An Introduction to Measure and Integration

0.0

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, Lp spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on. The text 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. Table of Contents Preface to the Second Edition  Preface to the First Edition  Recipe for One Semester Course and Interdependence of the Chapters  Notations used in the text  Prologue: The Length Function  Riemann Integration  Recipes for Extending The Riemann Integral  General Extension Theory  The Lebesgue Measure of R and Its Properties  Integration  Fundamental Theorem of The Integral Calculus for 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  Appendix A: Extended Real Numbers  Appendix B: Axiom of Choice  Appendix C: Continuum Hypothesis  Appendix D: Urysohn’s Lemma  Appendix E: Singular value decomposition of a Matrix  Appendix F: Functions of bounded variation  Appendix G: Differentiable Transformations  References Index  Index of Notations.