Integration Theory

307 Pages·1997·16.479 MB·\307

by  K. Weber (Author)

Integration Theory

9780412576805
Integration Theory
Size: 234x156 mm Spine size: 15 mm Color pages: Binding: Paperback
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Professor Keith Devlin Professor Derek Goldrei Dr James Montaldi
StMary's College Open University Universite de Lille
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Integration Theory
W. Filter
Professor of Analysis
University of Palermo
Italy
and
K. Weber
Professor of Mathematics
Technikum Winterthur
Switzerland
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First edition 1997
© 1997 W. Filter and K. Weber
Printed in Great Britain by St Edmundsbury Press Ltd, Bury St Edmunds,
Suffolk
ISBN 0 412 57680 5
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Contents
Preface vii
Introduction ix
1 Preliminaries 1
2 Function spaces and functionals 9
2.1 Ordered sets and lattices 9
2.2 The spaces JRX and i:X 17
2.3 Vector lattices of functions 23
2.4 Functionals 30
2.5 Daniell spaces 46
3 Extension of Daniell spaces 49
3.1 Upper functions 50
3.2 Lower functions 53
3.3 The closure of (X,£, f) 55
3.4 Convergence theorems for (X, l(f), f) 62
3.5 Examples 69
3.6 Null functions, null sets and integrability 75
3.7 Examples 86
3.8 The induction principle 88
3.9 Functionals on JRX 91
3.10 Summary 98
4 Measure and integral 101
4.1 Extensions of positive measure spaces 102
4.2 Examples 120
4.3 Locally integrable functions 124
vi Contents
4.4 JL-measurable functions 128
4.5 Product measures and Fubini's theorem 139
5 Measures on Hausdorff spaces 159
5.1 Regular measures 159
5.2 Measures on metric and locally compact spaces 176
5.3 The congruence invariance of the n-dimensional Lebesgue
measure 188
6 J:,P -spaces 195
6.1 The structure of .CP -spaces 195
6.2 Uniform integrability 209
7 Vector lattices, LP-spaces 221
7.1 Vector lattices 221
7.2 £P-spaces 235
8 Spaces of measures 241
8.1 The vector lattice structure and Hahn's theorem 241
8.2 Absolute continuity and the Radon-Nikodym theorem 255
9 Elements of the theory of real-valued functions on R 263
9.1 Functions of locally finite variation 263
9.2 Absolutely continuous functions 268
Symbol index 283
Subject index 287
Preface
This book contains the material from an introductory course on integration
theory taught at ETH (the Swiss Federal Institute of Technology) in Zurich.
Students taking the course are in their third or fourth year of tertiary
studies and therefore have had substantial prior exposure to mathematics.
The course assumes some familiarity with the concepts presented in the
preceding courses.
Since this book is addressed to a wider audience and since different in
stitutes have different programmes, the same assumptions cannot be made
here. As explaining everything in detail would have resulted in a book of
daunting dimensions, whose very size would discourage all but those of epic
heroism and dedication, we have chosen a compromise: we explain in detail
in the text itself only those ideas which are essential to the development
of the subject matter and we have appended a separate glossary of all def
initions used, adding explanations and examples as needed. The reader is,
however, expected to be familiar with the basic properties of the Riemann
integral as well as with basic facts from point-set topology; the latter are
especially needed for Chapter 5, 'Measures on Hausdorff Spaces'.
We have chosen this course in order to preserve the character of an intro
duction at an intermediate level, which should nevertheless be accessible to
those with limited prior knowledge, who are willing to postpone questions
on matters not central to the development of the theory.
Years of experience have convinced us of the importance to the student
of active learning, especially when confronted by new concepts from a still
unfamiliar theory. We have therefore included a large number of exercises.
These place less emphasis on originality and the majority of them have
been kept elementary, so that even an average student should be able to
complete them successfully.
The concept of the integral developed in this book is 'the optimal' one in
the sense explained in Integration Theory I by C. Constantinescu and K.
Weber (Wiley-Interscience, New York, 1985). (We shall occasionally point
viii Preface
readers who are interested in further pursuing a deep analysis to this book
and refer to it as [CW].) This concept of the integral was proposed by I.E.
Segal and R.A. Kunze in their book Integrals and Operators (McGraw-Hill,
New York, 1968), but they did not develop the theory. It results in a larger
class of integrable functions than the 'usual' integrals in general - it co
incides with them in the a-finite case - and unifies abstract integration
theory with the theory of integration on Hausdorff spaces. In fact, in the
context of this theory, the latter simply becomes a special case of the ab
stract theory. We also mention that in the topological case, we arrive at
Bourbaki's 'essential integral'.
There is methodical emphasis on the structural perspective. The fun
damental notion for our approach is that of a vector lattice. Naturally,
the basic properties of vector lattices which are needed in this book are
discussed in detail. For the sake of perspicuity, we first discuss vector lat
tices of real-valued functions, which is completely adequate for the first
chapters. Our discussion later moves to the abstract framework needed for
an adequate account of, for example, LP-spaces and the Radon-Nikodym
theorem.
We wish to express our deeply felt gratitude to our teacher, Professor
Corneliu Constantinescu. His enthusiasm for the theory of measure and
integration and his encompassing knowledge have inspired us and left their
indelible stamp upon us. We hope that we have succeeded in writing this
book in his spirit.
Our sincere thanks also go to Imre Bokor for his excellent translation
of the original German text and to Helmut Koditz for his expert help in
producing the :9-'JEX files.
Finally, we would like to thank our publishers, Chapman & Hall, for their
ever congenial cooperation.
Introduction
Many mathematical problems have contributed to the development of meas
ure and integration theory. One of the most significant of these is known
as the measure problem in ]Rn.
In essence, the problem is to assign to each element A of a sufficiently
broad set of subsets of JRn a positive number - the content or measure
of A - in such a way that several obvious conditions are satisfied:
(i) The content of the union of two disjoint sets is the sum of their
individual contents.
(ii) If A is a subset of B, then the content of A does not exceed that of
B.
(iii) Congruent sets have the same content.
Systems of sets for which such a content can be defined are not difficult
to find. Fix an orthogonal coordinate system in IRn. Then the set 9t of all
n-dimensional rectangular prisms whose faces are parallel to the coordinate
axes has this property. Recall that such a rectangular prism is the Cartesian
n;=l
product of closed intervals, that is, it is of the form [ak,.Bk], where
ak ~ .Bk for every k E {1, ... , n }. Note that because of our requirements,
for each content A on 9t there must be a number '"'/ 2:: 0 such that
IT
A(
='"'/IT
[ak,.Bk]) (.Bk- ak),
k=l k=l
and conversely, given any number '"'/ 2:: 0, this formula defines a content on
!Jt. Moreover, '"'/ > 0 is the only interesting case, and if we insist that the
n;=
unit cube [0, 1] have content 1, then'"'! must be 1. We shall use this
1
normalization henceforth.
More generally, we may consider sets of JRn which are unions of finitely
many sets of !Jt. Each such set A may be partitioned into a finite number of
rectangular prisms of 9t which have at most boundary points in common.