Measure and Integral: An Introduction to Real Analysis (2nd Edition)


Download Measure and Integral: An Introduction to Real Analysis (2nd Edition) written by Richard L. Wheeden, Antoni Zygmund in PDF format. This book is under the category Mathematics and bearing the isbn/isbn13 number 1498702899/9781498702898. You may reffer the table below for additional details of the book.

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Now considered a classic version on the topic; Measure and Integral: An Introduction to Real Analysis; 2nd Edition; (PDF) offers an introduction to real analysis by first forming the theory of measure and integration in the simple setting of Euclidean space; and then offering a more general treatment based on abstract notions categorized by axioms and with less geometric content.

Published almost forty years after the first edition; this long-awaited Second Edition also:

  • Analyzes the Fourier transform of functions in the spaces L1; L2; and Lp; 1 < p < 2
  • Demonstrates the existence of a tangent plane to the graph of a Lipschitz function of several variables
  • Lengthens the subrepresentation formula derived for smooth functions to functions with a weak gradient
  • Displays 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
  • Uses the norm estimates derived for fractional integral operators to get local and global first-order Poincaré–Sobolev inequalities; including endpoint cases
  • Develops a subrepresentation formula; which in higher dimensions plays a role quite similar to the one played by the fundamental theorem of calculus in one dimension
  • Includes fractional integration and some topics related to mean oscillation properties of functions; like the classes of Hölder continuous functions and the space of functions of bounded mean oscillation

Contains many new exercises not present in the first edition

This highly respected and extensively used textbook Measure and Integral: An Introduction to Real Analysis 2e for upper-division undergraduate and first-year graduate students of mathematics; probability; statistics; or engineering is updated for a new generation of college students and instructors. The ebook also serves as a handy reference for professional mathematicians.

NOTE: The product only includes the ebook; Measure and Integral: An Introduction to Real Analysis; 2nd Edition in PDF. No access codes are included.


Additional information


Richard L. Wheeden, Antoni Zygmund


Chapman and Hall/CRC; 2nd Edition




532 pages









Table of contents

Table of contents :
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 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]

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


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