Aspects of Integration

Novel Approaches to the Riemann and Lebesgue Integrals

Ronald B. Guenther, John W. Lee

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English

Chapman & Hall/CRC
24 August 2023

Discrete mathematics; Number theory; Functional analysis & transforms; Combinatorics & graph theory

Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals is comprised of two parts. The first part is devoted to the Riemann integral, and provides not only a novel approach, but also includes several neat examples that are rarely found in other treatments of Riemann integration. Historical remarks trace the development of integration from the method of exhaustion of Eudoxus and Archimedes, used to evaluate areas related to circles and parabolas, to Riemann’s careful definition of the definite integral, which is a powerful expansion of the method of exhaustion and makes it clear what a definite integral really is.

The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory. Our approach is novel in part because it uses integrals of continuous functions rather than integrals of step functions as its starting point. This is natural because Riemann integrals of continuous functions occur much more frequently than do integrals of step functions as a precursor to Lebesgue integration. In addition, the approach used here is natural because step functions play no role in the novel development of the Riemann integral in the first part of the book. Our presentation of the Riesz-Nagy approach is significantly more accessible, especially in its discussion of the two key lemmas upon which the approach critically depends, and is more concise than other treatments.

Features

Presents novel approaches designed to be more accessible than classical presentations A welcome alternative approach to the Riemann integral in undergraduate analysis courses Makes the Lebesgue integral accessible to upper division undergraduate students How completion of the Riemann integral leads to the Lebesgue integral Contains a number of historical insights Gives added perspective to researchers and postgraduates interested in the Riemann and Lebesgue integrals

By: Ronald B. Guenther, John W. Lee
Imprint: Chapman & Hall/CRC
Country of Publication: United Kingdom
Dimensions: Height: 234mm, Width: 156mm,
Weight: 400g
ISBN: 9781032481128
ISBN 10: 1032481129
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics
Pages: 160
Publication Date: 24 August 2023
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Hardback
Publisher's Status: Active

I. A Novel Approach to Riemann Integration. 1. Preliminaries. 1.1. Sums of Powers of Positive Integers. 1.2. Bernstein Polynomials. 2. The Riemann Integral. 2.1. Method of Exhaustion. 2.2. Integral of a Continuous Function. 2.3. Foundational Theorems of Integral Calculus. 2.4. Integration by Substitution. 3. Extension to Higher Dimensions. 3.1. Method of Exhaustion. 3.2. Bernstein Polynomials in 2 Dimensions. 3.3. Integral of a Continuous Function. 4. Extension to the Lebesgue Integral. 4.1. Convergence and Cauchy Sequences. 4.2. Completion of the Rational Numbers. 4.3. Completion of C in the 1-norm. II. Lebesgue Integration. 5. Riesz-Nagy Approach. 5.1. Null Sets and Sets of Measure Zero. 5.2. Lemmas A and B. 5.3. The Class C1. 5.4. The Class C2. 5.5. Convergence Theorems. 5.6. Completeness. 5.7. The C2-Integral is the Lebesgue Integral. 6. Comparing Integrals. 6.1. Properly Integrable Functions. 6.2. Characterization of the Riemann Integral. 6.3. Riemann vs. Lebesgue Integrals. 6.4. The Novel Approach. A. Dinis Lemma. B. Semicontinuity. C. Completion of a Normed Linear Space. Bibliography. Index.

Ronald B. Guenther is an emeritus professor in the Department of Mathematics at Oregon State University. His career began at the Marathon Oil Company where he served as an advanced research mathematician at its Denver Research Center. Most of his career was spent at Oregon State University, with visiting professorships at the Universities of Hamburg and Augsburg, and appointments at research laboratories in the United States and Canada, and at the Hahn-Meitner and Weierstrass Institutes in Berlin. His research interests include mathematically modeling deterministic systems and the ordinary and partial differential equations that arise from these models. John W. Lee is an Emeritus Professor in the Department of Mathematics at Oregon State University, where he spent his entire career with sabbatical leaves at Colorado State University, Montana State University, and many visits as a guest of Andrzej Granas at the University of Montreal. His research interests include topological methods use to study nonlinear differential and integral equations, oscillatory properties of problems of Sturm-Liouville type and related approximation theory, and various aspects of functional analysis and real analysis, especially measure and integration.