NOC:Measure and Integration

Lecture 1 - Preamble

Lecture 2 - Algebras of sets

Lecture 3 - Measures on rings

Lecture 4 - Outer-measure

Lecture 5 - Measurable sets

Lecture 6 - Caratheodory's method

Lecture 7 - Exercises

Lecture 8 - Exercises

Lecture 9 - Lebesgue measure: the ring

Lecture 10 - Construction of the Lebesgue measure

Lecture 11 - Errata

Lecture 12 - The Cantor set

Lecture 13 - Approximation

Lecture 14 - Approximation

Lecture 15 - Approximation

Lecture 16 - Translation Invariance

Lecture 17 - Non-measurable sets

Lecture 18 - Exercises

Lecture 19 - Measurable functions

Lecture 20 - Measurable functions

Lecture 21 - The Cantor function

Lecture 22 - Exercises

Lecture 23 - Egorov's theorem

Lecture 24 - Convergence in measure

Lecture 25 - Convergence in measure

Lecture 26 - Convergence in measure

Lecture 27 - Exercises

Lecture 28 - Integration: Simple functions

Lecture 29 - Non-negative functions

Lecture 30 - Monotone convergence theorem

Lecture 31 - Examples

Lecture 32 - Fatou's lemma

Lecture 33 - Integrable functions

Lecture 34 - Dominated convergence theorem

Lecture 35 - Dominated convergence theorem: Applications

Lecture 36 - Absolute continuity

Lecture 37 - Integration on the real line

Lecture 38 - Examples

Lecture 39 - Weierstrass' theorem

Lecture 40 - Exercises

Lecture 41 - Exercises

Lecture 42 - Vitali covering lemma

Lecture 43 - Monotonic functions

Lecture 44 - Functions of bounded variation

Lecture 45 - Functions of bounded variation

Lecture 46 - Functions of bounded variation

Lecture 47 - Differentiation of an indefinite integral

Lecture 48 - Absolute continuity

Lecture 49 - Exercises

Lecture 50 - Product spaces

Lecture 51 - Product spaces: measurable functions

Lecture 52 - Product measure

Lecture 53 - Fubini's theorem

Lecture 54 - Examples

Lecture 55 - Examples

Lecture 56 - Integration of radial functions

Lecture 57 - Measure of the unit ball in N dimensions

Lecture 58 - Exercises

Lecture 59 - Signed measures

Lecture 60 - Hahn and Jordan decompositions

Lecture 61 - Upper,lower and totaal variations of a signed measure; Absolute continuity

Lecture 62 - Absolute continuity

Lecture 63 - Radon-Nikodym theorem

Lecture 64 - Radon-Nikodym theorem

Lecture 65 - Exercises

Lecture 66 - Lebesgue spaces

Lecture 67 - Examples. Inclusion questions

Lecture 68 - Convergence in L^p

Lecture 69 - Approximation

Lecture 70 - Applications

Lecture 71 - Duality

Lecture 72 - Duality

Lecture 73 - Convolutions

Lecture 74 - Convolutions

Lecture 75 - Convolutions

Lecture 76 - Exercises

Lecture 77 - Exercises

Lecture 78 - Change of variable

Lecture 79 - Change of variable