I think that there must be hundreds of books that treat the Lebesgue integral, but I expect that very few of them are as readable as David Bressoud’s A Radical Approach to Lebesgue’s Theory of Integration. Several times, as I wrote this review, I found myself picking up the book, opening to a random page and reading — and continuing to read because I was captured by the story and I wanted to see what came next. How many analysis books do you know that invite this kind of browsing? Körner’s Fourier Analysis, a favorite of mine, is one that comes to mind, and I’m sure there are a few others — but not very many.
Why does the book have this appeal? For one thing, it is written with clarity and directness, one scholar to another. For another, it is historically savvy, sensitive to the context of the developments it describes. The author says, “…this is a textbook informed by history, attempting to communicate the motivations, uncertainties, and difficulties surrounding the key concepts.” We get a real sense of the challenges faced by Lebesgue, Riemann, Weierstrass, Cantor and a host of others. Pedagogically this makes a lot of sense. The author, paraphrasing Luzin, puts it this way: “…the ideas, methods, definitions, and theorems of this study are neither natural nor intuitive.” He goes on to say, “Here, more than anywhere else in the advanced undergraduate or beginning graduate curriculum, the historical context is critical to developing an understanding of the mathematics.”
The author’s approach focuses on what he calls the five big questions:
- When does a function have a Fourier series that converges (to that function)?
- What is integration?
- What is the relationship between integration and differentiation?
- What is the relationship between continuity and differentiability?
- When can an infinite series be integrated by integrating each term?
The first chapter describes what was known or believed about these questions by 1850, and the rest of the book moves on from there. We see — we participate in — the development of the Lebesgue integral on the real line up through the dominated convergence theorem and Egorov’s theorem. But this is not by any means a typical theorem-proof-remark approach. At several points in the historical process we stop with the author to review our progress on the five big questions. We take a hard look at the deficiencies of the Riemann integral, examine what we’ve perhaps forgotten about the amazing complexity of the real numbers, and see what difficulties nowhere dense sets can cause with the fundamental theorem of calculus. The final two chapters bring us to a conclusion with Lebesgue’s fundamental theorem of calculus and the convergence of Fourier series.
What I like best about Bressoud’s treatment is the way he conveys the sense that real analysis, and the Lebesgue theory in particular, developed in time, in fits and starts, with contributions from real people who sometimes made mistakes. We also see real analysis as more cohesive and less driven by bizarre counterexamples (as it must sometimes seem to students). Neither does Bressoud avoid the anomalies — after all, they were a driving force in the search for a better integral. The treatment of the interplay between analysis and set theory on the real line is important and is handled very well here. I particularly liked the discussion of non-measurable sets, which faces head-on issues related to the axiom of choice, offering illumination instead of mystification.
Together with the author’s earlier A Radical Approach to Real Analysis, this book offers a strong and approachable introduction to analysis. The current book is aimed at advanced undergraduates or beginning graduate students. It does not strive for the same breadth as other treatments of graduate-level real analysis; the development is limited to the real line and several of the usual topics from functional analysis are not discussed. Nonetheless, this would have been my first choice as a student.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.