Measure, Integral and Probability

229 Pages·1999·4.637 MB·English

by  Marek Capiński, Ekkehard Kopp

Springer U ndergraduate Mathematics Series
Springer-Verlag London Ltd.
Advisory Board
Professor P.J. Cameron Queen Mary and Westfieid College
Dr M.A.]. Chaplain University ofDundee
Dr K. Erdmann Oxford University
Professor L.c.G. Rogers University ofBath
Dr E. Süli Oxford University
Professor J.F. Toland University ofBath
Other books in this series
Basic Linear Algebra T.S. Blyth and B.P. Robertson (3-540-76122-5)
Elements ofLogic via Numbers and Sets D.L.Iohnson (3-540-76123-3)
Multivariate Calculus and Geometry S. Dineen (3-540-76176-4)
Elementary Number Theory G.A. Iones and I.M. Iones (3-540-76197-7)
Introductory Mathematics: Applications and Methods G.S. Marshall (3-540-76179-9)
Vector Calculus P.C. Matthews (3-540-76180-2)
Introductory Mathematics: Algebra and Analysis G. Smith (3-540-76178-0)
Groups, Rings and Fields D.A.R. Wallace (3-540-76177-2)
Marek Capinski and Ekkehard Kopp
Measure, Integral and
Probability
With 23 Figures
, Springer
Marek Capinski, Dr hab, PhD
Nowy Sacz Graduate School ofBusiness, 33-300 Nowy Sacz, ul.Zielona 27, Poland
Peter Ekkehard Kopp, DPhil
Department of Mathematics, U niversity of H ull, Cottingham Road, H ull HU 6 7RX, UK
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ISBN 978-3-540-76260-7
British Library Cataloguing in Publieation Data
CapiiIski, Marek
Measure, integral and prob ability. - (Springer undergraduate mathematics series)
1. Lebesque integral 2. Measure theory 3. Probabilities I. Title 11. Kopp, Ekkehard
515.4'3
ISBN 978-3-540-76260-7
Library of Congress Cataloging-in-Publication Data
CapiiIski, Marek, 1951-
Measure, integral, and prob ability I Marek CapiiIski and Ekkehard Kopp.
p. cm. -- (Springer undergraduate mathematies series) Includes index.
ISBN 978-3-540-76260-7 ISBN 978-1-4471-3631-6 (eBook)
DOI 10.1007/978-1-4471-3631-6
1. Measure theory. 2. Integrals, Generalized. 3. Probabilities.
I. Kopp, P.E., 1944- . 11. Tide. IIl. Series.
QA312.C36 1999 98-34763
515'.42--dc21 CIP
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Preface
The central concepts in this book are Lebesgue measure and the Lebesgue
integral. Their role as standard fare in UK undergraduate mathematics courses
is not wholly secure; yet they provide the principal model for the development of
the abstract measure spaces which underpin modern probability theory, while
the Lebesgue function spaces remain the main sour ce of examples on which
to test the methods of functional analysis and its many applications, such as
Fourier analysis and the theory of partial differential equations.
It follows that not only budding analysts have need of a clear understanding
of the construction and properties of measures and integrals, but also that those
who wish to contribute seriously to the applications of analytical methods in
a wide variety of areas of mathematics, physics, electronics, engineering and,
most recently, finance, need to study the underlying theory with some care.
We have found remarkably few texts in the current literature which aim
explicitly to provide for these needs, at a level accessible to current under
graduates. There are many good books on modern prob ability theory, and
increasingly they recognize the need for a strong grounding in the tools we
develop in this book, but all too often the treatment is either too advanced for
an undergraduate audience or else somewhat perfunctory. We hope therefore
that the current text will not be regarded as one which fills a much-needed gap
in the literat ure!
One fundamental decision in developing a treatment of integration is
whether to begin with measures or integrals, i.e. whether to start with sets or
with functions. Functional analysts have tended to favour the latter approach,
while the former is clearly necessary for the development of prob ability. We
have decided to side with the probabilists in this argument, and to use the
(reasonably) systematic development of basic concepts and results in proba
bility theory as the principal field of application - the order of topics and the
v
VI Preface
terminology we use reflect this choice, and each chapter concludes with further
development of the relevant probabilistic concepts. At times this approach may
seem less 'efficient' than the alternative, but we have opted for direct proofs
and explicit constructions, sometimes at the cost of elegance. We hope that it
will increase understanding.
The treatment of measure and integration is as self-contained as we could
make it within the space and time constraints: some sections may seem too
pedestrian for final-year undergraduates, but experience in testing much of the
material over a number of years at Rull University teaches us that familiar
ity and confidence with basic concepts in analysis can frequently seem some
what shaky among these audiences. Rence the preliminaries include a review
of Riemann integration, as well as areminder of some fundamental concepts of
elementary real analysis.
While prob ability theory is chosen here as the principal area of application
of measure and integral, this is not a text on elementary prob ability, of which
many can be found in the literat ure.
Though this is not an advanced text, it is intended to be studied (not
skimmed lightly) and it has been designed to be useful for directed self-study
as well as for a lecture course. Thus a significant proportion of results, labelled
'Proposition', are not proved immediately, but left for the reader to attempt
before proceeding further (often with a hint on how to begin), and there is
a generous helping of Exercises. To aid self-study, proofs of the Propositions
are given at the end of each chapter, and outline solutions of the Exercises are
given at the end of the book. Thus few mysteries should remain for the diligent.
After an introductory chapter, motivating and preparing for the principal
definitions of measure and integral, Chapter 2 provides a detailed construction
of Lebesgue measure and its properties, and proceeds to abstract the axioms ap
propriate for prob ability spaces. This sets a pattern for the remaining chapters,
where the concept of independence is pursued in ever more general contexts,
as a distinguishing feature of prob ability theory.
Chapter 3 develops the integral for non-negative measurable functions, and
introduces random variables and their induced probability distributions, while
Chapter 4 develops the main limit theorems for the Lebesgue integral and com
pares this with Riemann integration. The applications in prob ability lead to a
discussion of expectations, with a focus on densities and the role of character
istic functions.
In Chapter 5 the motivation is more functional-analytic: the focus is on the
Lebesgue function spaces, including a discussion of the special role of the space
L2 of square-integrable functions. Chapter 6 sees areturn to measure theory,
with the detailed development of product measure and Fubini's theorem, now
leading to the role of joint distributions and conditioning in probability. Finally,
Preface VII
following a discussion of the principal modes of convergence for sequences of
integrable functions, Chapter 7 adopts an unashamedly probabilistic bias, with
a treatment of the principallimit theorems, culminating in the Lindeberg-Feller
version of the Central Limit Theorem.
The treatment is by no means exhaustive, as this is a textbook, not a
treatise. Nonetheless the range of topics is probably slightly too extensive for
a one-semester course at third-year level: the first five chapters might provide
a useful course for such students, with the last two left for self-study or as
part of a reading course for students wishing to continue in prob ability theory.
Alternatively, students with astronger preparation in analysis might use the
first two chapters as background material and complete the remainder of the
book in a one-semester course.
May 1998 Marek Capinski
Ekkehard Kopp
Contents
1. Motivation and preliminaries ............................... 1
1.1 Notation and basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Countable and uncountable sets in IR ................. 4
1.1.3 Topological properties of sets in IR . . . . . . . . . . . . . . . . . . . . 5
1.2 The Riemann integral: scope and limitations . . . . . . . . . . . . . . . .. 7
1.3 Choosing numbers at random . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
2. Measure.................................................... 15
2.1 Null sets ................................................ 15
2.2 Outer measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
2.3 Lebesgue measurable sets and Lebesgue measure ............. 27
2.4 Basic properties of Lebesgue measure .. . . . . . . . . . . . . . . . . . . . .. 35
2.5 Borel sets ............................................... 40
2.6 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
2.6.1 Probability space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
2.6.2 Events: conditioning and independence . . . . . . . . . . . . . . .. 46
2.7 Proofs of propositions ..................................... 49
3. Measurable functions ....................................... 53
3.1 The extended realline .................................... 53
3.2 Definition............................................... 53
3.3 Examples................................................ 57
3.4 Properties............................................... 58
3.5 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
3.5.1 Random variables .................................. 64
ix
x Contents
3.5.2 Sigma fields generated by random variables ............ 65
3.5.3 Probability distributions ............................ 66
3.5.4 Independence of random variables .................... 68
3.6 Proofs of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
4. Integral..................................................... 71
4.1 Definition of the integral .................................. 71
4.2 Monotone Convergence Theorems .......................... 78
4.3 Integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
4.4 The Dominated Convergence Theorem ...................... 87
4.5 Relation to the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
4.6 Approximation of measurable functions ..................... 97
4.7 Probability .............................................. 100
4.7.1 Integration with respect to probability distributions .... 100
4.7.2 Absolutely continuous measures: examples of densities .. 102
4.7.3 Expectation of a random variable ..................... 109
4.7.4 Characteristic function .............................. 110
4.8 Proofs of propositions ..................................... 111
5. Spaces of integrable functions ............................... 115
5.1 The space LI ............................................ 116
5.2 The Hilbert space L2 ..................................... 121
5.2.1 Properties of the L2-norm ........................... 122
5.2.2 Inner product spaces ................................ 124
5.2.3 Orthogonality ...................................... 127
5.3 The LP spaces: completeness ............................... 128
5.4 Probability .............................................. 134
5.4.1 Moments .......................................... 134
5.4.2 Independence ...................................... 138
5.5 Proofs of propositions ..................................... 141
6. Product measures .......................................... 145
6.1 Multi-dimensional Lebesgue measure ........................ 145
6.2 Product a-fields .......................................... 146
6.3 Construction of the product measure ........................ 148
6.4 Fubini's Theorem ......................................... 155
6.5 Probability .............................................. 159
6.5.1 Joint distributions .................................. 159
6.5.2 Independence again ................................. 161
6.5.3 Conditional probability ............................. 163
6.5.4 Characteristic functions determine distributions ........ 165
6.6 Proofs of propositions ..................................... 168
Contents XI
7. Limit theorems ............................................. 171
7.1 Modes of convergence ..................................... 171
7.2 Probability .............................................. 173
7.2.1 Convergence in prob ability .......................... 175
7.2.2 Weak law of large numbers .......................... 179
7.2.3 Borel-Cantelli lemmas .............................. 184
7.2.4 Strong law of large numbers ......................... 188
7.2.5 Weak convergence .................................. 196
7.2.6 Central Limit Theorem ............................. 201
7.3 Proofs of propositions ..................................... 208
8. Solutions to exercises ....................................... 209
9. Appendix ................................................... 219
References ...................................................... 223
Index ........................................................... 225