A Modern View of the Riemann Integral

This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue's theory, the author embarks on an exploration rooted in Riemann's original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications.

This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor.

A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.

Showcases the full capabilities of the Riemann integral from Riemann's original viewpoint Establishes new results and methods for approaching computations and applications Offers numerous historical insights and details

Auteur

Alberto Torchinsky is Emeritus Professor of Mathematics at Indiana University Bloomington. His research interests are centered on harmonic and real analysis. He has authored several other books, including the widely cited LNM 1381, Weighted Hardy Spaces, with Jan-Olov Strömberg. Prior to Indiana University, he held positions at the University of Illinois and Cornell University, having received his PhD at the University of Chicago under A. P. Calderón.

Résumé

"There are 108 references. Most of them make use of the Riemann integral and a lot of them are related to deep questions. ... based on the results in this book, that new results will be obtained in the future and that this book will represent the state of the art in 2022." (Richard Becker, Mathematical Reviews, September, 2023)

"The book presents a detailed and well-written treatise on the Riemann integral and its many variants and properties. ... 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 ... . 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." (Sylvester Eriksson-Bique, zbMATH 1509.26003, 2023)

Contenu
Preface.- Chapter 1. Introduction.- Chapter 2. The Riemann Integral.- Chapter 3. A Convergence Theorem.- Chapter 4. The Modified Riemann Sums.- Chapter 5. The Pattern and Uniform Integrals.- Chapter 6. The Improper and Dominated Integrals.- Chapter 7. Coda.- Appendix I.- Appendix II.- References.- Index.