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Table of Contents
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
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