Measure and Integral: an Introduction to Real Analysis

289 Pages·1977·46.084 MB·English

by  Zygmund, Antoni, Wheeden, Richard L

MEASURE AND INTEGRAL
PURE AND APPLIED MATHEMATICS
A Program ofMonographs, Textbooks, andLecture Notes
EXECUTIVE EDITORS-MoNOGRAPHS, TEXTBOOKS, AND LECTURE NOTES
Earl J. Taft
Rutgers University
New Brunswick, NewJersey
Edwin Hewitt
University0/Washington
Seattle, Washington
CHAIRMAN OFTHE EDITORIAL BOARD
S. Kobayashi
University0/California, Berkeley
Berkeley, California
EDITORIAL BOARD
Masanao Aoki w. S. Massey
University0/California, LosAngeles Yale University
Glen E. Bredon Irving Reiner
Rutgers University UniversityofIllinoisat Urbana-Champaign
Sigurdur Helgason PaulJ. Sally, Jr.
Massachusetts InstituteofTechnology University0/Chicago
G. Leitmann Jane Cronin Scanlon
UniversityofCalifornia, Berkeley Rutgers University
Marvin Marcus Martin Schechter
University0/California, Santa Barbara Yeshiva University
Julius L. Shaneson
Rutgers University
MONOGRAPHS AND TEXTBOOKS IN
PURE AND APPLIED MATHEMATICS
1. K. Yano, Integral Formulasin RiemannianGeometry(1970) (outofprint)
.2-. S. Kobayashi, HyperbolicManifoldsandHolomorphicMappings
(1970) (outofprint)
3. v: S. V/adimirov, EquationsofMathematicalPhysics(A. Jeffrey, editor;
A. littlewood,translator)(1970) (outofprint)
4. B. N. Pshenichnyi, NecessaryConditionsforan Extremum(L. Neustadt,
translationeditor;K. Makowski, translator)(1971)
s. L. Narici, E. Beckenstein,andG. Bachman, FunctionalAnalysisand
ValuationTheory(1971)
6. D. S. Passrruzn, InfiniteGroup Rings(1971)
7. L. Dornhoff, Group RepresentationTheory(in two parts). PartA:
Ordinary RepresentationTheory. Part B: ModularRepresentationTheory
(1971,1972)
8. ~ BoothbyandG:L. Weiss(eds.), SymmetricSpaces: ShortCourses
PresentedatWashington University(1972) (outofprint)
9. Y: Matsushima, DifferentiableManifolds(E. T. Kobayashi, translator)
(1972)
10. L. E. Ward, Jr., Topology: AnOutline for a FirstCourse(1972) (outofprint}
11. A. Babakhanian, CohomologicalMethodsinGroupTheory (1972)
12. R. Gilmer, Multiplicative IdealTheory(1972) foutofprint)
13. J. Yeh, StochasticProcessesand theWienerIntegral(1973) (outofprint)
14. J. Barros-Nero, Introduction to the TheoryofDistributions(1973)(outofprint)
1S. R. Larsen, Functional Analysis: An Introduction(1973) (outofprint)
16. K. Yano andS. Ishihara, Tangent andCotangentBundles: Differential
Geometry(1973) (outofprint)
17. C Procesi, RingswithPolynomialIdentities(1973)
18. R. Hermann, Geometry,Physics,and Systems(1973) (outofprint)
19. N. R. Wallach, HarmonicAnalysisonHomogeneousSpaces(1973) (outofprint)
20. J. Dieudonne, Introduction tothe Theory ofFormal Groups(1973)
21. I. Vaisman, Cohomologyand Differential Forms(1973)
22. B.-Y. Chen, GeometryofSubmanifolds(1973) (outofprint)
23. M Marcus, Finite Dimensional MultilinearAlgebra(in two parts)
(1973, 1975)
24. R. Larsen, BanachAlgebras: An Introduction(1973)
25. R. O. KujalaandA. L. Vilterreds.), Value DistributionTheory: Part A;
Part B. Deficitand Bezout EstimatesbyWilhelmStoll(1973)
26. K. B. Stolarsky, Algebraic Numbersand Diophantine Approximation(1974)
27. A. R. Magid, The SeparableGaloisTheoryofCommutative Rings(1974)
28. B. R. McDonald, Finite Ringswith Identity(1974)
29. J. Satake, LinearAlgebra(S. Koh, T. Akiba,and S. Ihara, translators)
(1975)
30. J. S. Golan, LocalizationofNoncommutative Rings(1975)
31. G. Klambauer, Mathematical Analysis(1975)
32. M. K. Agoston, AlgebraicTopology: AFirstCourse(1976)
33. K. R. Goodearl, RingTheory: Nonsingular Ringsand Modules (1976)
34. L. E. Mansfield, LinearAlgebrawithGeometric Applications(1976)
35. N J. Pullman, MatrixTheory and itsApplications: .Selected Topics(1976)
36. B. R. McDonald, Geometric AlgebraOver Local Rings (1976)
37. C WGroetsch, Generalized InversesofLinearOperators: Representation
and Approximation(1977)
38. J. E. KuczkowskiandJ. L. Gersting, Abstract Algebra: AFirst Look
(1977)
39. C O. Christenson and W. L. Voxman, AspectsofTopology (1977)
40. M. Nagata, Field Theory (1977)
41. R. L. Long, Algebraic NumberTheory (1977)
42. WF. Pfeffer, Integralsand Measures(1977)
43. R. L. WheedenandA. Zygmund, Measureand Integral: An Introduction to
Real Analysis(1977)
MEASURE AND INTEGRAL
An Introduction to Real Analysis
Richard L. Wheeden Antoni Zygmund
Department ofMathematics Department ofMathematics
Rutgers" the State University University ofChicago
ofNew Jersey Chicago" Illinois
New Brunswick" New Jersey
o
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LibraryofCongressCataloging-in-PublicationData
Wheeden,RichardL.
Measureandintegral.
(Monographsandtextbooksinpureandappliedmathematics;43)
Includesindex.
ISBN0-8247-6499-4
1.Measuretheory. 2. Integrals,Generalized.I.Zygmund,Antoni,jointauthor. II.Title.
QA312.W43
515'.42 77-14167
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Introduction
The modern theory of measure and integration was created, primarily
through the work of Lebesgue, at the turn of this century. Although the
basic ideas are by now well established, there are ever widening applications
whichhavemadethetheoryoneofthecentralpartsofmathematicalanalysis.
However, differentapplicationsrequiredifferentemphasis on various aspects
ofthe theory. Forexample, certain facts are ofprimary interestfor real and
complex analysis, others for functional analysis, and still others for prob
ability and statistics. This text is written from the point of view of real
variables, and treats the theory primarily as a modern calculus.
The book presupposes that the reader has a feeling for rigor and some
knowledge of elementary facts from calculus. Some material which is no
doubt familiar to many readers has been included; its inclusion seemed
desirable in order to make the presentation clearand self-contained.
The approach of the book is to develop the theory of measure and
integration first in the simple setting of Euclidean space. In this case, there
is a rich theory having a close relation to familiar facts from calculus and
generalizing those facts. Later on, we introduce a more general treatment
based on abstract notions characterized by axioms and with less geometric
content. We have chosen this approach purposely, even though it leads to
some repetition, since considering a special case first usually helps in de
veloping a better understanding of the general situation. Anyway, we all
"learn by repetition."
The outline ofthe book is as follows. Chapter 1is primarily a collection
of various background information, including elementary definitions and
results that will be taken for granted later in the book; the reader should
already be familiar with most ofthis material. Very few proofs are given in
Chapter 1. Actual presentation of the theory begins in Chapter 2, which
treats notions associated with functions of bounded variation, such as the
Riemann-Stieltjes integral. Strictly speaking, a reading of Chapter 2 could
v