Measure and Integral: An Introduction to Real Analysis

Front Cover
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

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

Other editions - View all

Common terms and phrases

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.

Bibliographic information