Aspects of integration : novel approaches to the Riemann and Lebesgue integrals
著者
書誌事項
Aspects of integration : novel approaches to the Riemann and Lebesgue integrals
（Monographs and research notes in mathematics）
Chapman & Hall/CRC, 2024
- : hbk
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注記
Includes bibliographical references (p. 139-141) and index
内容説明・目次
内容説明
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.
目次
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.
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