HENSTOCK-KURZWEIL INTEGRATION: Its Relation to Topological Vector Spaces
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HENSTOCK-KURZWEIL INTEGRATION: Its Relation to Topological 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 by Jaroslav Kurzweil
Series in Real Analysis - Volume 7
HENSTOCK-KURZWEIL INTEGRATION: Its Relation to Topologicai Vector Spaces
Jaroslav Kurzweil Mathematical Institute of the Academy of Sciences the Czech Republic
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World Scientific Singapore • New Jersey • London • Hong Kong
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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
HENSTOCK-KURZWEIL INTEGRATION: ITS RELATION TO TOPOLOGICAL VECTOR SPACES Copyright © 2000 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
Since the Riemann approach to nonabsolutely convergent in tegration came into being, most efforts were aimed at exploring the scope of this approach through various definitions and their consequences, especially characterizations of the primitives and general formulations of the Stokes theorem. This monograph follows a different direction. Its object is the vector space of equivalence classes of functions which are Henstock-Kurzweil integrable on a compact one-dimensional in terval or equivalently, the vector space P of their primitives. There exists a convergence theorem for sequences of HenstockKurzweil integrable functions which is connected with the Rie mann approach and which is transferred into the space P in a natural way. The corresponding sequences of functions from P are called ^-convergent. In this book ^-convergence is studied in relation to topological vector spaces. Topics connected with the Riemann approach to integra tion 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 begin ning of this approach. I wish to thank the participants of the seminar for their con tributions and comments. I express sincere thanks to J. Jarnik and S. Schwabik, who read the manuscript and suggested several improvements. I am Typeset by V
AMS-T&.
vi
HENSTOCK-KURZWEIL INTEGRATION
grateful to S. Schwabik who encouraged me and transformed the manuscript into the camera ready form. The research which resulted in pubUshing this book was sup ported by the grant No. 210/97/0218 of the Grant Agency of the Czech Republic.
Prague, September 1999
Jaroslav Kurzweil
CONTENTS
Preface 0. Introduction 1. Integrable functions and their primitives 2. Gauges and Borel measurability 3. Convergence 4. An abstract setting 5. An abstract setting with D countable 6. Locally convex topologies tolerant to Q-convergence 7. Topological vector spaces tolerant to Q-convergence 8. P as a complete topological vector space 9. Open problems A. Appendix List of symbols Index References
vii 1 8 21 34 48 55 67 75 86 118 123 128 129 131
INTRODUCTION
The topic of this treatise are relations between integration, convergence and topology. The starting point is the vector space of Henstock-Kurzweil integrable functions / : I —> R, where J = [a, 6] is a compact interval in R. In the sequel the notion integrable and integration will be used instead of Henstock-Kurzweil integrable and HenstockKurzweil integration. It is well known that the integration is a true extension of Lebesgue integration. Let / : I —> R be integrable (cf. Definition 1.4). It is common to call G : I —► R, G(t) — j a fds the primitive of / . In this treatise the primitive of / is a function F which assigns to every interval J C I the value F(J) = fj fds. It follows that F is an element of A, the Banach space of additive and continuous functions which map intervals from / to the reals. (Obviously, A is isomorphic to the Banach space of continuous functions H : / -» R, H(a) = 0.) Let / , g : I —*• R, let / be integrable, / — g = 0 almost every where. Then g is integrable and / fds = jj gds, J C I being an interval. Therefore it is convenient to put the primitives of integrable functions to the foreground. One of the basic objects is the vector space P of F : / —> R such that F is the primitive of an integrable / . Obviously P C A. Let fi : I -> R for i € N, / : J -» R. l
2
HENSTOCK-KURZWEIL INTEGRATION
In the elementary convergence theorem (Theorem 3.1) it is assumed that (0.1) a uniformity condition is fulfilled by /, for i 6 N, (0.2) fi(t) -► f(t) for i — oo, t e / , which guarantees that (0.3) fi, i EN and / are integrable, (0.4) the primitives of fi converge (uniformly) to the primitive of/. The above elementary concept of convergence is transferred from the vector space of integrable functions to the space P as follows: Let Fi e P for i e N, F € P. A sequence F,, z e N is called F-convergent to F , shortly Fi -£♦ F if there exist /< : / -> R for t e N, / : I -» R such that (0.5) Fj is the primitive of fi for i e N and (0.1) and (0.2) are valid. By the elementary convergence theorem F is the primitive of / and Fi -> F . PROBLEM. Does there exist a topology T on P such that (0.6) Fi-^> F implies that F -» F in (P,T), (0.7) (P,T) is complete, (0.8) (P, T) is a topological vector space? M A I N RESULT. The answer is affirmative. If (0.8) is strenghtened to (0.9) (P, T) is a locally convex vector space, then the answer is negative. Note. Let PST be the set of primitives of stepfunctions on / . It can be proved that for every F € P there exists a sequence
INTRODUCTION
3
of F{ G PST such that Fi —► F. Therefore PST is dense in (P, T) and P is the completion of P$T- This is an analogy to the well known result that the space L (of equivalence classes) of Lebesgue integrable functions is the completion of the space of stepfunctions / : J —* R which is equipped with the norm
11/11 = /7l/|d*. In Chapter 1 basic concepts and results on integration are summarized. Let 9 : I —> (0, oo); 6 is called a gauge. Denote by D* the set of f : N x I —> (0, oo) so that £ represents a sequence of gauges eO',-)j'6N. Let G G A (i.e. G is an additive continuous function of inter val). Chapter 1 is concluded by a condition which is necessary and sufficient for G G P (Theorem 1.20). Since this condition plays an important part in the sequel, let us describe it in some detail. Denote by C(G, g, £, M) a predicate, the variables being G G A, g : / —> R, £ G D*, M e J\f, where M is the set of subsets of / of measure zero. The interpretation of C is not relevant in this place (but, in fact, C represents the couple of inequalities (1.16) and (1.17)). Now let G € A. Then (0.10)
GeP
if and only if (0.11) there exist g : / -»• R, £ € D* and M G M such that C{G,g,Z,M) is valid (Theorem 1.20). Denote by D the set od 6 G D* such that 6(j, •) is Borel mea surable for j e N . In Chapter 2 the above result is improved in the following way. Let G e A. Then (0.10) holds if and only if (0.12) there exist g : / -> R, r) G D and M* G M such that C(G,g,r),M*) is valid (Theorem 2.15).
4
HENSTOCK-KURZWEIL INTEGRATION
This improvement plays a crucial part in Chapter 8 in the proof that the answer to the PROBLEM is affirmative. In Chapter 3 the concept of Q-convergence is introduced and studied. For 8 € D denote by Q(6) the set of G e A such that there exist g : I —> R and M e M such that C(G, g, 6, M) is valid. It turns out that (0.13) Q(6) is a convex balanced and compact subset of A, (0.14) for 6i,82 G D there exists 63 e D such that
Q(h) +
(0.15)
Q{h)cQ(h),
P=\jQ(6) seD
(Theorems 3.9 and 3.10). Let Fi G P for i £ N, F e P. The sequence F{ is said to be Q- convergent to F, shortly Fi —► F, if (0.16) there exists 6 e D such that Fi e Q(6) for i € N, (0.17) Fi —► F for i —► 00 in the Banach space A. The relation between the concepts of .^-convergence and Qconvergence is described in (0.18) Fi-^+ F implies that Ft - ^ F and (0.19) if Fi —► F, then there exists a subsequence i(k), k 6 N such that Fi(k) - ^ F (Theorem 3.12). Let W b e a topology on P. It can be deduced from (0.18) and (0.19) that (0.20) Fi-^F
implies F{ -» F in (P,U),
INTRODUCTION
5
if and only if (0.21) Fi-^F
implies Ft -»• F in (P,U)
(Theorem 3.14). Therefore it is sufficient to study only the relation between topologies on P and Q-convergence. In Chapter 4 a more general setting is considered. Let X be a Banach space, let D be a set (of parameters d) and let Q be a map from D to 2X such that (0.22) Q(d) is a convex balanced and closed subset of X for de D, (0.23) if e?i, c?2 G D, then there exists cfe G JD such that Q(di) + Q ( d 2 ) c Q ( d 3 ) . It follows from (0.22) and (0.23) that
p= U w) is a vector space. Let Xi,x G P for i e N. The sequence X{ is called QQ
convergent to x, X{ —► x, if (0.24) there exists de D such that Xi G Q(d) for i G N, (0.25)
Hz. - x||-► 0.
Let W b e a topology on P such that (P,U) is a topological vector space. We present there (i) necessary and sufficient conditions for ( P , U) to be com plete (Theorems 4.5, 4.6), (ii) necessary and sufficient conditions that Xi —► x implies that Xi -v x in (P,W) (Theorem 4.7).
6
HENSTOCK-KURZWEIL
INTEGRATION
A 0- neighbourhood base in P is defined and the correspond ing topology is denoted by Ulc; (P,Ulc) is a locally convex topological vector space and Ulc is the finest of all topologies U on P such that (0.26) (P,U) is a locally convex vector space, (0.27) Xi -^* x implies that Xi -» x in (P,U) (Theorem 4.10). In Chapter 5 the setting presented in Chapter 4 is continued with the additional assumption (0.28)
D = N, 2Q(j) C QU + 1) for j G N.
(0.28) has remarkable consequences: (0.29) {P,U*LC)
is
complete
and (0.30) if Xi —> x in (P,U£C),
then Xi —► x
(Theorems 5.2 and 5.7). Chapter 5 is concluded by several examples; in one of them complete locally convex vector spaces are described the elements of which are primitives of integrable functions. In Chapter 6 we return to the space P of primitives to inte grable functions. It is proved that for any neighbourhood U of zero in (P,U) there exists a G R + such that {F e P; \\F\\
0, it follows that Ki £ K, for i sufficiently large (cf. [Cousin P., 1895], Lemme 10; see also [Jarnik J. and Kurzweil J., 1995], Lemma 1.1). D 1.3 Definition. Let D* be the set of 6 : N x I -> R+ such that 6(j,t)>6(j + l,t) for j€U,teI. A function C : I —* ^ + is often called a gauge, so that 6(j, •) : I -* R+ where 6(j,-)(t) = 6(j, t) for t G I, is a gauge and 6 represents a sequence of gauges. 1.4 Definition. Let K = [c,d] G Iv(J), / : K -► R. The function / is called integrable (I.e. Henstock-Kurzweil inte grate) on if, if there exist 7 G R, 6 G D* such that k
A
i=l
provided j G N, A = {(t,J)} = {(tuJi);i = 1,2,...,A:} G «S(A', # , 6(j, •)), A being a partition of K. The value 7 G R is unique by Lemma 1.2. The value 7 G R is called the integral of / and is denoted by fK fdt or / fdt.
10
HENSTOCK-KURZWEIL INTEGRATION
1.5 Note. Definition 1.4 is obviously equivalent to the tra ditional definition of integrability of a function / in which it is required that there exists 7 G R and for every e G R + there exists C : K —> (0,1] such that
l7-£/(WII<e A
for A = {(t, J)} G S(K, K,Q, A being a partition of K (see e.g. [Lee P.Y., 1989], Definitions 2.2 and 2.2a, [Pfeffer W. F., 1993], Definition 6.1.1). Definition 1.4 was introduced in [Kurzweil J., 1957], see Defi nition 1.2.1, where we have to put U(r, t) = f(r)t. It was shown that it is equivalent to the definition of Perron, which is based on major and minor functions and it was used to the study of continuous dependence of solutions of ordinary differential equations on a parameter. Independently it was introduced in [Henstock R., 1961]. It is also well known that Perron integration is equivalent to Denjoy integration in the restricted sense, which is based on the concept of ACG* functions (cf. [Saks S., 1937], Chapter VIII, Theorems 3.9 and 3.11). For a rather general result on integration by parts see [Kurzweil J., 1958]. 1.6 Definition. Let H : Iv(7) —► R be given. H is called additive (on Iv(I)) if H([cuc2])
+ H{[c2,c3]) = H([c1,c3])
for Ci,C2,C3 G / , ci < C2 < C3. Occasionally it is useful to assume that H($) = 0. H is called continuous, if for every e G R + there exists a G R+ such that \H(J)\ < e for J G Iv(7), | J | < a. Denote by A the set of H, which are additive and continuous.
1. INTEGRABLE FUNCTIONS, PRIMITIVES
11
1.7 N o t e . There is a natural one-to-one correspondence be tween additive functions H on Iv(J) and functions G : I —► K. which is given by the relation G(a) = 0, G(t) = H([a,i\) for t E (a, 6] where I = [a, 6]. The function G is continuous if and only if H is continuous. 1.8 L e m m a . Assume that I = [a, 6]. Let / : 7 —>■ R be integrabie. Then for every K E Iv(7) the restriction f\K is integrable. Put F(K) = JK f\Kdt (= JK fdt), F(0) = 0. Then F is additive on Iv(7). F is called the primitive of f. Moreover, let 6 E D* and let
|F(/)-£/(t)|J||
a\K(l,s)\
for
/ G N,
|/sf(/,s)| —> 0 for / —» oo. If (1.4) is false, then there exists a G R+, a < 1 such that | £ Q | > 0. Let j G N fulfil 2-^+ 2 < c*|£Q|. Let £ be the set of K(l,s) such that s G F a , if (s,/) C (5 — 8(j, s),s + 6(j, s)). K, is a covering of Ea in the sense of Vitali. Therefore there exist m G N, {(/i, «i); i = 1,2, . . . , m } such that the intervals K(li, Si) are pairwise disjoint and 771
\Ea\\jK(h,Si)\|£o|-2^>-|Ett|. z
1=1
By the definition of the intervals K(l,s) we have m
m
n
X) l * W i , »i)) - f(si)\K(h, 8i)\\ > a X TO, * ) | > ^a\Ea\. i=\
i=l
1. INTEGRABLE FUNCTIONS, PRIMITIVES
15
On the other hand, {{su K{li, Si)); i = 1, 2 , . . . , m} e <S(7, 7 \ TV, S(j, ■)) so that (cf.(1.3)) m
Y,\nK(ti,Si))-f{8i)\K(li,8i)\\ [0,1]. Then {tel;
tf(t)>^(t)}eJV.
Proof. Let t G I, 0(«) > ipt(t), (3 G (ip#(t),tf(t)), (3 rational. Then t G r(tf, /?) \ clessr(tf, (3). It follows that {*e/;i?(^(«)}c C | J ( r ( ^ ' 0) \ dessr(^, /?);/?€ (0,1), /? rational} and the proof is completed by Lemma 2.5.
□
2.11 Lemma. Let ■d, ( : I —> [0,1], iet £ be measurable and ((t) > fl(t) almost everywhere. Then ((t) > ip#(t) a.e. Proof. Put M = {t G 7;C(0 < #(«(.?' + 1,0 < t)>MJ + M ) < ^ ( i , *) for j 6 N, < G 7. 2.13 Theorem. Let *, G G A, # : 7 -> R, M G M. Assume that
Li G ( J )i^ 2 _ J
(2.i3)
forj G N, A = {(t, J)} €
(2.14)
+
E^7j
S(I,I,6(J,-)),
53|G(J)-»(t)|J||