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Measure and Integral

An Introduction to Real Analysis, Second Edition

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## Book Description

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
*L*,^{1}*L*, and^{2}*L*, 1 <^{p}*p*< 2 - Shows the Hilbert transform to be a bounded operator on
*L*, as an application of the^{2}*L*theory of the Fourier transform in the one-dimensional case^{2} - 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.

## Table of Contents

Preface to the Second Edition

Preface to the First Edition

Authors

Preliminaries

Points and Sets in R^{n}

R^{n} as a Metric Space

Open and Closed Sets in R^{n}, 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 R^{n}

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 *L ^{p}* 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 R^{n}

Exercises

*L ^{p}* Classes

Definition of *L ^{p}*

Hölder’s Inequality and Minkowski’s Inequality

Classes* l ^{p}*

Banach and Metric Space Properties

The Space *L ^{2}* 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 *L ^{p}*

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* ∈ *L ^{p}*, 1 <

*p*< ∞

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

Exercises

The Fourier Transform

The Fourier Transform on *L ^{1}*

The Fourier Transform on *L ^{2}*

The Hilbert Transform on *L ^{2}*

The Fourier Transform on *L ^{p}*, 1 <

*p*< 2

Exercises

Fractional Integration

Subrepresentation Formulas and Fractional Integrals

*L ^{1}, L^{1}* Poincaré Estimates and the Subrepresentation Formula; Hölder Classes

Norm Estimates for *I _{α}*

Exponential Integrability of *I _{α}f*

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

## Author(s)

### 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.