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Sets, Functions, Measures, Volume 2: Fundamentals of Functions and Measure Theory

Valeriy K. Zakharov, Timofey V. Rodionov, and Alexander V. Mikhalev
Walter de Gruyter
Publication Date: 
Number of Pages: 
Studies in Mathematics 68/2
[Reviewed by
Michael Berg
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Most suggestively, the authors’ dedication of this duet of books reads as follows: “The authors dedicate their book to the centenary of Felix Hausdorff’s outstanding book Set Theory. What they offer — and it is a lot! — is evidently intended to cover an orbit of comparable scope to Hausdorff’s breakthrough monograph, adjusted one century into the future, and therefore covering a broader and deeper swath. The book referenced above is actually the 1937 second edition of Hausdorff’s Grundzüge der Mengenlehre, which originally appeared in 1917 and is credited with the innovation of coherently presenting measure theory and topology (at first called analysis situs) as fitted properly under the umbrella of the theory of sets. The historical moment of this event can be gauged from the fact that Georg Cantor was still alive when Hausdorff’s book appeared: Cantor passed away in 1918. Additionally Poincaré had died only five years before, at only 58 years of age, Brouwer was only 38, and Lebesgue was 42: topology and measure theory were still very young at the time that Hausdorff presented his work. The book(s) under review seek to achieve something parallel, taking into account what has happened over the intervening one hundred years, and, again, it is a lot!

The first volume of the pair is focused on fundamentals of set theory and number theory. The latter is not to be confused with what we usually call by that name: there is nothing in these books about algebraic number theory, analytic number theory, geometric number theory, transcendental number theory, and so on. Instead, the concept on number itself is developed in very great detail, meaning that we meet ordinals and cardinals, Zermelo and Fraenkel, and a lot more. But don’t expect to find, say, the law of quadratic reciprocity.

In point of fact, set theory, in its broader sense, dominates all of Volume I, from a careful discussion of classes and sets to “the compactness theorem for generalized second-order language.” In my undergraduate days long ago, I did come across a compactness theorem of this flavor, but, as I recall, for first-order sentential calculus. Perhaps that result — as also the present one — qualifies as exotica, as far as mainstream mathematical studies go: it’s mathematical logic in action, and concerns the “fine structure” that we tend to take for granted or ignore because, after all, what does it have to do with the normal activities in, say, real analysis? But the devil is in the details, of course, and the book under review seeks to do justice to just these minutiae which hide in the shadows, or which we acknowledge as hiding there but often prudently choose to pass over.

Thus, the authors spend a great deal of time and effort on number systems, i.e. the bestiary we tend to take for granted, and this is part and parcel of doing justice to foundational questions. Specifically, after their first chapter they present three lengthy and detailed discussions, A, B, and C, dealing with, respectively, (A) characterizing von Neumann-Bernays-Hilbert and Zermelo-Fraenkel set theories, (B) what the authors call the local theory of sets, and (C) the aforementioned compactness theorem. They offer (B) as a foundation for category theory and connect it to Zermelo-Fraenkel set theory. Suffice it to say that, even now in this post-Grothendieck era, any discussion of foundational questions surrounding category theory is very important, both as scholarship and as high level pedagogy.

The attention paid to detail in Volume I is rather impressive. Early on, we encounter “logical axiom schemes on the theory of classes and sets,” which engenders a thorough development of the indicated fundamental notions, and thereafter there are major benefits to be reaped. We soon get to transfinite induction, a lot of Cantorian set theory (the theory of cardinal numbers), the inner life of the real numbers including Cantor completeness and Dedekind completeness, and even the theory of nets. Under heading (A) we encounter, besides ZF and NBG set theories, Tarski sets: we’re doing model theory now. Under heading (B) we encounter things that would warm Gödel’s heart: relative consistency, undeducibility, and finite axiomatizability, for example. Finally, under heading (C) we reach such arcana as “uncountable models of the second-order generalized Peano-Landau arithmetic.” To be sure, this is not for everyone, and the reader had better be prepared for some very serious mathematical logic and set theory. But it is clearly far-reaching material, truly geared to lay a foundation for function and measure theory in this millennium.

So we come to Volume II, dealing specifically with the two subjects just mentioned. The same thoroughness and attention to detail is found here, too, as well as topics that are unexpected to those of us who are, so to speak, mere users of the material, ubiquitous though it might be. For (an accessible) example, the authors explicitly address “the pointwise and uniform convergence of nets and sequences of functions,” rather than just looking at sequences. And, to drop a familiar name or two, we find material on “correlations between Baire’s and Borel’s functional collections” — a lot more than what one encounters in the regular scheme of things.

Volume II is split into two long chapters (Ch.2, Ch.3) and a discussion headed (D), covering, respectively, function theory, measure theory, and “Historical notes on the Riesz-Radon-Fréchet problem of characterization of Radon integrals as linear functionals.” The measure theory chapter is particularly well done, I think, with a lot of material on the Lebesgue integral, the Radon integral (in the context of topological spaces with measures). Special attention paid to the Riemann integral (only at the end of the chapter). Section (D), heading in the direction of functional analysis, is very impressive and informative. It sports such topics as Riesz’ original representation theorem, and then, in that context, material on BV (bounded variation) functions, the integral of Riemann and Stieltjes, and “[the] construction of the integral corresponding to a given functional.”

This set of two volumes evinces serious scholarship, and appears to do precisely what the authors set out to do, in homage to what Hausdorff did a century ago. It is a very valuable piece of work.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.