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Real Analysis: Theory of Measure and Integration

J. Yeh
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Jeff Ibbotson
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This rather voluminous selection is a compilation of theorems and proofs on measure theory and integration. According to the author they evolved from lecture notes for a course taught at UC Irvine.

The presentation begins with measures on sigma-algebras of sets and, after a strenuous workout with sets, segues to finitely and countably additive and sub-additive functions on such. There follows the usual stuff on outer measures, a construction of Lebesgue measure on \(\mathbb{R}\), measure spaces, and convergence in measure. This is then followed by an entire chapter devoted to the Lebesgue integral, signed measures and the Radon-Nikodym derivative.

A chapter on differentiation and integration follows. The presentation for this is very classical and oriented around the concepts of bounded variation and absolute continuity. Finally the classical Banach spaces appear in all their p-norm glory. All of the usual stuff is here: normed linear spaces, Hilbert Spaces, the triumvirate of the Uniform Boundedness Theorem, the Open Mapping Theorem and the Hahn-Banach Theorem. There is also a mountain of material on \(L^p\) spaces, Holder’s inequality, locally compact Hausdorff spaces, and Borel and Radon measures.

It’s all very complete, very dense. Every i is dotted and every t is crossed (as one of my old teachers used to say). At this point, after 500 pages of reading, the avid reader might wish to pause to catch her breath before plunging into the remaining 300 pages of this tome. They deal with extension of additive set functions, product measures, Lebesgue measure and integration on \(\mathbb{R}^n\), and Hausdorff measures and fractional dimension.

A word on exercises: this book has 394 of them and they run the gamut from nearly obvious to difficult. They are accessible for anyone who has done the reading and has a burning desire to learn the material. It was a pleasant surprise to me that a fair number of them actually bridged the yawning chasm between multivariable calculus and real analysis. These problems involved Cavalieri’s principle (stated in integral of measure form), various double integrals over domains in \(\mathbb{R}^2\), properties of the gamma function, and the spherical coordinate substitution for integrals over \(\mathbb{R}^n\). These would surely be wonderful motivation for anyone encountering this material for the first time.

I can easily imagine this book being used as a review source for graduate students before taking qualifying exams. The material is very dense, though, and I can only imagine a hapless undergraduate feeling much as if they were staring into the maw of Moby Dick as they page through this tome. On the other hand, if one is dedicated to learning all the details of measure theory and integration I can assert with absolute confidence that they are all here for the interested reader.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.

  • Measure Spaces
  • The Lebesgue Integral
  • Differentiation and Integration
  • The Classical Banach Spaces
  • Extension of Additive Set Functions to Measures
  • Measure and Integration on the Euclidean Space
  • Hausdorff Measures on the Euclidean Space
  • Appendices:
    • Digital Expansions of Real Numbers
    • Measurability of Limits and Derivatives
    • Lipschitz Condition and Bounded Derivatives
    • Uniform Integrability
    • Product-measurability and Factor-measurability
    • Functions of Bounded Oscillation