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.
General
Imprint: |
Taylor & Francis
|
Country of origin: |
United Kingdom |
Series: |
Chapman & Hall/CRC Monographs and Research Notes in Mathematics |
Release date: |
August 2023 |
Authors: |
Ronald B. Guenther
• John W. Lee
|
Dimensions: |
234 x 156mm (L x W) |
Pages: |
160 |
ISBN-13: |
978-1-03-248112-8 |
Categories: |
Books
|
LSN: |
1-03-248112-9 |
Barcode: |
9781032481128 |
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