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.

Other Resources

Course Curriculum

Introduction ,Extended Real numbers Details
Algebra and Sigma Algebra of a subset of a set Details
Sigma Algebra generated by a class Details
Monotone Class Details
Set function Details
The Length function and its properties Details
Countably additive set functions on intervals Details
Uniqueness Problem for Measure Details
Extension of measure Details
Outer measure and its properties Details
Measurable sets Details
Lebesgue measure and its properties Details
Characterization of Lebesque measurable sets Details
Measurable functions Details
Properties of measurable functions Details
Measurable functions on measure spaces Details
Integral of non negative simple measurable functions Details
Properties of non negative simple measurable functions Details
Monotone convergence theorem & Fatou’s Lemma Details
Properties of Integral functions & Dominated Convergence Theorem Details
Dominated Convergence Theorem and applications Details
Lebesgue Integral and its properties Details
Denseness of continuous function Details
Product measures, an Introduction Details
Construction of Product Measure Details
Computation of Product Measure-I Details
Computation of Product Measure-II Details
Integration on Product spaces Details
Fubini’s Theorems Details
Lebesgue Measure and integral on R2 Details
Properties of Lebesgue Measure and integral on Rn Details
Lebesgue integral on R2 Details
Integrating complex-valued functions Details
Lp – spaces Details
L2(X,S,mue) Details
Fundamental Theorem of calculas for Lebesgue Integral-I Details
Fundamental Theorem of calculas for Lebesgue Integral-II Details
Absolutely continuous measures Details
Modes of convergence Details
Convergence in Measure Details

This course is delivered by IIT Bombay , as part of NPTEL Free Courses sponsored by Govt of India.

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