Measure and Integral An Introduction to Real Analysis, Second Edition

Preface to the Second Edition

Preface to the First Edition

Authors

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

Rectifiable Curves

The Riemann–Stieltjes Integral

Further Results about Riemann–Stieltjes Integrals

Exercises

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

Elementary Properties of Measurable Functions

Semicontinuous Functions

Properties of Measurable Functions and Theorems of Egorov and Lusin

Convergence in Measure

Exercises

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 L^p Spaces, 0 < p < 8

Riemann and Lebesgue Integrals

Exercises

Fubini’s Theorem

Tonelli’s Theorem

Applications of Fubini’s Theorem

Exercises

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

Definition of L^p

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

Convolutions

Approximations of the Identity

The Hardy–Littlewood Maximal Function

The Marcinkiewicz Integral

Exercises

Additive Set Functions and Measures

Measurable Functions and Integration

Absolutely Continuous and Singular Set Functions and Measures

The Dual Space of L^p

Relative Differentiation of Measures

Exercises

Constructing Measures from Outer Measures

Metric Outer Measures

Lebesgue–Stieltjes Measure

Hausdorff Measure

Carathéodory–Hahn Extension Theorem

Exercises

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 ¿ L^p, 1 < p < 8

Application of Conjugate Functions to Partial Sums of S[f]

Exercises

The Fourier Transform on L¹

The Fourier Transform on L²

The Hilbert Transform on L²

The Fourier Transform on L^p, 1 < p < 2

Exercises

Subrepresentation Formulas and Fractional Integrals

L¹, L¹ Poincaré Estimates and the Subrepresentation Formula; Hölder Classes

Norm Estimates for I_a

Exponential Integrability of I_af

Bounded Mean Oscillation

Exercises

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.