Jewgeni Dshalalow states in the Preface to the book under review that his goal is to focus on the essentials of set theory, topology and measure theory for beginning graduate students in real analysis. He then notes that he emphatically wants to avoid the trap present in so many books that purportedly deal with “essentials,” viz. to provide the equivalent of lecture notes. Instead, he wishes to provide the reader with “a thorough and … rigorous treatment of key analysis subjects … by providing [him] with copious illustrations, examples, and exercises with selected solutions.” Additionally he presents “detailed discussions and blueprints” when the going is likely to get particularly sticky, “so that the reader will not get lost or intimidated in a long chain of proofs and notions.”
In other words, Dshalalow seeks to address a very real and widespread pedagogical problem in contemporary graduate schools. It is obviously the case that, in particular in a subject like real analysis, the rookie, fresh out of undergraduate school, is bound to reel uncontrollably under the psychological pressure that would be exercised by the Lebesgue integral, from Egorov’s theorem to the Riesz representation theorem, all in the span of mere weeks, with a lot more to follow (often functional analysis topics, for example) — as Maurice Chevalier used to croon, “Yes, I remember it well!”
Dshalalow goes against the grain in no uncertain terms, stating that “[b]ecause real analysis, in its proper form, is likely to be the first abstract mathematics course that many student take, the associated topics should be taught in a strict order starting with basic set theory followed by point set topology and then measure theory and integration … wasting no valuable time on Euclidean spaces [and thus reducing] Lebesgue measure and [the] Lebesgue integral to mere illustrations.” To be sure, this is, as the book’s title specifies, abstract analysis.
It is indeed a very different world than it was even (what seems) a short time ago, and there is really no arguing with Dshalalow’s premises: the first year graduate student of today is a very, very different animal than what he was even ten years back. Dshalalow notes (and I should like to suggest, laments) that “mathematical education has become increasingly more focused on applications and less on theory” and observes that accordingly “academics have repurposed courses in set theory, topology … and measure and integration as a real analysis course”; the present book is his attempt to make sure that in this brave new world of adaptation to all sorts of exigencies coming from applications, the sciences, pedagogical maneuvers, and who knows what else, the epsilons and deltas are still given their proper due.
With these goals in mind, then, Foundations of Abstract Analysis starts with the very basics of set theory, does some linear algebra, hits metric spaces with a lot of gusto (convergence, completeness, compactness), immediately followed by point-set topology (properly singling out Hausdorff spaces), and with this chapter closes the first part of the book. The subsequent second part is devoted entirely to measure and integration theory, the discussion culminating with a long and careful treatment of “integration in abstract spaces.” This (sixth) chapter includes a treatment of the Riemann and Lebesgue integrals in the same subsection, as well as treatments of, e.g., signed measures, complex measures, Radon-Nikodym for both of these, and finally the business of finite product theorem (Fubini, etc.).
Beyond this, the book’s third part, from p.535 to p.713, is devoted to detailed solutions to many of the problems strewn throughout the text, of which, indeed, there are very many. And these range from the elementary and routine to the more challenging if not downright sporty: if a student works these problems diligently (and not just those with “the answers in the back of the book,” of course: this is not freshman calculus for engineering students), he will indeed have gotten to where Dshalalow wants him: ready for real real analysis, so to speak, and certainly prepared for sundry ersatz enterprises that accrue to real analysis these days (or even more classically legitimate ones).
Foundations of Abstract Analysis is a herculean effort on the author’s part, and promises to be a very important solution to a huge pedagogical problem in graduate education today, i.e. to present very serious mathematics to an essentially underprepared population, certainly as compared to ages past when academic mathematics was so much less egalitarian. There is no question that Foundations of Abstract Analysis is a very timely book; it promises to be a very important player in the game.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.