A (Terse) Introduction to Lebesgue Integration
About this Title
John Franks, Northwestern University, Evanston, IL
Publication: The Student Mathematical Library
Publication Year 2009: Volume 48
ISBNs: 978-0-8218-4862-3 (print); 978-1-4704-1219-7 (online)
MathSciNet review: MR2514048
MSC: Primary 28A25; Secondary 26-01, 26A42, 42A16
This book provides a student's first encounter with the concepts of measure theory and functional analysis. Its structure and content reflect the belief that difficult concepts should be introduced in their simplest and most concrete forms.
Despite the use of the word “terse” in the title, this text might also have been called A (Gentle) Introduction to Lebesgue Integration. It is terse in the sense that it treats only a subset of those concepts typically found in a substantial graduate-level analysis course. The book emphasizes the motivation of these concepts and attempts to treat them simply and concretely. In particular, little mention is made of general measures other than Lebesgue until the final chapter and attention is limited to $R$ as opposed to $R^n$.
After establishing the primary ideas and results, the text moves on to some applications. Chapter 6 discusses classical real and complex Fourier series for $L^2$ functions on the interval and shows that the Fourier series of an $L^2$ function converges in $L^2$ to that function. Chapter 7 introduces some concepts from measurable dynamics. The Birkhoff ergodic theorem is stated without proof and results on Fourier series from Chapter 6 are used to prove that an irrational rotation of the circle is ergodic and that the squaring map on the complex numbers of modulus 1 is ergodic.
This book is suitable for an advanced undergraduate course or for the start of a graduate course. The text presupposes that the student has had a standard undergraduate course in real analysis.
Undergraduate and graduate students interested in analysis or its applications to other areas of mathematics.
Table of Contents
- Chapter 1. The regulated and Riemann integrals
- Chapter 2. Lebesgue measure
- Chapter 3. The Lebesgue integral
- Chapter 4. The integral of unbounded functions
- Chapter 5. The Hilbert space $L^2$
- Chapter 6. Classical Fourier series
- Chapter 7. Two ergodic transformations
- Appendix A. Background and foundations
- Appendix B. Lebesgue measure
- Appendix C. A non-measurable set