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.
Inder K. Rana.: Department of Mathematics Indian Institute of Technology Bombay
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.