This concise introduction to Lebesgue integration is geared toward advanced undergraduate math majors and may be read by any student possessing some familiarity with real variable theory and elementary calculus. The self-contained treatment features exercises at the end of each chapter that range from simple to difficult.
The approach begins with sets and functions and advances to Lebesgue measure, including considerations of measurable sets, sets of measure zero, and Borel sets and nonmeasurable sets. A two-part exploration of the integral covers measurable functions, convergence theorems, convergence in mean, Fourier theory, and other topics. A chapter on calculus examines change of variables, differentiation of integrals, and integration of derivatives and by parts. The text concludes with a consideration of more general measures, including absolute continuity and convolution products.
Dover (2014) republication of the edition originally published by Holt, Rinehart & Winston, New York, 1962.
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Author(s): J. H. Williamson
Publisher: Dover Publications
Year: 2014
Language: English
Pages: 123
Cover
Title Page
Copyright Page
Preface
Table of Contents
Chapter 1. Sets and Functions
1.1. Generalities
1.2. Countable and Uncountable Sets
1.3. Sets in R^n
1.4. Compactness
1.5. Functions
Chapter 2. Lebesgue Measure
2.1. Preliminaries
2.2. The Class
2.3. Measurable Sets
2.4. Sets of Measure Zero
2.5. Borel Sets and Nonmeasurable Sets
Chapter 3. The Integral I
3.1. Definition
3.2. Elementary Properties
3.3. Measurable Functions
3.4. Complex and Vector Functions
3.5. Other Definitions of the Integral
Chapter 4. The Integral II
4.1. Convergence Theorems
4.2. Fubinis Theorems
4.3. Approximations to Integrable Functions
4.4. The L p Spaces
4.5. Convergence in Mean
4.6. Fourier Theory
Chapter 5. Calculus
5.1. Change of Variables
5.2. Differentiation of Integrals
5.3. Integration of Derivatives
5.4. Integration by Parts
Chapter 6. More General Measures
6.1. Borel Measures
6.2. Signed Measures and Complex Measures
6.3. Absolute Continuity
6.4 Measures, Functions, and Functionals
6.5 Norms, Fourier Transforms, Convolution Products
Index
