This book provides an elementary, self-contained presentation of the integration processes developed by Lebesgue, Denjoy, Perron, and Henstock. The Lebesgue integral and its essential properties are first developed in detail. The other three integrals are all generalizations of the Lebesgue integral that satisfy the ideal version of the Fundamental Theorem of Calculus: if \(F\) is differentiable on the interval \([a,b]\), then \(F'\) is integrable on \([a,b]\) and \(\int _a^b F'= F(b) - F(a)\). One of the book's unique features is that the Denjoy, Perron, and Henstock integrals are each developed fully and carefully from their corresponding definitions. The last part of the book is devoted to integration processes which satisfy a theorem analogous to the Fundamental Theorem, in which \(F\) is approximately differentiable. This part of this book is preceded by a detailed study of the approximate derivative and ends with some open questions. This book contains over 230 exercises (with solutions) that illustrate and expand the material in the text. It would be an excellent textbook for first-year graduate students who have background in real analysis.

Readership

First year graduate students in mathematics.

Table of Contents

• Lebesgue measure
• Measurable functions
• The Lebesgue integral
• Bounded variation and absolute continuity
• Darboux and Baire class one functions
• Functions of generalized bounded variation
• The Denjoy integral
• The Perron integral
• The Henstock integral
• The McShane integral
• Equivalence of integrals
• Integration by parts
• Convergence theorems
• Approximate derivatives
• The Khintchine integral
• The approximately continuous Henstock integral
• The approximately continuous Perron integral
• Solutions to exercises
• References
• Notation index
• Subject index