Lectures
Week 1
- Finite Sets and Cardinality
- Infinite Sets and the Banach-Tarski Paradox - Part 1
- Infinite Sets and the Banach-Tarski Paradox - Part 2
- Elementary Sets and Elementary measure - Part 1
- Elementary Sets and Elementary measure - Part 2
- Properties of elementary measure - Part 1
- Properties of elementary measure - Part 2
Week 2
- Uniqueness of elementary measure and Jordan measurability - Part 1
- Uniqueness of elementary measure and Jordan measurability - Part 2
- Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 1
- Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 2
- Examples of Jordan measurable sets- I
- Examples of Jordan measurable sets- II - Part 1
- Examples of Jordan measurable sets- II - Part 2
- Jordan measure under Linear transformations - Part 1
- Jordan measure under Linear transformations - Part 2
- Connecting the Jordan measure with the Riemann integral - Part 1
- Connecting the Jordan measure with the Riemann integral - Part 2
Week 3
- Outer measure - Motivation and Axioms of outer measure
- Comparing Inner Jordan measure, Lebesgue outer measure and Jordan Outer measure
- Finite additivity of outer measure on Separated sets, Outer regularity - Part 1
- Finite additivity of outer measure on Separated sets, Outer regularity - Part 2
- Lebesgue measurable class of sets and their Properties - Part 1
- Lebesgue measurable class of sets and their Properties - Part 2
- Equivalent criteria for lebesgue measurability of a subset - Part 1
- Equivalent criteria for lebesgue measurability of a subset - Part 2
Week 4
- The measure axioms and the Borel-Cantelli Lemma
- Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 1
- Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 2
- Lebesgue measurability under Linear transformation, Construction of Vitali Set -Part 1
- Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 2
Week 5
- Abstract measure spaces: Boolean and Sigma-algebras
- Abstract measure and Caratheodory Measurability - Part 1
- Abstract measure and Caratheodory Measurability - Part 2
- Abstrsct measure and Hahn-Kolmogorov Extension
- Lebesgue measurable class vs Caratheodory extension of usual outer measure on R^d
- Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 1
- Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 2
Week 6
- Measurable functions: definition and basic properties - Part 1
- Measurable functions: definition and basic properties - Part 2
- Egorov's theorem: abstract version
- Lebesgue integral of unsigned simple measurable functions: definition and properties
- Lebesgue integral of unsigned measurable functions: motivation, definition and basic properties
- Fundamental convergence theorems in Lebesgue integration: Monotone convergence theorem, Tonelli's theorem and Fatou's lemma
Week 7
- Lebesgue integral for complex and real measurable functions: the space of L^1 functions
- Basic properties of L^1-functions and Lebesgue's Dominated convergence theorem
- L^1 functions on R^d: Egorov's theorem revisited (Littlewood's third principle)
- 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
- L^1 functions on R^d: Proof of Lusin's theorem, space of L^1 functions as a metric space
- L^1 functions on R^d: the Riesz-Fischer theorem
Week 8
- Various modes of convergence of measurable functions
- Easy implications from one mode of convergence to another
- Implication map for modes of convergence with various examples
- Uniqueness of limits across various modes of convergence
- Some criteria for reverse implications for modes of convergence
Week 9
- Riesz Representation theorem- Motivation
- Basics on Locally compact Hausdorff spaces
- Borel and Radon measures on LCH spaces
- Properties of Radon measures and Lusin's theorem on LCH spaces
- Riesz Representation theorem - Complete statement and proof - Part 1
- Riesz Representation theorem - Complete statement and proof - Part 2
- Examples of measures constructed using RRT
Week 10
- Theorems of Tonelli and Fubini- interchanging the order of integration for repeated integrals: motivation and discussion of product measure spaces
- Product measures
- Tonelli's theorem for sets - Part 1
- Tonelli's theorem for sets - Part 2
- Fubini-Tonelli theorem: interchanging order of integration for measurable and L^1 functions on sigma-finite measure spaces
Week 11
- Lebesgue's differentiation theorem: introduction and motivation
- Lebesgue's differentiation theorem: statement and proof - Part 1
- Lebesgue's differentiation theorem: statement and proof - Part 2
Week 12
- DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 1
- DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 2
- Riesz's Rising Sun Lemma
- Differentiation theorem for monone continuous functions
- Differentation theorem for general monotone functions and Second fundamental theorem of calculus for absolutely continuous functions
Live Session
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IMSc
- Department
- Mathematics
- Teacher
- Prof. Indrava Roy
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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