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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)

Introduction


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

References
Measure and Integration
Instructor: Prof. Inder K Rana, Department of Mathematics, IIT Bombay. This is an advanced-level course in Real Analysis.