Integration and Modern Analysis

Graduate level analysis books are as abundant as… Well, what should the mot juste be here? Guilt? Lust? Hair loss? No: let’s go parochial and pick a mathematically resonant simile: Graduate analysis books are as common as tea served at department colloquia! Therefore, Integration and Modern Analysis by Benedetto and Czaja had better have something special to distinguish itself. The authors take up the gauntlet as follows: “More than half of this book is a fundamental graduate real variables course as we now teach it [at U Maryland]. Since there are excellent textbooks that generally cover the course material herein, part of this Preface renders an apologia for our content, presentation, and existence… Chapters 1, 2, 3 and parts of Chapter 4 and Appendix A [Functional Analysis] are the usual content of the first-semester graduate course in real analysis… [but] The remaining chapters, Chapters 6 through 9… could be viewed as eclectic… by some readers. We view them as essential to our intellectual vision of twentieth-century real analysis.”

And so the veil is about to be lifted: we’re about to learn why Integration and Modern Analysis should be our cup of tea. To wit: “Chapter 6 is devoted to deep results by Vitali, Nikodym, Hahn-Saks, Dieudonné, Dunford-Pettis, and [Zowie!] Grothendieck as they relate to interchanging limits and integration, and to the characterization of weak sequential convergence of measures.” Then we get: “…Riesz representation (RRT) in Chapter 7 in the setting of [Yes!] locally compact Hausdorff spaces. We begin historically with Riesz’ original proof and are led to Radon measures and Laurent Schwartz’ theory of distributions… RRT establishes the equivalence of the set-theoretic measure theory of the previous chapters with the theory of Radon measures considered functional-analytically as continuous linear functionals. [This is very elegant, and very important, no? Well, it gets even better:]… on R… RRT asserts that a Schwartz distribution is a bounded Radon measure if and only if it is the first distributional derivative of a function of bounded variation. [Gorgeous!] We view this material as a quantitative approach to apply measure theory in harmonic analysis, partial differential equations, and distribution theory.”

This feature of the book already suffices to distinguish it and recommend it to any one interested in a cogent but rapid way to get from introductory graduate analysis to some of the hot topics of the day: the connections with harmonic analysis and PDE should be sufficient to tip the balance. But Benedetto and Czaja offer even more: “Chapter 8 develops differentiation on… R^d… This material has also had significant impact on other areas of mathematics [:] We close in Chapter 9 by analyzing self-similar sets and fractals…”

Finally, in addition to the aforementioned Appendix A, on functional analysis, the authors present their Appendix B on Fourier analysis with the following attractive rationale: “In discussing Grothendieck’s idea of bringing certain cohomological concepts into algebraic geometry… Mumford wrote, ‘It completely turned the field upside down. It’s like analysis before and after Fourier. Once you get Fourier techniques, suddenly you have this whole deep insight into looking at a function’… It is for this reason, and going beyond the notion of a function to differentiation and integration, where Fourier analysis was also a driving force, that we have included as much Fourier analysis as we have…”

And one more quote from Benedetto’s and Czaja’s Preface, to cap things off: “One of our main themes is the notion of absolute continuity and its role as the unifying concept for the major results of the theory, viz., the fundamental theorem of calculus, … the Lebesgue dominated convergence theorem, … and the Radon-Nikodym theorem… The problem of taking limits under the integral sign… is in a very real sense the fundamental problem in real analysis…”

Thus Integration and Modern Analysis is not just your average cup of tea. Its goals go well beyond the usual prosaic objective of presenting rookie graduate students with a certain standard set of tools and skills in real analysis; Benedetto and Czaja aim to persuade the reader to their particular point of view and, indeed, to enlist him in their enterprise as outlined through the preceding propaganda. I think this is a legitimate enterprise and well worth foisting on the impressionable young, modulo their predisposition to analysis. And being proselytes for their cause, the authors try to stack the deck even more by concluding each chapter with a section titled “Potpourri and titillation,” containing fascinating historical items (for example, p. 201 ff. deals with the Scottish Café, and with such folks as Steinhaus, Mazur, Orlicz, Banach, and Ulam, and also with Moisej Zaretsky (check it out!)).

Well, Benedetto and Czaja have a wonderful product to sell and are right in doing such enthusiastic preaching of their cause. Additionally, the exposition is solid, the book is loaded with exercises, and is dripping with the authors’ expertise. If you incline in this direction of analysis, Integration and Modern Analysis is unquestionably your cup of tea.

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