The integral
Subtitle: a crux for analysis
By: Steven G. Krantz
Series: Synthesis lectures on mathematics and statistics -- # 9
Publisher: Morgan & Claypool
Published: 2011
Place of publication: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA)
Language: eng
Format: [electronic resource] :
Number of pages: 93
ISBN-13: 9781608456147; 9781608456130
Subjects: Lebesgue integral; Integral theorems; Riemann integral
Other titles: Synthesis digital library of engineering and computer science.
Description: This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral, the Stieltjes integral, and other commonly used integrals. The purpose of this book is to provide a quick but accurate (and detailed) introduction to all aspects of modern integration theory. It should be accessible to any student who has had calculus and some exposure to upper division mathematics.
Notes: Part of: Synthesis digital library of engineering and computer science.

Series from website.

Includes bibliographical references (p. 87) and index.

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Table of contents: title: 1. Introduction; title: What is an integral; title: What is the Riemann integral; title: What is the Riemann integral good for; title: What is the Riemann integral not good for; title: What is the Lebesgue integral; title: What is the Lebesgue integral good for; title: What is the Lebesgue not good for; title: Exercises; title: 2. The Riemann integral; title: The definition; title: Properties of the Riemann integral; title: Characterization of Riemann integrability; title: The fundamental theorem of calculus; title: Numerical techniques of integration; title: Introduction; title: The method of rectangles; title: The trapezoidal rule; title: Simpson's rule; title: Integration by parts; title: Exercises; title: 3. The Lebesgue integral; title: Elementary measure theory; title: Measurable sets; title: The Lebesgue integral; title: Three big theorems about the Lebesgue integral; title: The Lebesgue spaces LP; title: The Riesz representation theorem; title: Product integration: Fubini's theorem; title: Three principles of Littlewood; title: Differentiation of integrals: covering lemmas and the Lebesgue theorem; title: Basic ideas; title: The maximal function; title: The concept of convergence in measure; title: Functions of bounded variation and absolute continuity; title: Exercises; title: 4. Comparison of the Riemann and Lebesgue integrals; title: Any Riemann integrable function is Lebesgue integrable; title: Exercises; title: 5. Other theories of the integral; title: The Daniell integral; title: The Riemann-Stieltjes integral; title: The Henstock-Kurzweil integral; title: Hausdorff measure; title: Haar measure; title: The fundamental theorem; title: Exercises; title: Bibliography; title: Author's biography; title: Index.