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Varieties of Integration

C. Ray Rosentrater
MAA Press
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 51
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on

Mathematicians typically encounter two or three versions of the integral during their training, no matter what their ultimate specialization turns out to be. We all probably see at least the Riemann, Lebesgue and Stieltjes integrals. Often enough the relationships between these integrals are not explored very deeply. But the current book does just that in a presentation aimed at undergraduates.

The author has three main goals. The first is to develop several versions of the integral and carefully compare one with another. The second is to explore the rich historical background, at least in a limited way. The third is to use the subject of the integral as a means of cultivating mathematical maturity in the students using his book. He never states it in those terms, but that’s clearly what’s going on.

The book begins with early history, with what the author calls Greek foreshadowing, and moves on to the contributions of the major figures: Newton and Leibnitz, Cauchy, Riemann and Darboux, then Lebesgue and finally Henstock and Kurzweil. It’s not a lengthy history and by no means a complete one, but it gives some valuable context.

The treatment of integrals begins naturally enough with Riemann. The author develops Riemann’s approach slowly and carefully, proves the basic theorems and investigates the integral’s convergence properties. Some of the most awkward features of the Riemann integral — partitions, partition refinement and the complications of computing Riemann sums — were avoided by Darboux who introduced suprema and infima into the definition of the integral.

With the Lebesgue integral we face a somewhat bigger step with the introduction of measure theory. The author wisely introduces the transition to Lebesgue with a discussion of the “functional zoo”: functions (including those of Dirichlet, Cantor and Volterra) that challenge more naïve notions of the integral. Then after Lebesgue’s integral we meet the gauge integral (maybe better known as Henstock-Kurzweil) followed by the Stieltjes integral and various extensions. The gauge integral extends the class of integrable functions beyond Riemann without measure theory and guarantees that every derivative is integrable.

Three questions provide a unifying thread as they are posed for each approach to the integral: What is the class of integrable functions? What are the convergence properties? Which gives the best form of the Fundamental Theorem of Calculus?

In his preface the author addresses the student reader and emphasizes that his goal is to help the student understand and appreciate what is needed to formalize and extend the ideas of integral calculus and in so doing to teach how to think as a mathematician and become a reader of professional mathematics.

Exercises are important here and they are used very effectively. “Filling the gap” exercises, as one might expect, ask students to go back and incorporate missing details in the author’s proofs. As the book proceeds, the author wants students to notice and fill the gaps as they read without being explicitly directed. “Deeper reflection” exercises are more varied. Some ask for analysis of definitions and structure of the proofs, some for counterexamples, others for extension of results in the text or for reconciling different approaches.

This would be a challenging text for undergraduates but a clearly valuable one for anyone contemplating a mathematical career. The book might work best in a second term of an introductory analysis course, or in a special topics course. 

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. 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: Ls Spaces and Fourier Series
Appendices: A Compendium of Definition and Results
About the Author