A Radical Approach to Lebesgue's Theory of Integration

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.

David M. Bressoud is the DeWitt Wallace Professor of Mathematics at Macalester College. He was a Peace Corps Volunteer in Antigua, West Indies, received his PhD from Temple University, and taught at The Pennsylvania State University before moving to Macalester, He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin, the University of Minnesota, and the University of Strasbourg, He has received a Sloan Fellowship, a Fulbright Fellowship, and the MAA Distinguished Teaching Award, He has published more than 50 research articles in number theory, partition theory, combinatorics, and the theory of special functions. His other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, A Radical Approach to Real Analysis, and Proofs and Confirmations, for which he won the MAA Beckenbach Book Prize.