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Aspects of Integration : Novel Approaches to the Riemann and Lebesgue Integrals

By: (Author) John W. Lee , (Author) Ronald B. Guenther

Manufacture on Demand

Ksh 29,700.00

Format: Hardback or Cased Book

ISBN-10: 1032481129

ISBN-13: 9781032481128

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

Publisher: Taylor & Francis Ltd

Imprint: Chapman & Hall/CRC

Country of Manufacture: GB

Country of Publication: GB

Publication Date: Aug 24th, 2023

Print length: 160 Pages

Weight: 453 grams

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  • Description

  • Reviews

This book is comprised of two parts. The first part is devoted to the Riemann integral, and provides a novel approach, that are rarely found in other treatments of Riemann integration. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory.

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

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