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A Course on Integration Theory

K. Chandrasekharan
Hindustan Book Agency
Publication Date: 
Number of Pages: 
Texts and Readings in Mathematics 8
[Reviewed by
Allen Stenger
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This is a conventional development of integration theory, starting with measure spaces. The Author’s Note (p. vii) states that “The goal here is concision, clarity, and accuracy,” and the book meets these goals. By American standards this is not a “Course”, as it has no exercises or motivation, and only a modest set of examples. The present volume is a reprint of the original 1996 publication.

I think this book works best as a review or as an introduction to the more abstract theories of integration. It is too abstract for a first course in integration, as it starts immediately with the most abstract theories and only (a great deal later) deals with integration on the real line. A book that takes a similar approach, but with more exercises and more concreteness, is Hewitt & Stromberg’s Real and Abstract Analysis.

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.

  • Author’s Note
  • Chapter I. Integration on a measure space
    • I.1 Measure spaces
    • I.2 Integration of simple functions
    • I.3 Integrable functions
    • I.4 Measurable functions
    • I.5 Non-negative-valued functions
    • I.6 Convergence Theorems
    • Appendix
  • Chapter II. The Lebesgue spaces
    • II.1 Definition
    • II.2 Inequalities of Schwarz, Hölder, and Minkowski
    • II.3 Completeness of L1(M)
  • Chapter III. The outer measure and its applications: the Lebesgue measure
    • III.1 Outer measure
    • III .2 Outer measurability
    • III.3 Lebesgue measure on the real line
    • III.4 Complete measure spaces
    • III.5 Riemann integrable functions
  • Chapter IV. Product measures and multiple integrals
    • IV.1 Existence of product measures
    • IV.2 Fubini’s theorem
  • Chapter V. Set functions and their derivatives
    • V.1 Set functions of bounded variation
    • V.2 Absolutely continuous set functions
    • V.3 Functions of bounded variation on the real line
    • V.4 Absolutely continuous functions on the real line
  • Index