Preface to the Second Edition

Preface to the First Edition

Authors

1.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

2.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

3.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

4.Lebesgue Measurable Functions

Elementary Properties of Measurable Functions

Semicontinuous Functions

Properties of Measurable Functions and Theorems of Egorov and Lusin

Convergence in Measure

Exercises

5.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

6.Repeated Integration

Fubini’s Theorem

Tonelli’s Theorem

Applications of Fubini’s Theorem

Exercises

7.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

8.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

9.Approximations of the Identity and Maximal Functions

Convolutions

Approximations of the Identity

The Hardy–Littlewood Maximal Function

The Marcinkiewicz Integral

Exercises

10.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

11.Outer Measure and Measure

Constructing Measures from Outer Measures

Metric Outer Measure

Lebesgue–Stieltjes Measure

Hausdorff Measure

Carathéodory–Hahn Extension Theorem

Exercises

12.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

13.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

14.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

15.Weak Derivatives and Poincaré–Sobolev Estimates

Weak Derivatives

Approximation by Smooth Functions and Sobolev Spaces

Poincaré–Sobolev Estimates

Exercises

Notations

Index