Lecture 1 - Finite Sets and Cardinality
Lecture 2 - Infinite Sets and the Banach-Tarski Paradox - Part 1
Lecture 3 - Infinite Sets and the Banach-Tarski Paradox - Part 2
Lecture 4 - Elementary Sets and Elementary measure - Part 1
Lecture 5 - Elementary Sets and Elementary measure - Part 2
Lecture 6 - Properties of elementary measure - Part 1
Lecture 7 - Properties of elementary measure - Part 2
Lecture 8 - Uniqueness of elementary measure and Jordan measurability - Part 1
Lecture 9 - Uniqueness of elementary measure and Jordan measurability - Part 2
Lecture 10 - Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 1
Lecture 11 - Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 2
Lecture 12 - Examples of Jordan measurable sets-I
Lecture 13 - Examples of Jordan measurable sets-II - Part 1
Lecture 14 - Examples of Jordan measurable sets-II - Part 2
Lecture 15 - Jordan measure under Linear transformations - Part 1
Lecture 16 - Jordan measure under Linear transformations - Part 2
Lecture 17 - Connecting the Jordan measure with the Riemann integral - Part 1
Lecture 18 - Connecting the Jordan measure with the Riemann integral - Part 2
Lecture 19 - Outer measure - Motivation and Axioms of outer measure
Lecture 20 - Comparing Inner Jordan measure, Lebesgue outer measure and Jordan Outer measure
Lecture 21 - Finite additivity of outer measure on Separated sets, Outer regularity - Part 1
Lecture 22 - Finite additivity of outer measure on Separated sets, Outer regularity - Part 2
Lecture 23 - Lebesgue measurable class of sets and their Properties - Part 1
Lecture 24 - Lebesgue measurable class of sets and their Properties - Part 2
Lecture 25 - Equivalent criteria for lebesgue measurability of a subset - Part 1
Lecture 26 - Equivalent criteria for lebesgue measurability of a subset - Part 2
Lecture 27 - The measure axioms and the Borel-Cantelli Lemma
Lecture 28 - Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 1
Lecture 29 - Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 2
Lecture 30 - Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 1
Lecture 31 - Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 2
Lecture 32 - Abstract measure spaces: Boolean and Sigma-algebras
Lecture 33 - Abstract measure and Caratheodory Measurability - Part 1
Lecture 34 - Abstract measure and Caratheodory Measurability - Part 2
Lecture 35 - Abstrsct measure and Hahn-Kolmogorov Extension
Lecture 36 - Lebesgue measurable class vs Caratheodory extension of usual outer measure on R^d
Lecture 37 - Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 1
Lecture 38 - Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 2
Lecture 39 - Measurable functions: definition and basic properties - Part 1
Lecture 40 - Measurable functions: definition and basic properties - Part 2
Lecture 41 - Egorov's theorem: abstract version
Lecture 42 - Lebesgue integral of unsigned simple measurable functions: definition and properties
Lecture 43 - Lebesgue integral of unsigned measurable functions: motivation, definition and basic properties
Lecture 44 - Fundamental convergence theorems in Lebesgue integration: Monotone convergence theorem, Tonelli's theorem and Fatou's lemma
Lecture 45 - Lebesgue integral for complex and real measurable functions: the space of L^1 functions
Lecture 46 - Basic properties of L^1-functions and Lebesgue's Dominated convergence theorem
Lecture 47 - L^1 functions on R^d: Egorov's theorem revisited (Littlewood's third principle)
Lecture 48 - L^1 functions on R^d: Statement of Lusin's theorem (Littlewood's second principle), Density of simple functions, step functions, and continuous compactly supported functions in L^1
Lecture 49 - L^1 functions on R^d: Proof of Lusin's theorem, space of L^1 functions as a metric space
Lecture 50 - L^1 functions on R^d: the Riesz-Fischer theorem
Lecture 51 - Various modes of convergence of measurable functions
Lecture 52 - Easy implications from one mode of convergence to another
Lecture 53 - Implication map for modes of convergence with various examples
Lecture 54 - Uniqueness of limits across various modes of convergence
Lecture 55 - Some criteria for reverse implications for modes of convergence
Lecture 56 - Riesz Representation theorem- Motivation
Lecture 57 - Basics on Locally compact Hausdorff spaces
Lecture 58 - Borel and Radon measures on LCH spaces
Lecture 59 - Properties of Radon measures and Lusin's theorem on LCH spaces
Lecture 60 - Riesz Representation theorem - Complete statement and proof - Part 1
Lecture 61 - Riesz Representation theorem - Complete statement and proof - Part 2
Lecture 62 - Examples of measures constructed using RRT
Lecture 63 - Theorems of Tonelli and Fubini- interchanging the order of integration for repeated integrals: motivation and discussion of product measure spaces
Lecture 64 - Product measures
Lecture 65 - Tonelli's theorem for sets - Part 1
Lecture 66 - Tonelli's theorem for sets - Part 2
Lecture 67 - Fubini-Tonelli theorem: interchanging order of integration for measurable and L^1 functions on sigma-finite measure spaces
Lecture 68 - Lebesgue's differentiation theorem: introduction and motivation
Lecture 69 - Lebesgue's differentiation theorem: statement and proof - Part 1
Lecture 70 - Lebesgue's differentiation theorem: statement and proof - Part 2
Lecture 71 - DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 1
Lecture 72 - DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 2
Lecture 73 - Riesz's Rising Sun Lemma
Lecture 74 - Differentiation theorem for monone continuous functions
Lecture 75 - Differentation theorem for general monotone functions and Second fundamental theorem of calculus for absolutely continuous functions
