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1st Edition

Measure and Integral
An Introduction to Real Analysis




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ISBN 9780429160868
Published November 1, 1977 by CRC Press
288 Pages

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Book Description

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.

Table of Contents

Preliminaries

Points and Sets in Rn
Rn as a Metric Space
Open and Closed Sets in Rn: Special Sets
Compact Sets; The Heine-Borel Theorem
Functions
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
Exercises

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
Exercises

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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Reviews

". . .well written and authoritative. "
---Choice

"The style is characterized by its clarity and concreteness. . . . this work constitutes an excellent introductory textbook for those who wish to get acquainted with the modern methods of real variables. "
---Mathematical Reviews