-differentiable at s to f3 3^-integral of / over K ^-integrable on K

C-fine

9 10 12 29 104 32 115 33 53 10 32 34 33 32 32 34 109 116 116 15 116 116 11 11 10

The Lebesgue Integral

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The Lebesgue Integral

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The Lebesgue integral

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Integral

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Lebesgue Integration and Measure

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Convex Analysis in General Vector Spaces

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