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Measure and Integration

IIT Bombay Course , Prof. Inder K Rana       Subscribe to feed

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Lecture 1: Mod-01 Lec-01 Introduction ,Extended Real numbers

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Measure and Integration by Prof. Inder K Rana ,Department of Mathematics, IIT Bombay. For more details on NPTEL visit http://nptel.iitm.ac.in

Course Description :

Contents:
Need to define Lebesgue Integral,Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theorem. Measure spaces.Extension of measures from algebras to the generated sigma-algebras: Measurable sets; Lebesgue Measure and its properties.

Measurable functions and their properties; Integration and Convergence theorems, Lebesgue integral, Fundamental Theorem of Calculus for Lebesgue Integrals (an outline)


Product measure spaces, Fubini's theorem
Absolute continuity of measures, Radon-Nikodym theorem Introduction
to Lp spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces.

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  1. Mod-01 Lec-01 Introduction ,Extended Real numbers
  2. Mod-01 Lec-02 Algebra and Sigma Algebra of a subset of a set
  3. Mod-01 Lec-03 Sigma Algebra generated by a class
  4. Mod-01 Lec-04 Monotone Class
  5. Mod-02 Lec-05 Set function
  6. Mod-02 Lec-06 The Length function and its properties
  7. Mod-02 Lec-07 Countably additive set functions on intervals
  8. Mod-02 Lec-08 Uniqueness Problem for Measure
  9. Mod-03 Lec-09 Extension of measure
  10. Mod-03 Lec-10 Outer measure and its properties
  11. Mod-03 Lec-11 Measurable sets
  12. Mod-04 Lec-12 Lebesgue measure and its properties
  13. Mod-04 Lec-13 Characterization of Lebesque measurable sets
  14. Mod-05 Lec-14 Measurable functions
  15. Mod-05 Lec-15 Properties of measurable functions
  16. Mod-05 Lec-16 Measurable functions on measure spaces
  17. Mod-06 Lec-17 Integral of non negative simple measurable functions
  18. Mod-06 Lec-18 Properties of non negative simple measurable functions
  19. Mod-06 Lec-19 Monotone convergence theorem & Fatou's Lemma
  20. Mod-06 Lec-20 Properties of Integral functions & Dominated Convergence Theorem
  21. Mod-06 Lec-21 Dominated Convergence Theorem and applications
  22. Mod-06 Lec-22 Lebesgue Integral and its properties
  23. Mod-06 Lec-23 Denseness of continuous function
  24. Mod-07 Lec-24 Product measures, an Introduction
  25. Mod-07 Lec-25 Construction of Product Measure
  26. Mod-07 Lec-26 Computation of Product Measure-I
  27. Mod-07 Lec-27 Computation of Product Measure-II
  28. Mod-07 Lec-28 Integration on Product spaces
  29. Mod-07 Lec-29 Fubini's Theorems
  30. Mod-08 Lec-30 Lebesgue Measure and integral on R2
  31. Mod-08 Lec-31 Properties of Lebesgue Measure and integral on Rn
  32. Mod-08 Lec-32 Lebesgue integral on R2
  33. Mod-09 Lec-33 Integrating complex-valued functions
  34. Mod-09 Lec-34 Lp - spaces
  35. Mod-09 Lec-35 L2(X,S,mue)
  36. Mod-10 Lec-36 Fundamental Theorem of calculas for Lebesgue Integral-I
  37. Mod-10 Lec-37 Fundamental Theorem of calculas for Lebesgue Integral-II
  38. Mod-10 Lec-38 Absolutely continuous measures
  39. Mod-10 Lec-39 Modes of convergence
  40. Mod-10 Lec-40 Convergence in Measure