Measure and Integral: An Introduction to Real Analysis

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CRC Press, 1 nov 1977 - 288 páginas
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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Índice

Introduction
1
Chapter
4
Functions of Bounded Variation the Riemann
15
Rectifiable Curves
21
Further Results About RiemannStieltjes Integrals
28
Lebesgue Measurable Functions
50
3
56
The Lebesgue Integral
64
Chapter 6
93
The Indefinite Integral
99
The Vitali Covering Lemma
109
Absolutely Continuous and Singular Functions
115
Convex Functions
121
Approximations of the Identity Maximal Functions
145
Abstract Integration
161
Outer Measure Measure
193

The Integral of an Arbitrary Measurable
71
Riemann and Lebesgue Integrals
80
Repeated Integration
87
A Few Facts From Harmonic Analysis
211
Notation
265
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Página 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.

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