An Introduction to Measure and Integration (Graduate Studies in Mathematics)

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, Lp 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. "Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been presented 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[subscript 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."--Jacket Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been presented 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. In this update of the 1997 edition, Rana, formerly a doctoral student at the U. of Bombay (Mumbia) and later affiliated with several schools including the Indian Institute of Technology-Bombay, presents a "motivated approach" to a core subject of the graduate math curriculum variously known as Lebesgue measure and integration, real analysis, measure theory, modern, or advanced analysis. The author treats modern extensions of Lebesque's theory developed to remove some drawbacks of the Riemann integral. Includes technical appendices (e.g., on the continuum hypothesis, Urysohn's lemma, singular value decomposition of a matrix, and differentiable transformations.) Indexed by subject and notation. The series is listed as "Graduate studies..." on the cover and title page, but as "Graduate texts...." per CiP. Annotation c. Book News, Inc., Portland, OR (booknews.com) Prologue: The length function Riemann integration Recipes for extending the Riemann integral General extension theory The Lebesgue measure on R and its properties Integration Fundamental theorem of calculus for the Lebesgue integral Measure and integration on product spaces Modes of convergence and Lp-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 The geometric problem that leads to the concept of Riemann integral is the following: given a bounded function f : [a,b] R, how to define the area of the region bounded by the graph of the function and the lines x = a and x = b?