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.
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978-0-8218-2974-5 (9780821829745)
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Schweitzer Klassifikation
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.
