A Radical Approach to Lebesgue's Theory of Integration

An introduction to measure theory and Lebesgue integration rooted in the historical questions that led to its development.

David M. Bressoud (Author)

9780521711838, Cambridge University Press

Paperback, published 14 January 2008

344 pages, 120 exercises
25.2 x 17.7 x 1.8 cm, 0.6 kg

'I find it difficult to think of a better introduction to this cornerstone of modern mathematics and highly recommend the book to a very broad readership of students and researchers alike.' Journal of the American Statistical Association

Meant for advanced undergraduate and graduate students in mathematics, this lively introduction to measure theory and Lebesgue integration is rooted in and motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems and highlights some of the difficulties that were encountered as these ideas were refined. The story begins with Riemann's definition of the integral, a definition created so that he could understand how broadly one could define a function and yet have it be integrable. The reader then follows the efforts of many mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work in the late 19th and early 20th centuries of Jordan, Borel, and Lebesgue, who finally broke with Riemann's definition. Ushering in a new way of understanding integration, they opened the door to fresh and productive approaches to many of the previously intractable problems of analysis.

1. Introduction
2. The Riemann integral
3. Explorations of R
4. Nowhere dense sets and the problem with the fundamental theorem of calculus
5. The development of measure theory
6. The Lebesgue integral
7. The fundamental theorem of calculus
8. Fourier series
9. Epilogue: A. Other directions
B. Hints to selected exercises.

Subject Areas: Probability & statistics [PBT], Calculus & mathematical analysis [PBK]