2nd Edition
Measure and Integral An Introduction to Real Analysis, Second Edition
Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content.
Published nearly forty years after the first edition, this long-awaited Second Edition also:
- Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 < p < 2
- Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case
- Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation
- Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension
- Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient
- 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
- Includes many new exercises not present in the first edition
This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.
Preface to the Second Edition
Preface to the First Edition
Authors
Preliminaries
Points and Sets in Rn
Rn as a Metric Space
Open and Closed Sets in Rn, and Special Sets
Compact Sets and the Heine–Borel Theorem
Functions
Continuous Functions and Transformations
The Riemann Integral
Exercises
Functions of Bounded Variation and the Riemann–Stieltjes Integral
Functions of Bounded Variation
Rectifiable Curves
The Riemann–Stieltjes Integral
Further Results about Riemann–Stieltjes Integrals
Exercises
Lebesgue Measure and Outer Measure
Lebesgue Outer Measure and the Cantor Set
Lebesgue Measurable Sets
Two Properties of Lebesgue Measure
Characterizations of Measurability
Lipschitz Transformations of Rn
A Nonmeasurable Set
Exercises
Lebesgue Measurable Functions
Elementary Properties of Measurable Functions
Semicontinuous Functions
Properties of Measurable Functions and Theorems of Egorov and Lusin
Convergence in Measure
Exercises
The Lebesgue Integral
Definition of the Integral of a Nonnegative Function
Properties of the Integral
The Integral of an Arbitrary Measurable f
Relation between Riemann–Stieltjes and Lebesgue Integrals, and the Lp Spaces, 0 < p < ∞
Riemann and Lebesgue Integrals
Exercises
Repeated Integration
Fubini’s Theorem
Tonelli’s Theorem
Applications of Fubini’s Theorem
Exercises
Differentiation
The Indefinite Integral
Lebesgue’s Differentiation Theorem
Vitali Covering Lemma
Differentiation of Monotone Functions
Absolutely Continuous and Singular Functions
Convex Functions
The Differential in Rn
Exercises
Lp Classes
Definition of Lp
Hölder’s Inequality and Minkowski’s Inequality
Classes l p
Banach and Metric Space Properties
The Space L2 and Orthogonality
Fourier Series and Parseval’s Formula
Hilbert Spaces
Exercises
Approximations of the Identity and Maximal Functions
Convolutions
Approximations of the Identity
The Hardy–Littlewood Maximal Function
The Marcinkiewicz Integral
Exercises
Abstract Integration
Additive Set Functions and Measures
Measurable Functions and Integration
Absolutely Continuous and Singular Set Functions and Measures
The Dual Space of Lp
Relative Differentiation of Measures
Exercises
Outer Measure and Measure
Constructing Measures from Outer Measures
Metric Outer Measures
Lebesgue–Stieltjes Measure
Hausdorff Measure
Carathéodory–Hahn Extension Theorem
Exercises
A Few Facts from Harmonic Analysis
Trigonometric Fourier Series
Theorems about Fourier Coefficients
Convergence of S[f] and SÞ[f]
Divergence of Fourier Series
Summability of Sequences and Series
Summability of S[f] and SÞ[f] by the Method of the Arithmetic Mean
Summability of S[f] by Abel Means
Existence of f Þ
Properties of f Þ for f ∈ Lp, 1 < p < ∞
Application of Conjugate Functions to Partial Sums of S[f]
Exercises
The Fourier Transform
The Fourier Transform on L1
The Fourier Transform on L2
The Hilbert Transform on L2
The Fourier Transform on Lp, 1 < p < 2
Exercises
Fractional Integration
Subrepresentation Formulas and Fractional Integrals
L1, L1 Poincaré Estimates and the Subrepresentation Formula; Hölder Classes
Norm Estimates for Iα
Exponential Integrability of Iαf
Bounded Mean Oscillation
Exercises
Weak Derivatives and Poincaré–Sobolev Estimates
Weak Derivatives
Approximation by Smooth Functions and Sobolev Spaces
Poincaré–Sobolev Estimates
Exercises
Notations
Index
Biography
Richard L. Wheeden is Distinguished Professor of Mathematics at Rutgers University, New Brunswick, New Jersey, USA. His primary research interests lie in the fields of classical harmonic analysis and partial differential equations, and he is the author or coauthor of more than 100 research articles. After earning his Ph.D. from the University of Chicago, Illinois, USA (1965), he held an instructorship there (1965–1966) and a National Science Foundation (NSF) Postdoctoral Fellowship at the Institute for Advanced Study, Princeton, New Jersey, USA (1966–1967).
Antoni Zygmund was Professor of Mathematics at the University of Chicago, Illinois, USA. He was earlier a professor at Mount Holyoke College, South Hadley, Massachusetts, USA, and the University of Pennsylvania, Philadelphia, USA. His years at the University of Chicago began in 1947, and in 1964, he was appointed Gustavus F. and Ann M. Swift Distinguished Service Professor there. He published extensively in many branches of analysis, including Fourier series, singular integrals, and differential equations. He is the author of the classical treatise Trigonometric Series and a coauthor (with S. Saks) of Analytic Functions. He was elected to the National Academy of Sciences in Washington, District of Columbia, USA (1961), as well as to a number of foreign academies.
