A modern view of the Riemann integral (Q2160161)

The book presents a detailed and well-written treatise on the Riemann integral and its many variants and properties. The Riemann integral often appears an as object feared by students in their Real Analysis classes. For this reason perhaps, it is often relegated to the end matter of the courses and is covered only to the extent absolutely necessary. Later, graduate courses focusing on the Lebesgue integral lead many students to believe that the Riemann integral is some sort of historical remnant which has long ago been superseded by later developments. However, for a number of reasons -- such as those coming from numerical analysis, harmonic analysis or in expressions concerning improper integrals - the Riemann integral remains an important concept, and its analysis is a source of several useful ideas. While the Riemann integral is important, it is challenging to find good and self-contained references. For example, Rudin presents the more technical Riemann-Stieltjes integral, and overlooks many basic results. Others develop so little of the theory, that the connections of the integrals to oscillations of functions is missed. This results in a problem for educators in giving good references on fundamental properites of the Riemann integral. This is a gap, which the present book is able to significantly patch up. Torchinsky does an admirable job of giving a comprehensive, modern and elementary treatise that includes much of the crucial facts: Riemann-Lebesuge Lemmas, monotonicy properties, (basic) numerical integration results with error bounds, improper integrals, patterned integrals, Stone-Weierstrass and a bit of the relationship to Fourier series. Additionally, the discussion is intertwined with intriguing historical discussion, many references, and identifying connections to applications -- such as the limnologist's problem of approximating the volume of a lake. Although, perhaps on some occasions the side tracks last a bit too long, and their connection to the main narrative is loose. The presentation is nearly completely self-contained, and quite flawless. It is easy for a graduate level student to follow, although quite a bit of mathematical maturity is presumed. Further, the text does assume that a student has seen the Riemann integral and studied its basic properties, since it does not cover them in much detail. Even some undergraduates may be able to follow the text, but this may need, as the author points out, appropriate guidance. One good example of the type of theorem covered in the book is the following. Consider the improper integral \[ \int_0^\infty f(x)dx, \] and the limits of the Riemann sums \[ \lim_{h\to 0}\sum_{n=0}^\infty h f(nh). \] When do they agree? This question is a bit subtle, and even the case where \(f:[0,\infty)\to[0,\infty)\) is monotone decreasing, is rarely proven in detail. The present book gives a clear theorem for when the equality holds, and a complete proof in Chapter 6. The book is not quite a text book, but an independent source book for results. It is very useful say for graduate students preparing for qualifying exams, and who need to strengthen their skills with the Riemann integral. It is also useful for a researcher interested if their research involves variants of the Riemann integral. The proofs could be used in instruction, and are often quite nice. At times the text maybe does a disservice by insisting too much on formally simple, but less intuitive, proofs. For example, the discussion on the Stone-Weierstrass theorem seems to the referee to be such. The error estimates for numerical integration methods are presented very concisely. However, perhaps this portion of the book is a bit harder to learn from, since the proofs appear a bit as tricks and without broader context. Further, the numerical section does not cover higher order methods, or other nice topics, such as Chebychev interpolation. To the reviewer, it seems the role of integration by parts could have been emphasized.