2nd Edition

Measure and Integral An Introduction to Real Analysis, Second Edition

By Richard L. Wheeden Copyright 2015
    536 Pages 8 B/W Illustrations
    by Chapman & Hall

    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 L1, L2, and Lp, 1 < p < 2
    • Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case
    • 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.

    Preface to the Second Edition

    Preface to the First Edition



    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


    Continuous Functions and Transformations

    The Riemann Integral


    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


    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


    Lebesgue Measurable Functions

    Elementary Properties of Measurable Functions

    Semicontinuous Functions

    Properties of Measurable Functions and Theorems of Egorov and Lusin

    Convergence in Measure


    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


    Repeated Integration

    Fubini’s Theorem

    Tonelli’s Theorem

    Applications of Fubini’s Theorem



    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


    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


    Approximations of the Identity and Maximal Functions


    Approximations of the Identity

    The Hardy–Littlewood Maximal Function

    The Marcinkiewicz Integral


    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


    Outer Measure and Measure

    Constructing Measures from Outer Measures

    Metric Outer Measures

    Lebesgue–Stieltjes Measure

    Hausdorff Measure

    Carathéodory–Hahn Extension Theorem


    A Few Facts from Harmonic Analysis

    Trigonometric Fourier Series

    Theorems about Fourier Coefficients

    Convergence of S[f] and [f]

    Divergence of Fourier Series

    Summability of Sequences and Series

    Summability of S[f] and [f] by the Method of the Arithmetic Mean

    Summability of S[f] by Abel Means

    Existence of f Þ

    Properties of f Þ for fLp, 1 < p < ∞

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


    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


    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


    Weak Derivatives and Poincaré–Sobolev Estimates

    Weak Derivatives

    Approximation by Smooth Functions and Sobolev Spaces

    Poincaré–Sobolev Estimates





    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.