This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral, the Stieltjes integral, and other commonly used integrals. The purpose of this book is to provide a quick but accurate (and detailed) introduction to all aspects of modern integration theory. It should be accessible to any student who has had calculus and some exposure to upper division mathematics. Table of Contents: Introduction / The Riemann Integral / The Lebesgue Integral / Comparison of the Riemann and Lebesgue Integrals / Other Theories of the Integral
Author(s): Steven G. Krantz
Series: Synthesis Lectures on Mathematics and Statistics
Edition: 1
Publisher: Morgan & Claypool Publishers
Year: 2011
Language: English
Pages: 106
Preface......Page 12
What is the Riemann Integral?......Page 14
What is the Riemann Integral Good For?......Page 15
Haar Measure......Page 0
The Definition......Page 20
Elementary Measure Theory......Page 48
Exercises......Page 81
Any Riemann Integrable Function is Lebesgue Integrable......Page 84
Exercises......Page 86
Other Theories of the Integral......Page 88
The Fundamental Theorem......Page 96
Exercises......Page 97
Bibliography......Page 100
Author's Biography......Page 102
Index......Page 104
