Lecture 55 of 64Radon-Nikodym theorem and applications
Continuous linear functionals on L^p spaces I
Prof. E. K. Narayanan · IISc Bangalore
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All the lectures
- Review of Riemann integration and introduction to sigma algebras
- Sigma algebras and measurability
- Measurable functions and approximation by simple functions
- Properties of countably additive measures
- Integration of positive measurable functions
- Some properties of integrals of positive simple functions
- Monotone convergence theorem and Fatou's lemma
- Integration of complex valued measurable functions
- Dominated convergence theorem
- Sets of measure zero and completion
- Consequences of MCT, Fatou's lemma and DCT
- Rectangles in R^n and some properties
- Outer measure on R^n
- Properties of outer measure on R^n
- Lebesgue measurable sets and Lebesgue measure on R^n
- Lebesgue sigma algebra
- Lebesgue measure.
- Fine properties of measurable sets
- Invariance properties of Lebesgue measure
- Non measurable set
- Measurable functions.
- Riemann and Lebesgue integrals
- Locally compact Hausdorff spaces
- Riesz representation theorem
- Positive Borel measures
- Lebesgue measure via Riesz representation theorem
- Construction of Lebesgue measure
- Invariance properties of Lebesgue measure.
- Linear transformations and Lebesgue measure
- Cantor set
- Cantor function
- Lebesgue set which is not Borel
- L^p spaces
- L^p norm
- Completeness of L^p
- Properties of L^p spaces
- Examples of L^p spaces
- Product sigma algebra
- Product measures I
- Product measures II
- Fubini's theorem I
- Fubini's theorem II
- Completeness of product measures
- Polar coordinates
- Applications of Fubini's theorem
- Complex measures I
- Complex measures II
- Absolutely continuous measures
- L^2 space
- Continuous linear functionals
- Radon-Nikodym theorem I
- Radon Nikodym theorem II
- Consequences of Radon-Nikodym theorem I
- Consequences of Radon-Nikodym theorem II
- Continuous linear functionals on L^p spaces I
- Continuous linear functionals on L^p spaces II
- Riesz representation theorem I
- Riesz representation theorem II
- Hardy-Littlewood maximal function
- Lebesgue differentiation theorem
- Absolutely continuous functions I
- Absolutely continuous functions II
- Live Session 09-03-2021
- Live Session 30-03-2021