A Modern View of the Riemann Integral

Description

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. Preface Contents 1 Introduction 2 The -Riemann Integral 2.1 The Volume of a Lake 2.2 Rate of Approximation 2.3 Quadrature Rules 2.4 Roots of Nonlinear Equations Monotonicity of Riemann Sums 3 A Convergence Theorem 3.1 The Riemann–Lebesgue Lemma 3.2 The Weierstrass Approximation Theorems 4 The Modified -Riemann Sums 4.1 Uniformly Distributed Sequences 5 The Pattern and Uniform Integrals 6 The Improper and Dominated Integrals 6.1 Improper Integrals of the First Type 6.2 The Dominated Integral 6.3 Improper Integrals of the Second Type 7 Coda Appendix I Appendix I Change of Variable Formulas for Riemann Integrals The Substitution Formula The Substitution Formula Change of Variable Formula, Riemann Integral Substitution Formula, Riemann–Stieltjes Integral Change of Variable Formula, Riemann–Stieltjes Integral Caveat Appendix II Appendix II Cauchy Integrability Implies Riemann Integrability References Index