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.

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

Table of Contents

• Gauges and integrals
• Some examples
• Basic properties of the integral
• The fundamental theorems of calculus
• The Saks-Henstock lemma
• Measurable functions
• Absolute integrability
• Convergence theorems
• Integrability and mean convergence
• Measure, measurability, and multipliers
• Modes of convergence
• Applications to calculus
• Substitution theorems
• Absolute continuity

• Introduction to Part 2
• Infinite intervals
• Further re-examination
• Measurable sets
• Measurable functions
• Sequences of functions
• Limits superior and inferior
• Unbounded sets and sequences
• The arctangent lemma
• Outer measure
• Lebesgue's differentiation theorem
• Vector spaces
• Semimetric spaces
• Riemann-Stieltjes integral
• Normed linear spaces
• Some partial solutions
• References
• Index
• Symbol index