Real Analysis: An Introduction to the Theory of Real Functions and Integration

582 Pages·2000·38.65 MB·English

by  David Mond

JEWGENI H. DSHALALOW
Real Analysis
An Introduction to
the Theory of
Real Functions
and Integration
I
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CHAPMAN & HALLICRC
Studies in Advanced Mathematics
Series Editor
STEVEN G. KRANTZ
Washiltgtorl University St. Louis
Editorial Board
R. Michael Beals Gerald B. Folland
University of Washington
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University of California at San Diego
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University of Kansas
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Lawrence C. Evans Michael E. Taylor
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H.
JEWGENZ DSHALALOW
Analysis
An Introduction to
the Theory of
Real Functions
and Integration
CHAPMAN & HALUCRC
Boca Raton London New York Washington, D.C.
Library of Congress Catalogingin-PublicationD ata
Dshalalow, Jewgeni H.
Real analysis : an introduction to the theory of real functions and integration / Jewgeni
H. Dshalalow.
--
p. cm. (Studies in advanced mathematics)
Includes bibliographical references and index.
ISBN 1-58488-073-2 (alk. paper)
1. Mathematical analysis. I. Title. 11. Series.
2. Biology-molecular. I. McLachlan, Alan. 11. Title.
QA300 .D742 000
5 15--dc2 1 00-058593
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To my Lord and Redeemer
Who made the supreme
sacrifice for me
and Who will come again

Preface
This book is intended to be an introductory two-semester course in abstract analy-
sis, which includes topology, measure theory, and integration, traditionally staff-
ing an assemblage of topics under the cognomen "Real Analysis," more common
in the United States. Most North American schools offer this as a graduate one- to
two-semester course for mathematics, physics, and engineering majors. Many
European schools, to the best of my knowledge, do not have such a course; they
have instead a sequence of separate courses such as Topology, Measure and Inte-
gration, and Functional Analysis. In some countries, such as Russia and former
Soviet Republics, they, additionally, have a Real Variables course, which is
somewhat similar to Real Analysis but is more specialized, and, its profile and
rigor vary fiom college to college.
A very good reason for learning real analysis is that not only is it a core
course for all mathematical disciplines, but it is absolutely mandatory for statistics
and probability, operations research, physics, and some engineering majors as
well. Hence, rephrasing an old adage, all routes of science and technology go
through real analysis.
This text predominantly targets the first year graduate students of mathemat-
ical science majors as well as the first and second year graduate students of engi-
neering, physics, and operations research majors. A stronger senior undergraduate
mathematics student can also benefit fiom the course. Some less theoretically
oriented programs or those with weaker mathematics course curricula may find it
reasonable to use the book for a three-semester course: with the first two semes-
ters of basics and the third semester of advanced topics. The course can always be
shortened to two semesters in such schools with the option to cover the fust seven
chapters, which are also quite sufficient for technical majors.
This book is destined primarily as a textbook and its purpose as a reference
is secondary. The reason for such a claim is a rather thorough elaboration of ma-
jor theorems, notions, and constructions, very often supplied with a blueprint and
sometimes a less formal introduction. The latter are then succeeded by detailed
treatments. For instance, the Radon Nikodym Theorem is first introduced in
Chapter 6, with a minimum of proofs and formalities, but with a number of exam-
ples and exercises. Then it is followed by a more abstract version later, in Chapter
8.
vii
PREFACE viii
The first three chapters of the book (Part I) include preliminaries on sets
theory and basics of metric spaces and topology. I have been using these three
chapters for the many years teaching a bilevel topology course at Florida Tech
during our quarter system. However, I would not be able to cover the present
version of the three chapters in one quarter, and one semester would be a more
appropriate term for the current program at our school. Hence, the first three
chapters can easily serve as a separate one quarter to one semester topology se-
nior undergraduate or beginning graduate course.
Chapters 4-7 (Part II) present basics of measure and integration and, again,
they can be offered as a separate measure theory (and integration) course. Con-
sequently, Parts I and II can become appealing to those programs with separate
named courses and, in particular, to European students. Part III (Chapters 8 and 9)
includes a more elaborate and abstract version of measure and integration, along
with their applications to functional analysis (LP spaces and Riesz Representation
Theorem for locally compact Hausdorff spaces), probability theory (conditional
expectation, uniform integrability, Lebesgue-Stieltjes integrals, decomposition of
distribution functions, stochastic convergence, and convergence of Radon mea-
sures), and conventional analysis on the real line (monotone and absolutely con-
tinuous functions, functions of bounded variations, and major theorems of calcu-
lus). Part 111 can be utilized for advanced topics, as well as an enlarged variant of
measure and integration. While the reader would be better off to have studied Part
I prior to Part II and the first six sections of Chapter 8, the latter can also be used
as an independent material with sufficient basics of topology drawn from any
generic advanced analysis course.
The book can also be used as a reference source for researchers in mathe-
matical and engineering sciences, and especially, operations research (such as
applied stochastic processes, queueing theory, and reliability). The reader should
understand, however, that the book is not intended to become an encyclopedia of
mathematics or to be any kind of a broad reference. I had to suppress my tempta-
tion to include some written chapters on Hilbert spaces, functional analysis, and
Fourier transforms, because of my motives to compile main topics of what consti-
tutes the real.analysis and to design a text by spending more time on details (with-
in the frameworks of the book size imposed by the publisher and buyers' afford-
ability). This text may be well suited for independent studies with or without in-
structors for which an abundance of examples and over 600 exercises provide a
pertinent support. While a solution manual is in preparation and will become
available soon (and it would be an additional studying aid), the publisher and I
have agreed on honoring only university instructors with this manual upon adop-
tion of the book for the course. The reader may also find the new terms subsect-
ions (at the end of each section) useful, especially considering a plethora of new
definitions and notations, which not only can be intimidating, but they can create
an additional memory burden and thereby slow down learning of the main
concepts.
PREFACE
Most of my thanks are due to my wife Irina for her ample support, encour-
agements, and overwhelming sacrifice. I would like to express my deep apprecia-
tion to Mr. Jiirgen Becker, for his constant guidance and countless ideas, Mr.
Donald Konwinski for his enormous editorial work on earlier versions of my
manuscript, Professors Gerald B. Folland and Ryszard Syski for their numerous
and very constructive remarks, as well as the kind assistance of Professors S.G.
Deo, Jean-B. Lassere, Jordan Stoyanov, Mr. Gary Russell, the project editor, Mr.
David Alliot, and anonymous reviewers who thoroughly read my manuscript and
made many helpful suggestions. My thanks are also due to the publisher, Mr.
Robert Stern for his help and extreme patience.
Jewgeni H. Dshalalow
Melbourne, Florida