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Course11110813564 lectures

Measure Theory

Prof. E. K. Narayanan · IISc Bangalore

Start with lecture 1 At NPTEL

Lectures

Sigma algebras, Measures and Integration

  1. Review of Riemann integration and introduction to sigma algebras Video
  2. Sigma algebras and measurability Video
  3. Measurable functions and approximation by simple functions Video
  4. Properties of countably additive measures Video
  5. Integration of positive measurable functions Video

Integration and convergence theorems

  1. Some properties of integrals of positive simple functions Video
  2. Monotone convergence theorem and Fatou's lemma Video
  3. Integration of complex valued measurable functions Video
  4. Dominated convergence theorem Video
  5. Sets of measure zero and completion Video

Outer measure

  1. Consequences of MCT, Fatou's lemma and DCT Video
  2. Rectangles in R^n and some properties Video
  3. Outer measure on R^n Video
  4. Properties of outer measure on R^n Video
  5. Lebesgue measurable sets and Lebesgue measure on R^n Video

Lebesgue measure and its properties

  1. Lebesgue sigma algebra Video
  2. Lebesgue measure. Video
  3. Fine properties of measurable sets Video
  4. Invariance properties of Lebesgue measure Video
  5. Non measurable set Video

Lebesgue measure and positive Borel measures on locally compact spaces

  1. Measurable functions. Video
  2. Riemann and Lebesgue integrals Video
  3. Locally compact Hausdorff spaces Video
  4. Riesz representation theorem Video
  5. Positive Borel measures Video

Lebesgue measure and invariance properties

  1. Lebesgue measure via Riesz representation theorem Video
  2. Construction of Lebesgue measure Video
  3. Invariance properties of Lebesgue measure. Video
  4. Linear transformations and Lebesgue measure Video
  5. Cantor set Video
  6. Cantor function Video

L^p spaces and completeness

  1. Lebesgue set which is not Borel Video
  2. L^p spaces Video
  3. L^p norm Video
  4. Completeness of L^p Video
  5. Properties of L^p spaces Video
  6. Examples of L^p spaces Video

Product spaces and Fubini's theorem

  1. Product sigma algebra Video
  2. Product measures I Video
  3. Product measures II Video
  4. Fubini's theorem I Video
  5. Fubini's theorem II Video

Applications of Fubini's theorem and complex measures

  1. Completeness of product measures Video
  2. Polar coordinates Video
  3. Applications of Fubini's theorem Video
  4. Complex measures I Video

Complex measures and Radon-Nikodym theorem

  1. Complex measures II Video
  2. Absolutely continuous measures Video
  3. L^2 space Video
  4. Continuous linear functionals Video
  5. Radon-Nikodym theorem I Video

Radon-Nikodym theorem and applications

  1. Radon Nikodym theorem II Video
  2. Consequences of Radon-Nikodym theorem I Video
  3. Consequences of Radon-Nikodym theorem II Video
  4. Continuous linear functionals on L^p spaces I Video
  5. Continuous linear functionals on L^p spaces II Video

Riesz representation theorem and Lebesgue differentiation theorem

  1. Riesz representation theorem I Video
  2. Riesz representation theorem II Video
  3. Hardy-Littlewood maximal function Video
  4. Lebesgue differentiation theorem Video
  5. Absolutely continuous functions I Video
  6. Absolutely continuous functions II Video

Live Session

  1. Live Session 09-03-2021 Video
  2. Live Session 30-03-2021 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Teacher
Prof. E. K. Narayanan
Level
Undergraduate
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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