Graduate Studies in Mathematics 2002; 424 pp; hardcover Volume: 45 ISBN10: 0821829742 ISBN13: 9780821829745 List Price: US$68 Member Price: US$54.40 Order Code: GSM/45
 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 RadonNikodym 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 selfstudy. 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 selfstudy."  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 onesemester or a oneyear 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 RadonNikodym 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
