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Book DetailsGraduate Studies in MathematicsVolume: 32; 2001; 458 ppMSC: Primary 26; Secondary 28
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 “HenstockKurzweil 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 firstyear 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 onethird of the exercises. A complete solutions manual is available separately.
ReadershipAdvanced undergraduates, graduate students and research mathematicians, physicists, and electrical engineers interested in real functions.

Table of Contents

Part 1. Integration on compact intervals

Chapter 1. Gauges and integrals

Chapter 2. Some examples

Chapter 3. Basic properties of the integral

Chapter 4. The fundamental theorems of calculus

Chapter 5. The SaksHenstock lemma

Chapter 6. Measurable functions

Chapter 7. Absolute integrability

Chapter 8. Convergence theorems

Chapter 9. Integrability and mean convergence

Chapter 10. Measure, measurability, and multipliers

Chapter 11. Modes of convergence

Chapter 12. Applications to calculus

Chapter 13. Substitution theorems

Chapter 14. Absolute continuity

Part 2. Integration on infinite intervals

Chapter 15. Introduction to Part 2

Chapter 16. Infinite intervals

Chapter 17. Further reexamination

Chapter 18. Measurable sets

Chapter 19. Measurable functions

Chapter 20. Sequences of functions

Appendixes

Appendix A. Limits superior and inferior

Appendix B. Unbounded sets and sequences

Appendix C. The arctangent lemma

Appendix D. Outer measure

Appendix E. Lebesgue’s differentiation theorem

Appendix F. Vector spaces

Appendix G. Semimetric spaces

Appendix H. RiemannStieltjes integral

Appendix I. Normed linear spaces

Some partial solutions

Solutions Manual


Additional Material

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 selfstudy 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 MathematikerVereinigung 
A comprehensive, beautifully written exposition of the HenstockKurzweil (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


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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 “HenstockKurzweil 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 firstyear 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 onethird of the exercises. A complete solutions manual is available separately.
Advanced undergraduates, graduate students and research mathematicians, physicists, and electrical engineers interested in real functions.

Part 1. Integration on compact intervals

Chapter 1. Gauges and integrals

Chapter 2. Some examples

Chapter 3. Basic properties of the integral

Chapter 4. The fundamental theorems of calculus

Chapter 5. The SaksHenstock lemma

Chapter 6. Measurable functions

Chapter 7. Absolute integrability

Chapter 8. Convergence theorems

Chapter 9. Integrability and mean convergence

Chapter 10. Measure, measurability, and multipliers

Chapter 11. Modes of convergence

Chapter 12. Applications to calculus

Chapter 13. Substitution theorems

Chapter 14. Absolute continuity

Part 2. Integration on infinite intervals

Chapter 15. Introduction to Part 2

Chapter 16. Infinite intervals

Chapter 17. Further reexamination

Chapter 18. Measurable sets

Chapter 19. Measurable functions

Chapter 20. Sequences of functions

Appendixes

Appendix A. Limits superior and inferior

Appendix B. Unbounded sets and sequences

Appendix C. The arctangent lemma

Appendix D. Outer measure

Appendix E. Lebesgue’s differentiation theorem

Appendix F. Vector spaces

Appendix G. Semimetric spaces

Appendix H. RiemannStieltjes integral

Appendix I. Normed linear spaces

Some partial solutions

Solutions Manual

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 selfstudy 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 MathematikerVereinigung 
A comprehensive, beautifully written exposition of the HenstockKurzweil (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