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The Joys of Haar Measure

Joe Diestel and Angela Spalsbury
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 150
[Reviewed by
Jeff Ibbotson
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The book under review is a serious adventure in Measure Theory and not for the faint of heart. It is a wonderfully structured guide to some deep exploration of continuous quantity.

At the heart of its exposition is the question of what it means to assign lengths, areas, volumes and hyper-volumes to abstract sets sitting somewhere in n-dimensional space. The concept of measure is built from the ground up, but the text will likely be a difficult read for anyone not already possessing one-year course in abstract analysis at the level of Rudin’s Real and Complex Analysis. As the merest example, the casual reader encounters the sentence “Any uncountable \(G_\delta\) subset of \(\mathbb{R}\) contains a homeomorphic copy of the Cantor set” on page 16, with precisely none of these terms defined beforehand (admittedly, this is in the “Notes and Remarks” section of Chapter 1 and is not used in what follows but….it’s hard for me to imagine a dedicated undergraduate nodding along in agreement).

I think the authors’ prescription that “This book is aimed at an audience of people who have been exposed to a basic course in real variables...” a bit of an understatement. And yet, I think I see what they mean. Nearly all the arguments are complete and almost seem built out of just basic set theory. There are no concepts used beyond this (even though some of the arguments are long and often involve developing equivalence classes of objects). It really does seem as though everything is built out of basic concepts involving counting!

An early note following the Brunn-Minkowski Theorem encourages the reader to see that it is just a generalization of the arithmetic-geometric mean inequality. And the chatty voice of the authors (“apply the usual epsilonics”, “isn’t life beautiful?”) invests the subject with a clear encouragement that one really ought to care about the material. Other notes introduce so many fascinating connections with things like measurable cardinals, the Banach-Tarski paradox, combinatorial cliques and the “marriage problem,” that one may often feel like they have happened into a special seminar where the speaker relays information directly from their personal experiences and beliefs about mathematics. The approach is virtually the opposite of a dry-as-dust presentation without motivation. However, don’t mistake this for a lack of rigor. There is none here, not in the slightest degree.

The chapters unfold the story of invariant measures; it is not until Chapter 4 that we even meet the afore-mentioned Haar measure. A nice, steady introduction to topological groups comes first, and many of the standard results (Birkhoff-Kakutani theorem, matrix groups as locally compact groups, products of Hausdorff spaces are Hausdorff) are here with proofs. Chapter 4, however, is the heart of the exposition. This chapter features the beautiful work of Stefan Banach and develops his approach to the Lebesgue integral. It utilizes the Hahn-Banach theorem, averages and Banach limits, a general theory of congruence in compact metrizable groups, positive linear functionals and upper and lower integrals. It’s difficult not to see similarities to the Daniell integral and to expect a parallel development. But then Banach cleverly uses equivalence classes called “hyperfunctions” to establish a connection between the Riemann integral and the Lebesgue integral. The flow of ideas is quite beautiful and elementary. In fact, I think the proof of the Hahn-Banach theorem is the clearest one I have ever seen. Its use blossoms forth in the development of the Lebesgue integral in a seamless way that really strengthens a sense of its importance. No quoting difficult big theorems just for heck of it here! The authors mention that heretofore this argument of Banach has languished in twilight as an appendix to Saks’s Theory of the Integral. Kudos for presenting such a beautiful piece of work!

The rest of the book develops invariant measures, homogeneous spaces, the Peter-Weyl theorem for unitary representations (but no other harmonic analysis), Haar measure on uniform spaces, G-invariance and finally invariant measures on Polish groups. All this in some 300 pages makes for a pretty heady brew.

While I am sure the hope of the authors is to create new lovers of measure theory, my fear is that none who are not already that will make it to the final chapters. A bit more motivation and a few well-placed problems would strengthen the use of this book for self-study purposes. The reader who can meet the evident joy of the authors with her own determination to hear their story will be amply rewarded through reading this book.

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy. He spends much of his time reading, playing ping pong and raising beagles.