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
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
Hints to Selected Exercises
About the Author
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. He served in the Peach Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before studying with Emil Grosswald at Temple and then teaching at Penn State for 17 years, eight of them as full professor. He chaired the mathematics department at Macalester from 1995 until 2001. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, Université Louis Pasteur (Strasbourg, France), and the State College Area High School. He has received the MAA Distinguished Teaching Award (Allegheny Mountain Section), the MAA Beckenbach Book Prize for Proofs and Confirmations, and has been a Pólya Lecturer for the MAA. He is a recipient of Macalester’s Jefferson Award. He has published over 50 research articles in number theory, combinatorics, and special functions. Other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, the first and second editions of A Radical Approach to Real Analysis, and, with Stan Wagon, A Course in Computational Number Theory. David Bressoud chairs the MAA Committee on the Undergraduate Program in Mathematics. He has chaired the AP Calculus Development Committee and has served as Director of FIPSE-supported program Quantitative Methods for Public Policy. He has been active in the activities and programs of both the Mathematical Association of America and the American Mathematical Society.