A Radical Approach to Lebesgue's Theory of Integration

by David M. Bressoud

• ISBN13: 9780521711838
• ISBN10: 0521711835
• Edition: 1st
• Format: Paperback
• Copyright: 2008-01-14
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Introduction

The Riemann integral

Explorations of R

Nowhere dense sets and the problem with the fundamental theorem of calculus

The development of measure theory

The Lebesgue integral

The fundamental theorem of calculus

Fourier series

Epilogue

Other directions

Hints to selected exercises

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