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Theories of Integration

Douglas S. Kurtz and Charles W. Swartz
World Scientific
Publication Date: 
Number of Pages: 
Series in Real Analysis 13
[Reviewed by
Allen Stenger
, on

This advanced undergraduate text gives an in-depth look at the theory of the Lebesgue and Henstock-Kurzweil integrals, and a less-detailed but still thorough look at the Riemann and McShane integrals. The flavor of the book is to consider different definitions of integrals and investigate the consequences in depth. Each integral has an independent development in its own chapter, and there are references to the other chapters to compare and contrast the integrals (although a centralized comparison table would have been very helpful). There are numerous exercises at the end of each chapter.

The book begins with some historical remarks on area and integrals, followed by a chapter developing the Riemann integral, focusing on criteria for integrability. The development of the Lebesgue integral in the following chapter is very conventional, based on outer measure and approximation by simple functions. The discussion includes all the theorems that would be included in a real analysis course, including the Bounded and Dominated Convergence Theorems, the Fubini theorem, and the Tonelli theorem. One peculiarity of the treatment is that, although it includes the Riesz-Fischer theorem and the Riemann-Lebesgue lemma, there is no mention of Fourier series. Putting Fourier series on a solid base was an early triumph of the Lebesgue theory.

The Henstock-Kurzweil integral has a number of equivalent formulations under the names Denjoy integral, Perron integral, gauge integral, and generalized Riemann integral. Ralph Henstock (1923–2007) did the best job of publicizing the integral, which he pushed for pedagogical reasons because this integral has the intuitive appeal of the Riemann integral and allows one to achieve the power of the Lebesgue integral without all the measure-theory infrastructure. The approach here, though, is closer to Denjoy, where the motivation was to preserve the Fundamental Theorem of Calculus: every derivative should be integrable (and produce the original function). The development here roughly parallels that of the Lebesgue integral chapter, with the differences being pointed out as you go along. A final chapter covers the McShane integral, a different generalization of the Riemann integral that is equivalent to the Lebesgue integral.

A somewhat similar book is Burk’s A Garden of Integrals. That book is also aimed at undergraduates and covers more types of integrals, although in less depth. A different approach is in Bressoud’s A Radical Approach to Lebesgue’s Theory of Integration, that gives a much better idea of how the Lebesgue theory developed, and that includes a comparative discussion of the generalized Riemann integral.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • Introduction:
    • Areas
    • Exercises
  • Riemann Integral:
    • Riemann's Definition
    • Basic Properties
    • Cauchy Criterion
    • Darboux's Definition
    • Fundamental Theorem of Calculus
    • Characterizations of Integrability
    • Improper Integrals
    • Exercises
  • Convergence Theorems and the Lebesgue Integral:
    • Lebesgue's Descriptive Definition of the Integral
    • Measure
    • Lebesgue Measure in ℝn
    • Measurable Functions
    • Lebesgue Integral
    • Riemann and Lebesgue Integrals
    • Mikusinski's Characterization of the Lebesgue Integral
    • Fubini's Theorem
    • The Space of Lebesgue Integrable Functions
    • Exercises
  • Fundamental Theorem of Calculus and the Henstock–Kurzweil Integral:
    • Denjoy and Perron Integrals
    • A General Fundamental Theorem of Calculus
    • Basic Properties
    • Unbounded Intervals
    • Henstock's Lemma
    • Absolute Integrability
    • Convergence Theorems
    • Henstock–Kurzweil and Lebesgue Integrals
    • Differentiating Indefinite Integrals
    • Characterizations of Indefinite Integrals
    • The Space of Henstock–Kurzweil Integrable Functions
    • Henstock–Kurzweil Integrals on ℝn
    • Exercises
  • Absolute Integrability and the McShane Integral:
    • Defintions
    • Basic Properties
    • Absolute Integrability
    • Convergence Theorems
    • The McShane Integral as a Set Function
    • The Space of McShane Integrable Functions
    • McShane, Henstock–Kurzweil and Lebesgue Integrals
    • McShane Integrals on ℝn
    • Fubini and Tonelli Theorems
    • McShane, Henstock–Kurzweil and Lebesgue Integrals in ℝn
    • Exercises