Measure and integral : an introduction to real analysis

524 Pages·2015·4.72 MB·English

by  Wheeden

Mathematics
Measure and Integral
M
Now considered a classic text on the topic, Measure and Integral: An
e
Introduction to Real Analysis provides an introduction to real analysis by
a
An Introduction to Real Analysis
first developing the theory of measure and integration in the simple setting s
of Euclidean space, and then presenting a more general treatment based on u
r
abstract notions characterized by axioms and with less geometric content. e Second Edition

Published nearly forty years after the first edition, this long-awaited Second a
n
Edition also:
d
• Studies the Fourier transform of functions in the spaces L1, L2, and Lp,
I
n
1 < p < 2
t
• Shows the Hilbert transform to be a bounded operator on L2, as e
an application of the L2 theory of the Fourier transform in the one- g
r
dimensional case
a
• Covers fractional integration and some topics related to mean l

oscillation properties of functions, such as the classes of Hölder
S
continuous functions and the space of functions of bounded mean e
c
oscillation
o
• Derives a subrepresentation formula, which in higher dimensions plays n
d
a role roughly similar to the one played by the fundamental theorem of
Richard L. Wheeden
E
calculus in one dimension
d
• Extends the subrepresentation formula derived for smooth functions to i
t
functions with a weak gradient io Antoni Zygmund
n
• Applies the norm estimates derived for fractional integral operators

to obtain local and global first-order Poincaré–Sobolev inequalities,
including endpoint cases
• Proves the existence of a tangent plane to the graph of a Lipschitz
function of several variables W
h
• Includes many new exercises not present in the first edition e
e
d
This widely used and highly respected text for upper-division undergraduate e
n
and first-year graduate students of mathematics, statistics, probability, or •
Z
engineering is revised for a new generation of students and instructors. The y
g
m
book also serves as a handy reference for professional mathematicians.
u
n
d
K24634
www.crcpress.com
K24634_cover.indd 1 3/20/15 10:44 AM
Measure and Integral
An Introduction to Real Analysis
Second Edition
PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J. Taft Zuhair Nashed
Rutgers University University of Central Florida
Piscataway, New Jersey Orlando, Florida
EDITORIAL BOARD
Jane Cronin Freddy van Oystaeyen
Rutgers University University of Antwerp,
S. Kobayashi Belgium
University of California, Donald Passman
Berkeley University of Wisconsin,
Marvin Marcus Madison
University of California, Fred S. Roberts
Santa Barbara Rutgers University
W. S. Massey David L. Russell
Yale University Virginia Polytechnic Institute
and State University
Anil Nerode
Walter Schempp
Cornell University
Universität Siegen
MONOGRAPHS AND TEXTBOOKS IN
PURE AND APPLIED MATHEMATICS
Recent Titles
John Dauns and Yiqiang Zhou, Classes of Modules (2006)
N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados,
Frontiers in Interpolation and Approximation (2006)
Luca Lorenzi and Marcello Bertoldi, Analytical Methods for Markov Semigroups (2006)
M. A. Al-Gwaiz and S. A. Elsanousi, Elements of Real Analysis (2006)
Theodore G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite
Rank Groups (2007)
R. Sivaramakrishnan, Certain Number-Theoretic Episodes in Algebra (2006)
Aderemi Kuku, Representation Theory and Higher Algebraic K-Theory (2006)
Robert Piziak and P. L. Odell, Matrix Theory: From Generalized Inverses to
Jordan Form (2007)
Norman L. Johnson, Vikram Jha, and Mauro Biliotti, Handbook of Finite
Translation Planes (2007)
Lieven Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders (2008)
Fritz Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations (2008)
Jane Cronin, Ordinary Differential Equations: Introduction and Qualitative Theory,
Third Edition (2008)
Su Gao, Invariant Descriptive Set Theory (2009)
Christopher Apelian and Steve Surace, Real and Complex Analysis (2010)
Norman L. Johnson, Combinatorics of Spreads and Parallelisms (2010)
Lawrence Narici and Edward Beckenstein, Topological Vector Spaces, Second Edition (2010)
Moshe Sniedovich, Dynamic Programming: Foundations and Principles, Second Edition (2010)
Drumi D. Bainov and Snezhana G. Hristova, Differential Equations with Maxima (2011)
Willi Freeden, Metaharmonic Lattice Point Theory (2011)
Murray R. Bremner, Lattice Basis Reduction: An Introduction to the LLL Algorithm and
Its Applications (2011)
Clifford Bergman, Universal Algebra: Fundamentals and Selected Topics (2011)
A. A. Martynyuk and Yu. A. Martynyuk-Chernienko, Uncertain Dynamical Systems: Stability and
Motion Control (2012)
Washek F. Pfeffer, The Divergence Theorem and Sets of Finite Perimeter (2012)
Willi Freeden and Christian Gerhards, Geomathematically Oriented Potential Theory (2013)
Anatoly Martynyuk, Larisa Chernetskaya, and Vladislav Martynyuk, Weakly Connected Nonlinear
Systems: Boundedness and Stability of Motion (2013)
Franz Halter-Koch, Quadratic Irrationals: An Introduction to Classical Number Theory (2013)
Łukasz Piasecki, Classification of Lipschitz Mappings (2014)
Andy R. Magid, The Separable Galois Theory of Commutative Rings, Second Edition (2015)

Measure and Integral
An Introduction to Real Analysis
Second Edition
Richard L. Wheeden
Rutgers University
Antoni Zygmund
Late of the University of Chicago
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2015 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20150318
International Standard Book Number-13: 978-1-4987-0290-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copy-
right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
vides licenses and registration for a variety of users. For organizations that have been granted a photo-
copy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Toourfamilies
RichardL.WheedenandAntoniZygmund
Tomywife,Sharon,whoseloveandsupporthavebeenvitaltomeduringthe
revisionprocessandthroughoutourmarriage.
RichardL.Wheeden

Contents
PrefacetotheSecondEdition.................................................... xiii
PrefacetotheFirstEdition.........................................................xv
Authors .............................................................................xvii
1. Preliminaries....................................................................1
1.1 PointsandSetsinRn................................................. 1
1.2 RnasaMetricSpace ................................................. 2
1.3 OpenandClosedSetsinRn,andSpecialSets ................... 5
1.4 CompactSetsandtheHeine–BorelTheorem .................... 9
1.5 Functions .............................................................. 10
1.6 ContinuousFunctionsandTransformations..................... 11
1.7 TheRiemannIntegral................................................ 12
Exercises....................................................................... 13
2. FunctionsofBoundedVariationandtheRiemann–Stieltjes
Integral..........................................................................17
2.1 FunctionsofBoundedVariation ................................... 17
2.2 RectifiableCurves.................................................... 25
2.3 TheRiemann–StieltjesIntegral..................................... 26
2.4 FurtherResultsaboutRiemann–StieltjesIntegrals.............. 34
Exercises....................................................................... 37
3. LebesgueMeasureandOuterMeasure ...................................41
3.1 LebesgueOuterMeasureandtheCantorSet .................... 41
3.2 LebesgueMeasurableSets .......................................... 45
3.3 TwoPropertiesofLebesgueMeasure ............................. 50
3.4 CharacterizationsofMeasurability................................ 52
3.5 LipschitzTransformationsofRn ................................... 54
3.6 ANonmeasurableSet................................................ 57
Exercises....................................................................... 58
4. LebesgueMeasurableFunctions ...........................................63
4.1 ElementaryPropertiesofMeasurableFunctions ................ 64
4.2 SemicontinuousFunctions.......................................... 69
4.3 PropertiesofMeasurableFunctionsandTheoremsof
EgorovandLusin..................................................... 71
4.4 ConvergenceinMeasure............................................ 73
Exercises....................................................................... 76
ix