Measure and Integral: An Introduction to Real Analysis

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

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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.

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