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Measure and Integration

Measure and Integration. Instructor: Prof. Inder K Rana, Department of Mathematics, IIT Bombay. "Measure and Integration" is an advanced-level course in Real Analysis, followed by a basic course in Real Analysis. The aim of this course is to give an introduction to the theory of measure and integration with respect to a measure. The material covered lays foundations for courses in "Functional Analysis", "Harmonic Analysis" and "Probability Theory". Starting with the need to define Lebesgue Integral, extension theory for measures will be covered. Abstract theory of integration with respect to a measure and introduction to Lp spaces, product measure spaces, Fubini's theorem, absolute continuity and Radon-Nikodym theorem will be covered. (from nptel.ac.in)

Lecture 36 - Fundamental Theorem of Calculus for Lebesgue Integral I

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Introduction and Classes of Sets

Lecture 01 - Introduction, Extended Real Numbers

Lecture 02 - Algebra and Sigma Algebra of a Subset of a Set

Lecture 03 - Sigma Algebra Generated by a Class

Lecture 04 - Monotone Class

Measure

Lecture 05 - Set Function

Lecture 06 - The Length Function and its Properties

Lecture 07 - Countably Additive Set Functions on Intervals

Lecture 08 - Uniqueness Problem for Measure

Extension of Measures

Lecture 09 - Extension of Measure

Lecture 10 - Outer Measure and its Properties

Lecture 11 - Measurable Sets

The Lebesgue Measure and its Properties

Lecture 12 - Lebesgue Measure and its Properties

Lecture 13 - Characterization of Lebesgue Measurable Sets

Measurable Functions

Lecture 14 - Measurable Functions

Lecture 15 - Properties of Measurable Functions

Lecture 16 - Measurable Functions on Measure Spaces

Integrations

Lecture 17 - Integral of Nonnegative Simple Measurable Functions

Lecture 18 - Properties of Nonnegative Simple Measurable Functions

Lecture 19 - Monotone Convergence Theorem and Fatou's Lemma

Lecture 20 - Properties of Integral Functions and Dominated Convergence Theorem

Lecture 21 - Dominated Convergence Theorem and Applications

Lecture 22 - Lebesgue Integral and its Properties

Lecture 23 - Denseness of Continuous Function

Measure and Integration on Product Spaces

Lecture 24 - Product Measures: An Introduction

Lecture 25 - Construction of Product Measure

Lecture 26 - Computation of Product Measure I

Lecture 27 - Computation of Product Measure II

Lecture 28 - Integration on Product Spaces

Lecture 29 - Fubini's Theorems

Lebesgue Measure on Rn

Lecture 30 - Lebesgue Measure and Integral on R2

Lecture 31 - Properties of Lebesgue Measure and Integral on Rn

Lecture 32 - Lebesgue Integral on R2

Lp-Spaces

Lecture 33 - Integrating Complex-Valued Functions

Lecture 34 - Lp-Spaces

Lecture 35 - L2 (X, S, μ)

Special Topics

Lecture 36 - Fundamental Theorem of Calculus for Lebesgue Integral I

Lecture 37 - Fundamental Theorem of Calculus for Lebesgue Integral II

Lecture 38 - Absolutely Continuous Measures

Lecture 39 - Modes of Convergence

Lecture 40 - Convergence in Measure