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Measure and Integral


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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 Hoelder'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 Caratheodory-Hahn Extension Theorem Exercises A Few Facts from Harmonic Analysis Trigonometric Fourier Series Theorems about Fourier Coefficients Convergence of S[f] and STH[f] Divergence of Fourier Series Summability of Sequences and Series Summability of S[f] and STH[f] by the Method of the Arithmetic Mean Summability of S[f] by Abel Means Existence of f TH Properties of f TH 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 Poincare Estimates and the Subrepresentation Formula; Hoelder Classes Norm Estimates for I Exponential Integrability of I f Bounded Mean Oscillation Exercises Weak Derivatives and Poincare-Sobolev Estimates Weak Derivatives Approximation by Smooth Functions and Sobolev Spaces Poincare-Sobolev Estimates Exercises Notations Index

About the Author

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.

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