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Last updated:

September 23, 2023

Duration:

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FREE

This course includes:

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Description

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)

Course Curriculum

    • Lecture 01 – Introduction, Extended Real Numbers Unlimited
    • Lecture 02 – Algebra and Sigma Algebra of a Subset of a Set Unlimited
    • Lecture 03 – Sigma Algebra Generated by a Class Unlimited
    • Lecture 04 – Monotone Class Unlimited
    • Lecture 05 – Set Function Unlimited
    • Lecture 06 – The Length Function and its Properties Unlimited
    • Lecture 07 – Countably Additive Set Functions on Intervals Unlimited
    • Lecture 08 – Uniqueness Problem for Measure Unlimited
    • Lecture 09 – Extension of Measure Unlimited
    • Lecture 10 – Outer Measure and its Properties Unlimited
    • Lecture 11 – Measurable Sets Unlimited
    • Lecture 12 – Lebesgue Measure and its Properties Unlimited
    • Lecture 13 – Characterization of Lebesgue Measurable Sets Unlimited
    • Lecture 15 – Properties of Measurable Functions Unlimited
    • Lecture 16 – Measurable Functions on Measure Spaces Unlimited
    • Lecture 17 – Integral of Nonnegative Simple Measurable Functions Unlimited
    • Lecture 18 – Properties of Nonnegative Simple Measurable Functions Unlimited
    • Lecture 19 – Monotone Convergence Theorem and Fatou’s Lemma Unlimited
    • Lecture 20 – Properties of Integral Functions and Dominated Convergence Theorem Unlimited
    • Lecture 21 – Dominated Convergence Theorem and Applications Unlimited
    • Lecture 22 – Lebesgue Integral and its Properties Unlimited
    • Lecture 23 – Denseness of Continuous Function Unlimited
    • Lecture 24 – Product Measures: An Introduction Unlimited
    • Lecture 25 – Construction of Product Measure Unlimited
    • Lecture 26 – Computation of Product Measure I Unlimited
    • Lecture 27 – Computation of Product Measure II Unlimited
    • Lecture 28 – Integration on Product Spaces Unlimited
    • Lecture 29 – Fubini’s Theorems Unlimited
    • Lecture 30 – Lebesgue Measure and Integral on R2 Unlimited
    • Lecture 31 – Properties of Lebesgue Measure and Integral on Rn Unlimited
    • Lecture 32 – Lebesgue Integral on R2 Unlimited
    • Lecture 33 – Integrating Complex-Valued Functions Unlimited
    • Lecture 34 – Lp-Spaces Unlimited
    • Lecture 35 – L2 (X, S, μ) Unlimited
    • Lecture 36 – Fundamental Theorem of Calculus for Lebesgue Integral I Unlimited
    • Lecture 37 – Fundamental Theorem of Calculus for Lebesgue Integral II Unlimited
    • Lecture 38 – Absolutely Continuous Measures Unlimited
    • Lecture 39 – Modes of Convergence Unlimited
    • Lecture 40 – Convergence in Measure Unlimited

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