A Modern Theory of Integration

by Bartle, Robert G.

• ISBN13: 9780821808450
• ISBN10: 0821808451
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: Amer Mathematical Society

Summary

The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is "better" because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with "improper" integrals. This book is an introduction to a relatively new theory of the integral (called the "generalized Riemann integral" or the "Henstock-Kurzweil integral") that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral. Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader's understanding. Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given full exposure to the main classical results. The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises.

Table of Contents

Preface	ix
Part 1. Integration on compact intervals	1	(246)
Gauges and integrals	3	(20)
Some examples	23	(18)
Basic properties of the integral	41	(14)
The Fundamental Theorems of Calculus	55	(20)
The Saks-Henstock lemma	75	(14)
Measurable functions	89	(12)
Absolute integrability	101	(14)
Convergence theorems	115	(20)
Integrability and mean convergence	135	(16)
Measure, measurability and multipliers	151	(20)
Modes of convergence	171	(16)
Applications to calculus	187	(22)
Substitution theorems	209	(20)
Absolute continuity	229	(18)
Part 2. Integration on infinite intervals	247	(166)
Introduction to Part 2	249	(6)
Infinite intervals	255	(20)
Further re-examination	275	(24)
Measurable sets	299	(24)
Measurable functions	323	(24)
Sequences of functions	347	(66)
Appendixes
A Limits superior and inferior	365	(6)
B Unbounded sets and sequences	371	(2)
C The arctangent lemma	373	(2)
D Outer measure	375	(4)
E Lebesgue's differentiation theorem	379	(4)
F Vector spaces	383	(4)
G Semimetric spaces	387	(4)
H The Riemann-Stieltjes integral	391	(10)
I Normed linear spaces	401	(12)
Some partial solutions	413	(30)
References	443	(6)
Index	449	(8)
Symbol Index	457