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Course11110110062 lectures

Measure Theory

Prof. Inder Kumar Rana · IIT Bombay

Start with lecture 1 At NPTEL

About

This is a course on the concepts of Measure and Integration. Normally, this is a core course for M,.Sc. Mathematics and Statistics students. The concepts find applications in advance Analysis Courses, Signal Processing, Financial Mathematics courses.

INTENDED AUDIENCE : B.Tech Dual degree in Electrical, M.Sc. Physics, mathematics

PREREQUISITES : Basic Course in Real Analysis

Lectures

Week 1

  1. Lecture 1A: Introduction, Extended Real Numbers YouTube
  2. Lecture 1B: Introduction, Extended Real Numbers YouTube
  3. Lecture 2A: Algebra and Sigma Algebra of Subsets of a Set YouTube
  4. Lecture 2B: Algebra and Sigma Algebra of Subsets of a Set YouTube
  5. Lecture 3A: Sigma Algebra generated by a Class YouTube
  6. Lecture 3B: Sigma Algebra generated by a Class YouTube

Week 2

  1. Lecture 4A: Monotone Class YouTube
  2. Lecture 4B: Monotone Class YouTube
  3. Lecture 5A: Set Functions YouTube
  4. Lecture 5B: Set Functions YouTube
  5. Lecture 6A: The Length Function and its Properties YouTube
  6. Lecture 6B: The Length Function and its Properties YouTube

Week 3

  1. Lecture 7A: Countably Additive Set Functions on Intervals YouTube
  2. Lecture 7B: Countably Additive Set Functions on Intervals YouTube
  3. Lecture 8A: Uniqueness Problem for Measure YouTube
  4. Lecture 8B: Uniqueness Problem for Measure YouTube

Week 4

  1. Lecture 9A: Extension of Measure YouTube
  2. Lecture 9B: Extension of Measure YouTube
  3. Lecture 10A: Outer Measure and its Properties YouTube
  4. Lecture 10B: Outer Measure and its Properties YouTube
  5. Lecture 11A: Measurable Sets YouTube
  6. Lecture 11B: Measurable Sets YouTube

Week 5

  1. Lecture 12A: Lebesgue Measure and its Properties YouTube
  2. Lecture 12B: Lebesgue Measure and its Properties YouTube
  3. Lecture 13A: Characterization of Lebesgue Measurable Sets YouTube
  4. Lecture 13B: Characterization of Lebesgue Measurable Sets YouTube

Week 6

  1. Lecture 14A: Measurable Functions YouTube
  2. Lecture 14B: Measurable Functions YouTube
  3. Lecture 15A: Properties of Measurable Functions YouTube
  4. Lecture 15B: Properties of Measurable Functions YouTube
  5. Lecture 16A: Measurable Functions on Measure Spaces YouTube
  6. Lecture 16B: Measurable Functions on Measure Spaces YouTube

Week 7

  1. Lecture 17A: Integral of Nonnegative Simple Measurable Functions YouTube
  2. Lecture 17B: Integral of Nonnegative Simple Measurable Functions YouTube
  3. Lecture 18A: Properties of Nonnegative Simple Measurable Functions YouTube
  4. Lecture 18B: Properties of Nonnegative Simple Measurable Functions YouTube
  5. Lecture 19A: Monotone Convergence Theorem and Fatou's Lemma YouTube
  6. Lecture 19B: Monotone Convergence Theorem and Fatou's Lemma YouTube

Week 8

  1. Lecture 20A: Properties of Integrable Functions and Dominated Convergence Theorem YouTube
  2. Lecture 20B: Properties of Integrable Functions and Dominated Convergence Theorem YouTube
  3. Lecture 21A: Dominated Convergence Theorem and Applications YouTube
  4. Lecture 21B: Dominated Convergence Theorem and Applications YouTube

Week 9

  1. Lecture 22A: Lebesgue Integral and its Properties YouTube
  2. Lecture 22B: Lebesgue Integral and its Properties YouTube
  3. Lecture 23A: Product Measure, an Introduction YouTube
  4. Lecture 23B: Product Measure, an Introduction YouTube
  5. Lecture 24A: Construction of Product Measures YouTube
  6. Lecture 24B: Construction of Product Measures YouTube

Week 10

  1. Lecture 25A: Computation of Product Measure - I YouTube
  2. Lecture 25B: Computation of Product Measure - I YouTube
  3. Lecture 26A: Computation of Product Measure - II YouTube
  4. Lecture 26B: Computation of Product Measure - II YouTube

Week 11

  1. Lecture 27A: Integration on Product Spaces YouTube
  2. Lecture 27B: Integration on Product Spaces YouTube
  3. Lecture 28A: Fubini's Theorems YouTube
  4. Lecture 28B: Fubini's Theorems YouTube

Week 12

  1. Lecture 29A: Lebesgue Measure and Integral on R2 YouTube
  2. Lecture 29B: Lebesgue Measure and Integral on R2 YouTube
  3. Lecture 30A: Properties of Lebesgue Measure on R2 YouTube
  4. Lecture 30B: Properties of Lebesgue Measure on R2 YouTube
  5. Lecture 31A: Lebesgue Integral on R2 YouTube
  6. Lecture 31B: Lebesgue Integral on R2 YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Bombay
Department
Mathematics
Teacher
Prof. Inder Kumar Rana
Level
Graduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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