Measure Theory D3Q2WH8CK5

URI	`m2://res/D3Q2WH8CK5`
Kind	course
Creators	Indrava Roy (instructor)
Date	2020-09-28
Scope	in

Expressions

e1: edition

Language	en
Date	2020-09-28

m1: page, IMSc (record). nptel 111106140
m2: page, 2020-09-28 (record). nptel 111106140

Contents

part 1: Week 1
session 1: Finite Sets and Cardinality
session 2: Infinite Sets and the Banach-Tarski Paradox - Part 1
session 3: Infinite Sets and the Banach-Tarski Paradox - Part 2
session 4: Elementary Sets and Elementary measure - Part 1
session 5: Elementary Sets and Elementary measure - Part 2
session 6: Properties of elementary measure - Part 1
session 7: Properties of elementary measure - Part 2
part 2: Week 2
session 8: Uniqueness of elementary measure and Jordan measurability - Part 1
session 9: Uniqueness of elementary measure and Jordan measurability - Part 2
session 10: Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 1
session 11: Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 2
session 12: Examples of Jordan measurable sets- I
session 13: Examples of Jordan measurable sets- II - Part 1
session 14: Examples of Jordan measurable sets- II - Part 2
session 15: Jordan measure under Linear transformations - Part 1
session 16: Jordan measure under Linear transformations - Part 2
session 17: Connecting the Jordan measure with the Riemann integral - Part 1
session 18: Connecting the Jordan measure with the Riemann integral - Part 2
part 3: Week 3
session 19: Outer measure - Motivation and Axioms of outer measure
session 20: Comparing Inner Jordan measure, Lebesgue outer measure and Jordan Outer measure
session 21: Finite additivity of outer measure on Separated sets, Outer regularity - Part 1
session 22: Finite additivity of outer measure on Separated sets, Outer regularity - Part 2
session 23: Lebesgue measurable class of sets and their Properties - Part 1
session 24: Lebesgue measurable class of sets and their Properties - Part 2
session 25: Equivalent criteria for lebesgue measurability of a subset - Part 1
session 26: Equivalent criteria for lebesgue measurability of a subset - Part 2
part 4: Week 4
session 27: The measure axioms and the Borel-Cantelli Lemma
session 28: Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 1
session 29: Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 2
session 30: Lebesgue measurability under Linear transformation, Construction of Vitali Set -Part 1
session 31: Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 2
part 5: Week 5
session 32: Abstract measure spaces: Boolean and Sigma-algebras
session 33: Abstract measure and Caratheodory Measurability - Part 1
session 34: Abstract measure and Caratheodory Measurability - Part 2
session 35: Abstrsct measure and Hahn-Kolmogorov Extension
session 36: Lebesgue measurable class vs Caratheodory extension of usual outer measure on R^d
session 37: Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 1
session 38: Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 2
part 6: Week 6
session 39: Measurable functions: definition and basic properties - Part 1
session 40: Measurable functions: definition and basic properties - Part 2
session 41: Egorov's theorem: abstract version
session 42: Lebesgue integral of unsigned simple measurable functions: definition and properties
session 43: Lebesgue integral of unsigned measurable functions: motivation, definition and basic properties
session 44: Fundamental convergence theorems in Lebesgue integration: Monotone convergence theorem, Tonelli's theorem and Fatou's lemma
part 7: Week 7
session 45: Lebesgue integral for complex and real measurable functions: the space of L^1 functions
session 46: Basic properties of L^1-functions and Lebesgue's Dominated convergence theorem
session 47: L^1 functions on R^d: Egorov's theorem revisited (Littlewood's third principle)
session 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
session 49: L^1 functions on R^d: Proof of Lusin's theorem, space of L^1 functions as a metric space
session 50: L^1 functions on R^d: the Riesz-Fischer theorem
part 8: Week 8
session 51: Various modes of convergence of measurable functions
session 52: Easy implications from one mode of convergence to another
session 53: Implication map for modes of convergence with various examples
session 54: Uniqueness of limits across various modes of convergence
session 55: Some criteria for reverse implications for modes of convergence
part 9: Week 9
session 56: Riesz Representation theorem- Motivation
session 57: Basics on Locally compact Hausdorff spaces
session 58: Borel and Radon measures on LCH spaces
session 59: Properties of Radon measures and Lusin's theorem on LCH spaces
session 60: Riesz Representation theorem - Complete statement and proof - Part 1
session 61: Riesz Representation theorem - Complete statement and proof - Part 2
session 62: Examples of measures constructed using RRT
part 10: Week 10
session 63: Theorems of Tonelli and Fubini- interchanging the order of integration for repeated integrals: motivation and discussion of product measure spaces
session 64: Product measures
session 65: Tonelli's theorem for sets - Part 1
session 66: Tonelli's theorem for sets - Part 2
session 67: Fubini-Tonelli theorem: interchanging order of integration for measurable and L^1 functions on sigma-finite measure spaces
part 11: Week 11
session 68: Lebesgue's differentiation theorem: introduction and motivation
session 69: Lebesgue's differentiation theorem: statement and proof - Part 1
session 70: Lebesgue's differentiation theorem: statement and proof - Part 2
part 12: Week 12
session 71: DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 1
session 72: DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 2
session 73: Riesz's Rising Sun Lemma
session 74: Differentiation theorem for monone continuous functions
session 75: Differentation theorem for general monotone functions and Second fundamental theorem of calculus for absolutely continuous functions
part 13: Live Session
session 76: Live Session 20-11-2020

Catalogue records

nptel: 111106140 (manifestation)
nptel_video: 111106140 (manifestation)