# Measure and Integral: An Introduction to Real Analysis

CRC Press, Nov 1, 1977 - Mathematics - 288 pages
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.

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### Contents

 Introduction Chapter 1 Preliminaries 1 Points and Sets in 1 R as a Metric Space 2 Open and Closed Sets in R Special Sets 5 Compact Sets the HeineBorel Theorem 8 Functions 6 Continuous Functions and Transformations 10 The Riemann Integral Exercises 11 5 12 11 13
 Characterizations of Measurability 42 Lipschitz Transformations of R 44 A Nonmeasurable Set 46 Exercises 47 Lebesgue Measurable Functions 50 Egorovs 56 The Lebesgue Integral 64 Repeated Integration 87

 Functions of Bounded Variation the Riemann Stieltjes Integral 15 Rectifiable Curves 21 The RiemannStieltjes Integral 23 Further Results About RiemannStieltjes Integrals 28 Exercises 31 Lebesgue Measure and Outer Measure 33 Lebesgue Measurable Sets 37 Two Properties of Lebesgue Measure 40
 If Classes 125 Approximations of the Identity Maximal Functions 145 Abstract Integration 161 Outer Measure Measure 193 A Few Facts From Harmonic Analysis 211 Notation 265 130 270 Copyright

### Popular passages

Page ix - ... properties (i) ||x|| > 0 and ||x|| = 0 if and only if x = 0. (ii) ||ax|| = a| ||x||, where |a| is the absolute value of the complex scalar a.