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An Introduction to Measure and Integration: Second Edition
Inder K. Rana, Indian Institute of Technology, Powai, Mumbai, India
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Graduate Studies in Mathematics
2002; 424 pp; hardcover
Volume: 45
ISBN-10: 0-8218-2974-2
ISBN-13: 978-0-8218-2974-5
List Price: US$68
Member Price: US$54.40
Order Code: GSM/45
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Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, \(L_p\) spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on.

The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study.

For this edition, more exercises and four appendices have been added.

The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.

Readership

Graduate students and research mathematicians interested in mathematical analysis.

Reviews

From reviews for the first edition:

"Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral ... 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables ... the conversational tone and helpful insights make this a useful introduction to the topic ... The material is presented with generous details and helpful examples at a level suitable for an introductory course or for self-study."

-- Zentralblatt MATH

"A special feature [of the book] is the extensive historical and motivational discussion ... At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally ... The book attempts to be comprehensive and largely succeeds ... The text can be used for either a one-semester or a one-year course at M.Sc. level ... The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that ... All in all, the book is highly recommended to serious and demanding students."

-- Resonance -- journal of science education

Table of Contents

  • Prologue: The length function
  • Riemann integration
  • Recipes for extending the Riemann integral
  • General extension theory
  • The Lebesgue measure on \(\mathbb{R}\) and its properties
  • Integration
  • Fundamental theorem of calculus for the Lebesgue integral
  • Measure and integration on product spaces
  • Modes of convergence and \(L_p\)-spaces
  • The Radon-Nikodym theorem and its applications
  • Signed measures and complex measures
  • Extended real numbers
  • Axiom of choice
  • Continuum hypotheses
  • Urysohn's lemma
  • Singular value decomposition of a matrix
  • Functions of bounded variation
  • Differentiable transformations
  • References
  • Index
  • Index of notations
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