Foundations of abstract analysis

754 Pages·2013·3.5 MB·English

by  Jewgeni H Dshalalow

Foundations of Abstract Analysis

Jewgeni H. Dshalalow
Foundations of Abstract
Analysis
Second Edition
Jewgeni H. Dshalalow
Mathematical Sciences
Florida Institute of Technology
Melbourne 32901
Florida
USA
ISBN978-1-4614-5961-3 ISBN9 78-1-4614-5962-0 (eBook)
DOI10.1007/978-1-4614-5962-0
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012951666
Mathematics Subject Classification (2010): 54-01, 28A05, 28A10, 28A12, 28A20, 28A25, 28A33,
28A35, 26-XX, 26-01
2nd Ed of a book previously published by Chapman and Hall/CRC Press in 2000, Real Analysis: An
Introduction to the Theory of Real Functions and Integration
© Springer Science+Business Media, LLC 2013
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or
information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purpose of being entered and executed on a computer system, for exclusive use by the purchaser of
the work. Duplication of this publication or parts thereof is permitted only under the provisions of the
Copyright Law of the Publisher’s location, in its current version, and permission for use must always
be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright
Clearance Center. Violations are liable to prosecution under the respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the
relevant protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of
publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for
any errors or omissions that may be made. The publisher makes no warranty, express or implied, with
respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)
To Irina

Preface
This is the first of a two book series written for beginning graduate-
level real analysis students and it is focused on essentials of set theo-
ry, topology, and measure theory. The word “essentials” is often as-
sociated with lecture notes, but this is exactly opposite to my inten-
tion. While there are quite a few fine abstract analysis books, they
are difficult to read, and an enthusiastic reader needs more time and
effort to master the subject, as opposed to elementary level books
that only in part address the needs of forthcoming mathematics and
advancedengineeringcourses.
That is why this book offers a thorough and yet rigorous treat-
ment of key analysis subjects in abstract spaces by providing the
reader with copious illustrations, examples, and exercises with se-
lected solutions. Some difficult to understand topics are preceded by
detaileddiscussions andblueprints, so that thereaderwill not get lost
or intimidated in a long chain of proofs and notions. Furthermore, at
the end of each section there is a summaryof new terms and notation
in the chronological order in which they appeared in the text. This
should be helpful and relieve students of the potential overload of
new words, definitions, and concepts. So, there is an increased likeli-
hood of success byusingthis text in a course or bymeans of individ-
ual self-study.
Myprevious text, Real Analysis, published byChapman and Hall
in 2000, was the first effort to createthiskindofbook.However,this
book only partially accomplished the goal I was striving to achieve.
To fully realize that goal, it was necessary to write a new and ex-
panded edition, including more topics and details, and it had to be
produced as two books. The companion book, Advances in Abstract
Analysis and Applications, includes further topics in topology and
vii
viii Preface
measure theory, which justifies and rewards the reader for investing
thetimespent on“essentials.”
As mathematical education has become increasingly more fo-
cused on applications and less on theory, and in order to save them
from extinction, academics have repurposed courses in set theory, to-
pology, abstract algebra, and measureandintegrationas areal analy-
sis course. At the same time, mathematical research, driven by seri-
ous applications to other sciences, continued to require sound foun-
dations. The pertinent precedents include physics, stochastic finance,
mechanics, and now biology. There was even a time when some pro-
ponents called real analysis “the single most important graduate
courseinmathematics toprepareforacareerinoperations research.”
Today, real analysis is still very much alive, although it has un-
dergone some significant modifications. One of these changes is that
contemporary real analysis books include various, sometimes exotic,
applications ranging from partial differentials equations to wavelet
analysis, probability, and even physics. While such connections
might be justified, one has to ensure that this propensity to connect
analysis with remote disciplines does not relax its very substance.
Consequently, facinga challenge of two alternatives to yieldanover-
sized or abridged book, a broad-spectrum project (under strong en-
couragement from Springer) got bifurcated into two entirely differ-
entlyfocusedtexts.
Because real analysis, in its proper form, is likely to be the first
abstract mathematics course that many students take, the associated
topics should be taught in a strict order starting with basic set theory
followed by point-set topology and then measure theory and integra-
tion. Throughout my book I follow these principles. I strongly advo-
cate the idea of introducing measure and integration in abstract
spaces wasting no valuable time on Euclidean spaces. Consequently,
Lebesgue measure and Lebesgue integral are reduced to mere illus-
trations. The topology part, to be necessarily preceded by metric
spaces, contains mostly fundamentals (such as bases, subbases,
Hausdorff spaces, Tychonov product, and compactness). In partic-
ular, old good sequential convergence is enough to proceed with rig-
orous and comprehensive measure and integration (which accounts
Preface ix
to over 55% of my book). Such topics as filters and nets, locally
compact Hausdorff spaces, Radon measures, and Hilbert spaces
(measure-theoretic version) I consider relatively advanced and there-
fore treat them in my forthcoming sequel to this book, along with
various applications tostochasticanalysis.
It is absolutely impossible to produce a sound text without
building on the foundations of my predecessors' important scholar-
ship. I am very grateful for the valuable remarks and suggestions,
made to the present and earlier edition, by Gustave Choquet, Jerald
Folland, Jordan Stoyanov, Jürgen Becker, Richard Syski, Jean
Lasserre, Donald Konwinski, and Dean Spitzer. I am much indebted
to Simon Smith, the creator of the EXP word processor, for his gen-
erous and timelysupport.
Mydeep appreciationalso goes toMs.Vaishali Damle,whois an
extremely competent and efficient mathematics editor. Her constant
help and patience was made available from the verybeginning of the
project and receiving her feedback was one of the key benefits of
writing the book. Finally, I want to thank the Springer copyeditor,
Ms. Valerie Greco, who did an excellent job in polishing the final
draft of my manuscript. Because books are often written at home, I
would be remiss in not ending with profound gratitude to my wife,
Irina, who createdanenvironment conducivetowritingthis book.
J.H.Dshalalow
Melbourne,Florida