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A Radical Approach to Lebesgue's Theory of Integration

Part of Mathematical Association of America Textbooks

Author: David M. Bressoud, Macalester College, Minnesota

Date Published: January 2008
availability: Available
format: Paperback
isbn: 9780521711838

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About the Authors

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.
- Exercises at the end of each section, allowing students to explore their understanding
- Hints to help students get started on challenging problems
- Boxed definitions make it easier to identify key definitions
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Reviews & endorsements
'This introduction to measure theory and Lebesgue integration is intended for advancerd undergraduate and graduate students in mathematics, and is rooted in and motivated by the historical questions that led to its development.' The Times Higher Education Supplement
'Bressoud is an insightful writer and he presents this material in an enchanting fashion. The writing is scholarly but inviting, rigourous but readable.' The UMAP Journal
'The way that facts are presented makes the book accessible for graduate or advanced undergraduate students as an alternative to the standard approach of teaching real analysis. The book will be interesting for teachers as well.' EMS Newsletter
'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
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Product details
- Date Published: January 2008
- format: Paperback
- isbn: 9780521711838
- length: 344 pages
- dimensions: 252 x 177 x 18 mm
- weight: 0.6kg
- contains: 120 exercises
- availability: Available
Table of Contents
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.
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Instructors have used or reviewed this title for the following courses
- Mathematics of Networks
- Real Analysis ll
- Real Variables I and Real Variables ll
- Topics in Mathematics: Analysis
Author
David M. Bressoud, Macalester College, Minnesota