Varieties of Integration
Dolciani Mathematical Expositions Series

By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation, with little attention paid to the relationships between them or to the historical issues that motivated their definitions. This book redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals using a common set of examples. This allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus. With a thorough set of appendices and exercises, and interesting historical context, this book is equally useful as a reference for mathematicians or as a text for a second undergraduate course in real analysis.

Preface; 1. A historical introduction; 2. The Riemann integral; 3. The Darboux integral; 4. A functional zoo; 5. Another approach: measure theory; 6. The Lebesgue integral; 7. The gauge integral; 8. Stieltjes-type integrals and extensions; 9. A look back; 10. Afterword: L2 spaces and Fourier series; Appendices: a compendium of definitions and results; Index.

C. Ray Rosentrater is a Professor of Mathematics at Westmont College, California, where he has also served as Department Chair and Associate Dean for Curriculum. He has been recognized as Westmont's Teacher of the Year in the Natural and Behavioral Sciences and has received the Faculty Research Award.