Table Of ContentHENSTOCK-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
v>
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HENSTOCK-KURZWEIL INTEGRATION: ITS RELATION TO TOPOLOGICAL
VECTOR SPACES
Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.
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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 Henstock-
Kurzweil 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 AMS-T&.
V
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 vii
0. Introduction 1
1. Integrable functions and their primitives 8
2. Gauges and Borel measurability 21
3. Convergence 34
4. An abstract setting 48
5. An abstract setting with D countable 55
6. Locally convex topologies tolerant to Q-convergence 67
7. Topological vector spaces tolerant to Q-convergence 75
8. P as a complete topological vector space 86
9. Open problems 118
A. Appendix 123
List of symbols 128
Index 129
References 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 Henstock-
Kurzweil 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 fds the primitive of /. In this
a
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?
MAIN 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 = / l/|d*.
7
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).
