Measure and Integral: An Introduction to Real Analysis / Edition 1

by Richard Wheeden, Richard L. Wheeden, Antoni Zygmund
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Measure and Integral: An Introduction to Real Analysis / Edition 1

This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given.

Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis—harmonic analysis—are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function.

Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas.

Product Details

ISBN-13: 9780824764999
Publisher: Taylor & Francis
Publication date: 11/01/1977
Series: Chapman & Hall/CRC Pure and Applied Mathematics Series , #308
Edition description: New Edition
Pages: 288
Product dimensions: 6.00(w) x 9.00(h) x 0.80(d)

Table of Contents


Points and Sets in Rn
Rn as a Metric Space
Open and Closed Sets in Rn: Special Sets
Compact Sets; The Heine-Borel Theorem
Continuous Functions and Transformations
The Riemann Integral
Exercises Function of Bounded Variation; The Riemann-Stieltjes Integral Functions of Bounded Variation
Rectifiable Curves
The Reiman-Stieltjes Integral
Further Results About the Reimann-Stieltjes Integrals

Lebesgue Measure and Outer Measure Lebesgue Outer Measures; 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; Egorov's Theorem and Lusin's Theorem
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
A Relation Between Riemann-Stieltjes and Lebesgue Integrals; the LP Spaces, 0

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