Table Of ContentAn Introduction to Measure
and Probability
Springer Science+Business Media, LLC
J.C. Taylor
An Introduction to Measure
and Probability
With 12Illustrations
, Springer
J.c. Taylor
Department of Mathematics and Statistics
McGill University
805 Sherbrooke Street West
Montreal, Quebec H3A 2K6
Canada
Library of Congress Cataloging-in-Publication Data
Taylor, J.C. (John Christopher), 1936-
An introduction to measure and probability/J.C. Taylor.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-387-94830-0 ISBN 978-1-4612-0659-0 (eBook)
DOI 10..1007/978-1-4612-0659-0
1. Measure theory. 2. Probabilities. 1. Title.
QA325.T39 1996
519.2-dc20 96-25447
Printed on acid-free paper.
© 1997 Springer Science+Business Media New York
OriginaJlypublished by Springer-Verlag New York in 1997
All rights reserved. This work may not be translated or copied in whole or in part without the
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Production managed by Hal Henglein; manufacturing supervised by Jeffrey Taub.
Camera-ready copy prepared from the author's A.MS-1EX files.
9 8 7 6 5 4 3 2 (Corrected second printing. 1998)
ISBN 978-0-387-94830-0
I DEDICATE THIS BOOK TO ALL MY TEACHERS
IN GRATITUDE AND THANKFULNESS
Teach me Your way, 0 Lord
and I will walk in Your truth.
Psalm 86, verse 11
PREFACE
The originalversionofthis bookwasan attempt to present the basicsof
measure theory, as it relates to probability theory, in a form that might be
accessibletostudentswithout the usual background inanalysis. Thissetof
notes was intended to be a "primer" for measure theory and probability
across betweenaworkbookanda text- ratherthanastandard "manual".
They were intended to give the reader a "hands-on" understanding ofthe
basic techniques of measure theory together with an illustration of their
use in probability as illustrated by an introduction to the laws of large
numbers and martingale theory. The sections ofthe first five chapters not
marked by an asterisk are the current version of the original set of notes
and constitute the essential core ofthe book.
Since the main goal of the core of this book is to be a "fast-track",
self-contained approach to martingale theory for the reader without a solid
technical background, manydetailsare put intoexercises, which the reader
is advised to do ifa basic mastery ofthe subject is the goal.
Thepresentbookretainsthebasicpurposeandphilosophyoftheoriginal
set of notes but has been expanded to make it ofmore use to statisticians
by adding a sixth chapter on weak convergence, which is an introduction
to this very large circle ofideas as well as to the central limit theorem.
In addition, having in mind its possible use by students of analysis,
severalsections in thisbook branchofffrom thecentralcoretoexaminethe
Radon-Nikodym theorem, functions of bounded variation, Fourier series
(very briefly), and differentiation theory and maximal functions. Measure
theory for a-finite measures reduces in many respects to the studyoffinite
measures and hence (after normalization) to the study of probabilities.
From this point ofview, this book canserve as an introduction to measure
theory.
Each chapter, except for Chapter V, concludes with a section of addi
tional exercises in which various ancillary topics are explored. They may
be omitted on first reading and are not used in the coreofthis text. Much
ofthe materialin theexercisesmay befound in thestandard textsonmea
sure theory or probability that are cited. The author is ofthe opinion that
it is more useful to fill in details for many ofthese items than to read com
pleted arguments but is well aware that it is more time-consuming! Note
that the additional exercises and other sections marked by asterisks may
be omitted without prejudice to understanding the core ofthese notes.
There aresix chapters: the first two chapters constitute the basic intro
duction; and the remaining four explain some uses ofmeasure theory and
vii
viii PREFACE
integration in probability. They have been written to be "self-contained",
and as a result, some background information from analysis is scattered
throughout. While this may no doubt appear to be excessive to the "well
prepared" reader, itismyhopethat withthisapproachthetechnicaldetails
ofthe subject will be made accessible to the "well-motivated" reader who
may not have all the appropriate background. In other words, I hope to
empower readers who wish to learn the subject but who have not come to
this point by a standard route. As in many things, a user-friendly atti
tude seems to me to be very desirable. Whether these notes exhibit that
is obviously up to you, the reader, to judge.
Asstated before, the main purposeofthese notes is to explain and then
make use of the main tools and techniques of the theory of Lebesgue in
tegration as they apply to probability. To begin, Chapter I shows how to
construct a probability measure on the a-algebra s.B(IR) of Borel subsets
of the real line from a distribution function F. Then Chapter II defines
integration with respect to a probability measure and proves the main the
orems ofintegration theory (e.g., the theorem ofdominated convergence).
Also, it defines Lebesgue measure on the real line and explains the rela
tion between the Lebesgue integral and the Riemann integral. These two
chapters (excepting for §2.7 and §2.8 on the Radon-Nikodym theorem and
functions of bounded variation, respectively) constitute the essential part
of the central core of the book, and the reader is advised to be familiar
with them before proceeding further.
Chapter III is concerned with independence and product measures. It
dealsmainlywithfinite products, although theexistenceofcountableprod
ucts of probability measures is proved following ideas of Kakutani (and
independently Jessen) as an illustration of the use of measure-theoretic
techniques. This is the first technically difficult result in these notes, and
the reader can skip the details (as indicated) without any loss ofcontinu
ity. The extension of these ideas to prove the existence of Markov chains
associated with transition kernels on the real line is made in §3.5.
Chapter IV discusses the main convergence concepts for random vari
ables (Le., measurable functions) and proves Khintchine's weak law and
Kolmogorov's strong law of large numbers for sums of independent, iden
tically distributed (Li.d.) integrable random variables. The chapter closes
with a discussion of uniform integrability and truncation.
The last chapter of the original version, Chapter V, covers conditional
expectation and martingales. While sufficient statistics are discussed in
§5.3 as an applicationofconditional expectation, the bulkofthe chapter is
an introduction to martingales. Doob's optional stopping theorem (finite
case) andthemartingaleconvergencetheorem (fordiscretemartingales) are
proved. The chapter concludes with Doob's proofofthe Radon-Nikodym
theorem on countably generated a-algebras and the backward martingale
proofofKolmogorov's strong law.
The final chapter, ChapterVI, examines weakconvergence and the cen-
PREFACE ix
tral limit theorem. It is independent of Chapter V and can be read af
ter having worked through §4.1 and §4.3 of Chapter IV together with the
beginning of §4.2. Tightness is emphasized, and an attempt is made to
enhance the mathematical focus ofthe standard discussions on weak con
vergence. The discussion ofuniqueness and the continuity theorem for the
characteristicfunction follows Feller'sand Lamperti'sapproachofusingthe
Gaussian or normal densities. The central limit theorem is first discussed
in the Li.d. case and concludes with the case of a sequence or triangular
array satisfying the Lindebergh condition.
The reader may find the references to standard texts and sources on
probability distracting. In that case, please ignore them. They are there
to encourage one to look at and explore other treatments as well as to ac
knowledge what Ihave learned from them. Inaddition, ifyouarecurrently
using one of them as a text, it may be helpful to have some connections
pointedout. Referencesare mainlyto bookslisted in the Bibliography, and
the occasional reference to an article is done with a footnote. The book
by LOEwe [L2] has many references in it and will be helpful if the reader
wants to explore the classical literature. For information and references on
martingales, Doob's book [Dl] has lots of historical notes in an appendix
and an extensive bibliography. Since these notes emphasize the connection
between analysis and probability, the reader wishing to go further into the
subject should consult the recent book [S2] by Stroock, which has a decid
edly analytic flavour. Finally, for a more abstract version of many of the
topics covered in these notes and for detailed historical notes, the book by
Dudley [D2] is recommended.
For several years since 1986, versions of these notes have been used
at McGill University (successfully, from my perspective) to teach a one
semester course in probability to a mixed audience of statisticians and
electrical engineers. Dependingon the preparationofthe class, most ofthe
original version of this book was covered in a one-semester course. These
notes have also been used to teach an introductory course on measure the
ory to honours students (in this case, the course was restricted to the first
four chapters with the sections on infinite product measures and Markov
chains omitted as well as §4.5 and §4.6).
I would like to thank all the students who have (unwittingly) partici
pated in the process that has led to the creation of this set of notes; my
wife, Brenda MacGibbon of UQAM, for her enthusiastic encouragement,
my colleagues Minggao Gu and Jim Ramsay ofMcGill University and Ah
mid Bose of Carleton University for their support and encouragement, as
well as Regina Liu of Rutgers and D. Phong of Columbia University. I
thank Jal Choksi ofMcGill University and Miklos Cs6rg6ofCarleton Uni
versity for useful conversations on various points. In addition, I wish to
thank all my colleagues and friends who have helped me in many, many
ways over the years. I also wish to thank Mr. Masoud Asgharian, Mr.
Jun Cai, and Ms. Gulhan Alpargu for their diligent search for misprints
x PREFACE
and errors, and Mr. Erick Lamontagne for help in final editing. Finally,
I also wish to thank my copy editor, Kristen Cassereau, and my editor at
Springer, John Kimmel, for their patience and professional help.
J.C. Taylor
CONTENTS
Preface vii
List ofSymbols xiii
Chapter I. Probability Spaces 1
1. Introduction to IR 1
2. What is a probability space? Motivation 5
3. Definition ofa probability space 9
4. Construction ofa probability from a distribution
function 16
5. Additional exercises* 26
Chapter II. Integration 29
1. Integration on a probability space 29
2. Lebesgue measure on IR and Lebesgue integration 44
3. The Riemann integral and the Lebesgue integral 49
4. Probability density functions 55
5. Infinite series again 58
6. Differentiation under theintegral sign 59
7. Signed measures and the Radon-Nikodym theorem* 60
8. Signed measures on IR and functions ofbounded
variation* 71
9. Additional exercises* 78
Chapter III. Independence and Product
Measures 86
1. Random vectors and Borel sets in IRn 86
2. Independence 89
3. Product measures 95
4. Infinite products 110
5. Some remarks on Markov chains* 119
6. Additional exercises* 131
Chapter IV. Convergence of Random Variables
and Measurable Functions 137
1. Norms for random variables and measurable functions 137
2. Continuous functions and LP* 149
3. Pointwise convergence and convergence in measure or
probability 167
4. Kolmogorov's inequality and the strong law of large
oom~m lW
5. Uniform integrability and truncation* 180
xi
