دوره‌ای کوتاه در مورد انتگرال لبه‌س و نظریه اندازه

This article develops the basics of the Lebesgue integral and measure theory. In terms of content, it adds nothing new to any of the existing textbooks on the subject. But our approach here will be to avoid unduly abstractness and absolute generality, instead focusing on producing proofs of useful results as quickly as possible. Much of the material here comes from lecture notes from a short real analysis course I had taken, and the rest are well-known results whose proofs I had worked out myself with hints from various sources. I typed this up mainly for my own benefit, but I hope it will be interesting for anyone curious about the Lebesgue integral (or higher mathematics in general). I will be providing proofs of every theorem. If you are bored reading them, you are invited to do your own proofs. The bibliography outlines the background you need to understand this article. A substantial course in real analysis is an essential part of the preparation of any potential mathematician. Analysis on Manifolds is a thorough, class-tested approach that begins with the derivative and the Riemann integral for functions of several variables, followed by a treatment of differential forms and a proof of Stokes' theorem for manifolds in euclidean space. The book includes careful treatment of both the inverse function theorem and the change of variables theorem for n-dimensional integrals, as well as a proof of the Poincare lemma. Intended for students at the senior or first-year graduate level, this text includes more than 120 illustrations and exercises that range from the straightforward to the challenging . The book evolved from courses on real analysis taught by the author at the Massachusetts Institute of Technology. --back cover Motivation for the Lebesgue integral......Page 2 Basic measure theory......Page 4 Measurable functions......Page 8 Definition of the Lebesgue Integral......Page 11 Convergence theorems......Page 15 Some Results of Integration Theory......Page 18 Lp spaces......Page 23 Construction of Lebesgue Measure......Page 28 Lebesgue Measure in Rn......Page 32 Riemann integrability implies Lebesgue integrability......Page 34 Product measures and Fubini's Theorem......Page 36 Change of variables in Rn......Page 39 Vector-valued integrals......Page 41 C0 functions are dense in Lp(Rn)......Page 42 Other examples of measures......Page 47 Egorov's Theorem......Page 50 Exercises......Page 51 Bibliography......Page 52 1. The Algebra And Topology Of R(n) -- 2. Differentiation -- 3. Integration -- 4. Change Of Variables -- 5. Manifolds -- 6. Differential Forms -- 7. Stokes' Theorem -- 8. Closed Forms And Exact Forms -- 9. Epilogue : Life Outside R(n). James R. Munkres. Includes Bibliographical References (p. 359-360) And Index. A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.