# InfoCoBuild

## 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.