# Measure Theory

- Institution: IMSc
- Department: Mathematics
- Teacher: Prof. Indrava Roy
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106140)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106140)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [Finite Sets and Cardinality](https://invisiblenetwork.ai/learn/nptel/111106140/8a28b4404daf) (Video)
2. [Infinite Sets and the Banach-Tarski Paradox - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/9ac2962efe30) (Video)
3. [Infinite Sets and the Banach-Tarski Paradox - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/c2b17a531e2e) (Video)
4. [Elementary Sets and Elementary measure - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/64956b487925) (Video)
5. [Elementary Sets and Elementary measure - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/f8de3a7e847b) (Video)
6. [Properties of elementary measure - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/56faae960239) (Video)
7. [Properties of elementary measure - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/85dc34fce5e6) (Video)
8. [Uniqueness of elementary measure and Jordan measurability - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/b927bd74d4e2) (Video)
9. [Uniqueness of elementary measure and Jordan measurability - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/b1e27836fd77) (Video)
10. [Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/a0f88b17cb87) (Video)
11. [Characterization of Jordan measurable sets and basic properties of Jordan measure - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/8c647bfc7e88) (Video)
12. [Examples of Jordan measurable sets- I](https://invisiblenetwork.ai/learn/nptel/111106140/1bec9ac4ff02) (Video)
13. [Examples of Jordan measurable sets- II - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/c7e0390dc7f6) (Video)
14. [Examples of Jordan measurable sets- II - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/ba2fa8a5c0a6) (Video)
15. [Jordan measure under Linear transformations - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/f2d8e50955c6) (Video)
16. [Jordan measure under Linear transformations - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/d0784614f592) (Video)
17. [Connecting the Jordan measure with the Riemann integral - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/a33ec4afce07) (Video)
18. [Connecting the Jordan measure with the Riemann integral - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/4175f4773772) (Video)
19. [Outer measure - Motivation and Axioms of outer measure](https://invisiblenetwork.ai/learn/nptel/111106140/46b375b691ab) (Video)
20. [Comparing Inner Jordan measure, Lebesgue outer measure and Jordan Outer measure](https://invisiblenetwork.ai/learn/nptel/111106140/3d9730efea31) (Video)
21. [Finite additivity of outer measure on Separated sets, Outer regularity - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/f68875c4c4cf) (Video)
22. [Finite additivity of outer measure on Separated sets, Outer regularity - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/970425b6e120) (Video)
23. [Lebesgue measurable class of sets and their Properties - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/33c57ed7d2db) (Video)
24. [Lebesgue measurable class of sets and their Properties - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/8c92aa359144) (Video)
25. [Equivalent criteria for lebesgue measurability of a subset - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/c7805d06a929) (Video)
26. [Equivalent criteria for lebesgue measurability of a subset - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/90ec18803431) (Video)
27. [The measure axioms and the Borel-Cantelli Lemma](https://invisiblenetwork.ai/learn/nptel/111106140/c9c140868a6a) (Video)
28. [Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/dd8a6909c7a1) (Video)
29. [Properties of the Lebesgue measure: Inner regularity,Upward and Downwar Monotone convergence theorem, and Dominated convergence theorem for sets - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/2befbdd5b2ca) (Video)
30. [Lebesgue measurability under Linear transformation, Construction of Vitali Set -Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/dedc070b1a9b) (Video)
31. [Lebesgue measurability under Linear transformation, Construction of Vitali Set - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/ca754f225b20) (Video)
32. [Abstract measure spaces: Boolean and Sigma-algebras](https://invisiblenetwork.ai/learn/nptel/111106140/440086de976f) (Video)
33. [Abstract measure and Caratheodory Measurability - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/027efbb71d0f) (Video)
34. [Abstract measure and Caratheodory Measurability - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/0c0b6575f818) (Video)
35. [Abstrsct measure and Hahn-Kolmogorov Extension](https://invisiblenetwork.ai/learn/nptel/111106140/2448871a0813) (Video)
36. [Lebesgue measurable class vs Caratheodory extension of usual outer measure on R^d](https://invisiblenetwork.ai/learn/nptel/111106140/f4f2a3db1ddc) (Video)
37. [Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/5339516ed635) (Video)
38. [Examples of Measures defined on R^d via Hahn Kolmogorov extension - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/2693c08e0c95) (Video)
39. [Measurable functions: definition and basic properties - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/c6a12384ceba) (Video)
40. [Measurable functions: definition and basic properties - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/c65118bda2c8) (Video)
41. [Egorov's theorem: abstract version](https://invisiblenetwork.ai/learn/nptel/111106140/4f1c7fdcce14) (Video)
42. [Lebesgue integral of unsigned simple measurable functions: definition and properties](https://invisiblenetwork.ai/learn/nptel/111106140/1d7200f0f877) (Video)
43. [Lebesgue integral of unsigned measurable functions: motivation, definition and basic properties](https://invisiblenetwork.ai/learn/nptel/111106140/e330b9787be5) (Video)
44. [Fundamental convergence theorems in Lebesgue integration: Monotone convergence theorem, Tonelli's theorem and Fatou's lemma](https://invisiblenetwork.ai/learn/nptel/111106140/d136ccb4ec26) (Video)
45. [Lebesgue integral for complex and real measurable functions: the space of L^1 functions](https://invisiblenetwork.ai/learn/nptel/111106140/866266cc7f56) (Video)
46. [Basic properties of L^1-functions and Lebesgue's Dominated convergence theorem](https://invisiblenetwork.ai/learn/nptel/111106140/e07d1b47027a) (Video)
47. [L^1 functions on R^d: Egorov's theorem revisited (Littlewood's third principle)](https://invisiblenetwork.ai/learn/nptel/111106140/b6538147930f) (Video)
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](https://invisiblenetwork.ai/learn/nptel/111106140/7e691754e3e1) (Video)
49. [L^1 functions on R^d: Proof of Lusin's theorem, space of L^1 functions as a metric space](https://invisiblenetwork.ai/learn/nptel/111106140/7f3dcd35efa8) (Video)
50. [L^1 functions on R^d: the Riesz-Fischer theorem](https://invisiblenetwork.ai/learn/nptel/111106140/9029ca9d3108) (Video)
51. [Various modes of convergence of measurable functions](https://invisiblenetwork.ai/learn/nptel/111106140/7b18be268695) (Video)
52. [Easy implications from one mode of convergence to another](https://invisiblenetwork.ai/learn/nptel/111106140/03ee53930249) (Video)
53. [Implication map for modes of convergence with various examples](https://invisiblenetwork.ai/learn/nptel/111106140/0c5d6af770e9) (Video)
54. [Uniqueness of limits across various modes of convergence](https://invisiblenetwork.ai/learn/nptel/111106140/5b6dbb4d0f04) (Video)
55. [Some criteria for reverse implications for modes of convergence](https://invisiblenetwork.ai/learn/nptel/111106140/db7291375e28) (Video)
56. [Riesz Representation theorem- Motivation](https://invisiblenetwork.ai/learn/nptel/111106140/1b298b0b0d11) (Video)
57. [Basics on Locally compact Hausdorff spaces](https://invisiblenetwork.ai/learn/nptel/111106140/1bd3fc897e55) (Video)
58. [Borel and Radon measures on LCH spaces](https://invisiblenetwork.ai/learn/nptel/111106140/c825677bfeda) (Video)
59. [Properties of Radon measures and Lusin's theorem on LCH spaces](https://invisiblenetwork.ai/learn/nptel/111106140/4086fee7db2b) (Video)
60. [Riesz Representation theorem - Complete statement and proof - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/b3fa1ee01226) (Video)
61. [Riesz Representation theorem - Complete statement and proof - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/8f30b4cf85a7) (Video)
62. [Examples of measures constructed using RRT](https://invisiblenetwork.ai/learn/nptel/111106140/29b1ca2ce045) (Video)
63. [Theorems of Tonelli and Fubini- interchanging the order of integration for repeated integrals: motivation and discussion of product measure spaces](https://invisiblenetwork.ai/learn/nptel/111106140/64b83d789bc2) (Video)
64. [Product measures](https://invisiblenetwork.ai/learn/nptel/111106140/266a130af105) (Video)
65. [Tonelli's theorem for sets - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/b825f73df047) (Video)
66. [Tonelli's theorem for sets - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/f53bf2a76624) (Video)
67. [Fubini-Tonelli theorem: interchanging order of integration for measurable and L^1 functions on sigma-finite measure spaces](https://invisiblenetwork.ai/learn/nptel/111106140/35e6417d614a) (Video)
68. [Lebesgue's differentiation theorem: introduction and motivation](https://invisiblenetwork.ai/learn/nptel/111106140/2510be19c9da) (Video)
69. [Lebesgue's differentiation theorem: statement and proof - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/246df3ca4e40) (Video)
70. [Lebesgue's differentiation theorem: statement and proof - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/1f908693b448) (Video)
71. [DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 1](https://invisiblenetwork.ai/learn/nptel/111106140/7db25494280c) (Video)
72. [DIfferentiation theorems: Almost everywhere differentiability for Monotone and Bounded Variation functions - Part 2](https://invisiblenetwork.ai/learn/nptel/111106140/02745327c86f) (Video)
73. [Riesz's Rising Sun Lemma](https://invisiblenetwork.ai/learn/nptel/111106140/1d2f11885854) (Video)
74. [Differentiation theorem for monone continuous functions](https://invisiblenetwork.ai/learn/nptel/111106140/8288b5123285) (Video)
75. [Differentation theorem for general monotone functions and Second fundamental theorem of calculus for absolutely continuous functions](https://invisiblenetwork.ai/learn/nptel/111106140/826820d9bdca) (Video)
76. [Live Session 20-11-2020](https://invisiblenetwork.ai/learn/nptel/111106140/8a28b4404daf-2) (Video)