# 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

 Chapter 1 Chapter 9 Stieltjes Integral 15 i 33 12 48 Lebesgue Measurable Functions 50 23 60 The Lebesgue Integral 64
 LP Classes 125 Approximations of the Identity Maximal Functions 145 The HardyLittlewood Maximal Function 155 Abstract Integration 161 Measurable Functions Integration 167 Absolutely Continuous and Singular 174 The Dual Space of L 182 Outer Measure Measure 193

 31 75 Exercises 85 Repeated Integration 87 Differentiation 98
 A Few Facts From Harmonic Analysis 211 Notation 265 145 268 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.