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Measure Theory

Measure Theory. Instructor: Prof. Inder K. Rana, Department of Mathematics, IIT Bombay. This is a course on the concepts of Measure and Integration. The course covers lessons in Extended real numbers, Algebra and sigma algebra of a subsets of a set, Sigma algebra generated by a class, Monotone class, Set functions, Countably additive set functions on intervals, Uniqueness problem for measure, Extension of measure, Outer measure and its properties, Measurable sets, Lebesgue measure and its properties, Measurable functions, Monotone convergence theorem and Fatou's lemma, Dominated convergence theorem, Lebesgue integral and its properties, Product measure, Computation of product measure, Fubini's theorems, and Lebesgue measure and integral on R2 (from nptel.ac.in)

Lecture 43 - Lebesgue Integral and its Properties

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Lecture 01 - Introduction, Extended Real Numbers

Lecture 02 - Introduction, Extended Real Numbers (cont.)

Lecture 03 - Algebra and Sigma Algebra of a Subsets of a Set

Lecture 04 - Algebra and Sigma Algebra of a Subsets of a Set (cont.)

Lecture 05 - Sigma Algebra Generated by a Class

Lecture 06 - Sigma Algebra Generated by a Class (cont.)

Lecture 07 - Monotone Class

Lecture 08 - Monotone Class (cont.)

Lecture 09 - Set Functions

Lecture 10 - Set Functions (cont.)

Lecture 11 - The Length Function and its Properties

Lecture 12 - The Length Function and its Properties (cont.)

Lecture 13 - Countably Additive Set Functions on Intervals

Lecture 14 - Countably Additive Set Functions on Intervals (cont.)

Lecture 15 - Uniqueness Problem for Measure

Lecture 16 - Uniqueness Problem for Measure (cont.)

Lecture 17 - Extension of Measure

Lecture 18 - Extension of Measure (cont.)

Lecture 19 - Outer Measure and its Properties

Lecture 20 - Outer Measure and its Properties (cont.)

Lecture 21 - Measurable Sets

Lecture 22 - Measurable Sets (cont.)

Lecture 23 - Lebesgue Measure and its Properties

Lecture 24 - Lebesgue Measure and its Properties (cont.)

Lecture 25 - Characterization of Lebesgue Measurable Sets

Lecture 26 - Characterization of Lebesgue Measurable Sets (cont.)

Lecture 27 - Measurable Functions

Lecture 28 - Measurable Functions (cont.)

Lecture 29 - Properties of Measurable Functions

Lecture 30 - Properties of Measurable Functions (cont.)

Lecture 31 - Measurable Functions on Measure Spaces

Lecture 32 - Measurable Functions on Measure Spaces (cont.)

Lecture 33 - Integral of Nonnegative Simple Measurable Functions

Lecture 34 - Integral of Nonnegative Simple Measurable Functions (cont.)

Lecture 35 - Properties of Nonnegative Simple Measurable Functions

Lecture 36 - Properties of Nonnegative Simple Measurable Functions (cont.)

Lecture 37 - Monotone Convergence Theorem and Fatou's Lemma

Lecture 38 - Monotone Convergence Theorem and Fatou's Lemma (cont.)

Lecture 39 - Properties of Integrable Functions and Dominated Convergence Theorem

Lecture 40 - Properties of Integrable Functions and Dominated Convergence Theorem (cont.)

Lecture 41 - Dominated Convergence Theorem and Applications

Lecture 42 - Dominated Convergence Theorem and Applications (cont.)

Lecture 43 - Lebesgue Integral and its Properties

Lecture 44 - Lebesgue Integral and its Properties (cont.)

Lecture 45 - Product Measure, an Introduction

Lecture 46 - Product Measure, an Introduction (cont.)

Lecture 47 - Construction of Product Measures

Lecture 48 - Construction of Product Measures (cont.)

Lecture 49 - Computation of Product Measure

Lecture 50 - Computation of Product Measure (cont.)

Lecture 51 - Computation of Product Measure (cont.)

Lecture 52 - Computation of Product Measure (cont.)

Lecture 53 - Integration on Product Spaces

Lecture 54 - Integration on Product Spaces (cont.)

Lecture 55 - Fubini's Theorem

Lecture 56 - Fubini's Theorem (cont.)

Lecture 57 - Lebesgue Measure and Integral on R2

Lecture 58 - Lebesgue Measure and Integral on R2 (cont.)

Lecture 59 - Properties of Lebesgue Measure on R2

Lecture 60 - Properties of Lebesgue Measure on R2 (cont.)

Lecture 61 - Lebesgue Integral on R2

Lecture 62 - Lebesgue Integral on R2 (cont.)