**Graduate Studies in Mathematics**

Volume: 32;
2001;
458 pp;
Hardcover

MSC: Primary 26;
Secondary 28

Print ISBN: 978-0-8218-0845-0

Product Code: GSM/32

List Price: $72.00

Individual Member Price: $57.60

**Electronic ISBN: 978-1-4704-2086-4
Product Code: GSM/32.E**

List Price: $72.00

Individual Member Price: $57.60

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#### Supplemental Materials

# A Modern Theory of Integration

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*Robert G. Bartle*

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. A complete solutions manual is available separately.

#### Table of Contents

# Table of Contents

## A Modern Theory of Integration

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Part 1 Integration on Compact Intervals 116 free
- 1. Gauges and Integrals 318
- 2. Some Examples 2338
- 3. Basic Properties of the Integral 4156
- 4. The Fundamental Theorems of Calculus 5570
- 5. The Saks-Henstock Lemma 7590
- 6. Measurable Functions 89104
- 7. Absolute Integrability 101116
- 8. Convergence Theorems 115130
- 9. Integrability and Mean Convergence 135150
- 10. Measure, Measurability, and Multipliers 151166
- 11. Modes of Convergence 171186
- 12. Applications to Calculus 187202
- 13. Substitution Theorems 209224
- 14. Absolute Continuity 229244

- Part 2 Integration on Infinite Intervals 247262
- Appendixes 365380
- A: Limits superior and inferior 365380
- B: Unbounded sets and sequences 371386
- C: The arctangent lemma 373388
- D: Outer measure 375390
- E: Lebesgue's differentiation theorem 379394
- F: Vector spaces 383398
- G: Semimetric spaces 387402
- H: The Riemann-Stieltjes integral 391406
- I: Normed linear spaces 401416

- Some partial solutions 413428
- References 443458
- Index 449464
- Symbol Index 457472
- Back Cover Back Cover1474

#### Readership

Advanced undergraduates, graduate students and research mathematicians, physicists, and electrical engineers interested in real functions.

#### Reviews

The book presents its subject in a pleasing, didactically surprising, and well worth reading exposition. The proofs are, as a rule, easily understandable and the significance of the theorems that are worked through is illustrated by means of numerous examples. It can be recommended as a self-study book to every student with a basic foundation in analysis. It is also very suitable as a supplementary text for a course on integration on \(\mathbf{R}\).

-- Translated fromJahresbericht der Deutschen Mathematiker-Vereinigung

A comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral … There is an abundant supply of exercises which serve to make this book an excellent choice for a text for a course which would contain an elementary introduction to modern integration theory.

-- Zentralblatt MATH