Learn · Measure and Integration
Lecture 33 of 79Week 5
5.8 - Integrable functions
Prof. S. Kesavan · IMSc
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- Measure and Integration, IMSc (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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All the lectures
- 1.1 - Preamble
- 1.2 - Algebras of sets
- 1.3 - Measures on rings
- 1.4 - Outer-measure
- 1.5 - Measurable sets
- 2.1 - Caratheodory's method
- 2.2 - Exercises
- 2.3 - Exercises
- 2.4 - Lebesgue measure: the ring
- 2.5 - Construction of the Lebesgue measure
- 2.6 - Errata
- 2.7 - The Cantor set
- 2.8 - Approximation
- 3.1 - Approximation
- 3.2 - Approximation
- 3.3 - Translation Invariance
- 3.4 - Non-measurable sets
- 3.5 - Exercises
- 3.6 - Measurable functions
- 4.1 - Measurable functions
- 4.2 - The Cantor function
- 4.3 - Exercises
- 4.4 - Egorov's theorem
- 4.5 - Convergence in measure
- 4.6 - Convergence in measure
- 5.1 - Convergence in measure
- 5.2 - Exercises
- 5.3 - Integration: Simple functions
- 5.4 - Non-negative functions
- 5.5 - Monotone convergence theorem
- 5.6 - Examples
- 5.7 - Fatou's lemma
- 5.8 - Integrable functions
- 6.1 - Dominated convergence theorem
- 6.2 - Dominated convergence theorem: Applications
- 6.3 - Absolute continuity
- 6.4 - Integration on the real line
- 6.5 - Examples
- 6.6 - Weierstrass' theorem
- 6.7 - Exercises
- 6.8 - Exercises
- 7.1 - Vitali covering lemma
- 7.2 - Monotonic functions
- 7.3 - Functions of bounded variation
- 7.4 - Functions of bounded variation
- 7.5 - Functions of bounded variation
- 7.6 - Differentiation of an indefinite integral
- 7.7 - Absolute continuity
- 8.1 - Exercises
- 8.2 - Product spaces
- 8.3 - Product spaces: measurable functions
- 8.4 - Product measure
- 8.5 - Fubini's theorem
- 8.6 - Examples
- 8.7 - Examples
- 9.1 - Integration of radial functions
- 9.2 - Measure of the unit ball in N dimensions
- 9.3 -Exercises
- 9.4 - Signed measures
- 9.5 - Hahn and Jordan decompositions
- 9.6 - Upper,lower and totaal variations of a signed measure; Absolute continuity
- 9.7 - Absolute continuity
- 10.1 - Radon-Nikodym theorem
- 10.2 - Radon-Nikodym theorem
- 10.3 - Exercises
- 10.4 - Lebesgue spaces
- 10.5 - Examples. Inclusion questions
- 10.6 - Convergence in L^p
- 11.1 - Approximation
- 11.2 - Applications
- 11.3 - Duality
- 11.4 - Duality
- 11.5 - Convolutions
- 11.6 - Convolutions
- 12.1 - Convolutions
- 12.2 - Exercises
- 12.3 - Exercises
- 12.4 - Change of variable
- 12.5 - Change of variable