# Measure and Integration

- Institution: IMSc
- Department: Mathematics
- Teacher: Prof. S. Kesavan
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106161)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106161)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

The theory of measure and integration is now an integral part of any masters or graduate program in mathemat ics in Indian Universities. It is an important prerequisite for most analysis based courses. In this course, we will start with a study of abstract measures with the Lebesgue measure being the most important example.

After looking at measurable functions and some types of convergences for such functions, we will introduce the notion of the Lebesgue integral in an abstract measure space and study its properties. We will relate the two important processes of the calculus – differentiation and integration – in the context of the Lebesgue integral. We will also study measures and integrals on product spaces and also signed measures. Finally we will study the important properties of L^p – spaces.

Intended audience: MSc (Mathematics) and above

Prerequisites: MSc Real Analysis, Topology, Linear AlgebraINTENDED AUDIENCE: MSc (Mathematics) and above

## Lectures

1. [1.1 - Preamble](https://invisiblenetwork.ai/learn/nptel/111106161/1d1b584b6430) (YouTube)
2. [1.2 - Algebras of sets](https://invisiblenetwork.ai/learn/nptel/111106161/54c41195a63b) (YouTube)
3. [1.3 - Measures on rings](https://invisiblenetwork.ai/learn/nptel/111106161/eacc5e8f2c73) (YouTube)
4. [1.4 - Outer-measure](https://invisiblenetwork.ai/learn/nptel/111106161/d92de8045fc2) (YouTube)
5. [1.5 - Measurable sets](https://invisiblenetwork.ai/learn/nptel/111106161/1180d770ee51) (YouTube)
6. [2.1 - Caratheodory's method](https://invisiblenetwork.ai/learn/nptel/111106161/3713c91afa4f) (YouTube)
7. [2.2 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/55364a0e1165) (YouTube)
8. [2.3 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/86a76f407147) (YouTube)
9. [2.4 - Lebesgue measure: the ring](https://invisiblenetwork.ai/learn/nptel/111106161/56cb8901010d) (YouTube)
10. [2.5 - Construction of the Lebesgue measure](https://invisiblenetwork.ai/learn/nptel/111106161/cea043848a4f) (YouTube)
11. [2.6 - Errata](https://invisiblenetwork.ai/learn/nptel/111106161/d98dd70395eb) (YouTube)
12. [2.7 - The Cantor set](https://invisiblenetwork.ai/learn/nptel/111106161/28575339bf62) (YouTube)
13. [2.8 - Approximation](https://invisiblenetwork.ai/learn/nptel/111106161/87ccd5f852e3) (YouTube)
14. [3.1 - Approximation](https://invisiblenetwork.ai/learn/nptel/111106161/1d801d5d7e8b) (YouTube)
15. [3.2 - Approximation](https://invisiblenetwork.ai/learn/nptel/111106161/038e86e710e5) (YouTube)
16. [3.3 - Translation Invariance](https://invisiblenetwork.ai/learn/nptel/111106161/3b56d08ccca5) (YouTube)
17. [3.4 - Non-measurable sets](https://invisiblenetwork.ai/learn/nptel/111106161/18ca2fed00f0) (YouTube)
18. [3.5 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/3cbae8e827de) (YouTube)
19. [3.6 - Measurable functions](https://invisiblenetwork.ai/learn/nptel/111106161/f5ff510a2128) (YouTube)
20. [4.1 - Measurable functions](https://invisiblenetwork.ai/learn/nptel/111106161/5ea7edd41e4a) (YouTube)
21. [4.2 - The Cantor function](https://invisiblenetwork.ai/learn/nptel/111106161/1b046f48f49c) (YouTube)
22. [4.3 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/eee309babea6) (YouTube)
23. [4.4 - Egorov's theorem](https://invisiblenetwork.ai/learn/nptel/111106161/ec54b7fa9ab9) (YouTube)
24. [4.5 - Convergence in measure](https://invisiblenetwork.ai/learn/nptel/111106161/6de745e50e85) (YouTube)
25. [4.6 - Convergence in measure](https://invisiblenetwork.ai/learn/nptel/111106161/0a4984b16a4d) (YouTube)
26. [5.1 - Convergence in measure](https://invisiblenetwork.ai/learn/nptel/111106161/975554330cee) (YouTube)
27. [5.2 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/b5e289de66f5) (YouTube)
28. [5.3 - Integration: Simple functions](https://invisiblenetwork.ai/learn/nptel/111106161/8fccb21bd2d3) (YouTube)
29. [5.4 - Non-negative functions](https://invisiblenetwork.ai/learn/nptel/111106161/39dc6d98e3e7) (YouTube)
30. [5.5 - Monotone convergence theorem](https://invisiblenetwork.ai/learn/nptel/111106161/9ddb1503afd4) (YouTube)
31. [5.6 - Examples](https://invisiblenetwork.ai/learn/nptel/111106161/8c358d3e01ed) (YouTube)
32. [5.7 - Fatou's lemma](https://invisiblenetwork.ai/learn/nptel/111106161/8be3f417e708) (YouTube)
33. [5.8 - Integrable functions](https://invisiblenetwork.ai/learn/nptel/111106161/b8ef7df3d1ff) (YouTube)
34. [6.1 - Dominated convergence theorem](https://invisiblenetwork.ai/learn/nptel/111106161/d95358811789) (YouTube)
35. [6.2 - Dominated convergence theorem: Applications](https://invisiblenetwork.ai/learn/nptel/111106161/ef5235d1d733) (YouTube)
36. [6.3 - Absolute continuity](https://invisiblenetwork.ai/learn/nptel/111106161/b999831711d1) (YouTube)
37. [6.4 - Integration on the real line](https://invisiblenetwork.ai/learn/nptel/111106161/dcd2cc196972) (YouTube)
38. [6.5 - Examples](https://invisiblenetwork.ai/learn/nptel/111106161/c7e98f46c652) (YouTube)
39. [6.6 - Weierstrass' theorem](https://invisiblenetwork.ai/learn/nptel/111106161/d85a5be090eb) (YouTube)
40. [6.7 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/d5fc6969fcc1) (YouTube)
41. [6.8 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/dde7159a7418) (YouTube)
42. [7.1 - Vitali covering lemma](https://invisiblenetwork.ai/learn/nptel/111106161/181b2addbed2) (YouTube)
43. [7.2 - Monotonic functions](https://invisiblenetwork.ai/learn/nptel/111106161/27be93b1b3b3) (YouTube)
44. [7.3 - Functions of bounded variation](https://invisiblenetwork.ai/learn/nptel/111106161/01e92215a097) (YouTube)
45. [7.4 - Functions of bounded variation](https://invisiblenetwork.ai/learn/nptel/111106161/7b75881f2bab) (YouTube)
46. [7.5 - Functions of bounded variation](https://invisiblenetwork.ai/learn/nptel/111106161/da279e6569c4) (YouTube)
47. [7.6 - Differentiation of an indefinite integral](https://invisiblenetwork.ai/learn/nptel/111106161/08abe605d164) (YouTube)
48. [7.7 - Absolute continuity](https://invisiblenetwork.ai/learn/nptel/111106161/6aaf9fe4ad17) (YouTube)
49. [8.1 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/e69fd7718757) (YouTube)
50. [8.2 - Product spaces](https://invisiblenetwork.ai/learn/nptel/111106161/c9ebe14b3fb0) (YouTube)
51. [8.3 - Product spaces: measurable functions](https://invisiblenetwork.ai/learn/nptel/111106161/03b07e98b4b5) (YouTube)
52. [8.4 - Product measure](https://invisiblenetwork.ai/learn/nptel/111106161/14c0ef7e23e9) (YouTube)
53. [8.5 - Fubini's theorem](https://invisiblenetwork.ai/learn/nptel/111106161/689b3a11704b) (YouTube)
54. [8.6 - Examples](https://invisiblenetwork.ai/learn/nptel/111106161/de774b2d8e07) (YouTube)
55. [8.7 - Examples](https://invisiblenetwork.ai/learn/nptel/111106161/42abb1ffb586) (YouTube)
56. [9.1 - Integration of radial functions](https://invisiblenetwork.ai/learn/nptel/111106161/616340b436dc) (YouTube)
57. [9.2 - Measure of the unit ball in N dimensions](https://invisiblenetwork.ai/learn/nptel/111106161/d20a8ecaeef1) (YouTube)
58. [9.3 -Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/f4fcc902c0b5) (YouTube)
59. [9.4 - Signed measures](https://invisiblenetwork.ai/learn/nptel/111106161/5167a71fe98e) (YouTube)
60. [9.5 - Hahn and Jordan decompositions](https://invisiblenetwork.ai/learn/nptel/111106161/9ccd0b9917ac) (YouTube)
61. [9.6 - Upper,lower and totaal variations of a signed measure; Absolute continuity](https://invisiblenetwork.ai/learn/nptel/111106161/3cf7b34614d6) (YouTube)
62. [9.7 - Absolute continuity](https://invisiblenetwork.ai/learn/nptel/111106161/d3c251b05ca7) (YouTube)
63. [10.1 - Radon-Nikodym theorem](https://invisiblenetwork.ai/learn/nptel/111106161/610cd175bc7a) (YouTube)
64. [10.2 - Radon-Nikodym theorem](https://invisiblenetwork.ai/learn/nptel/111106161/635e517d9235) (YouTube)
65. [10.3 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/c65443c1c78a) (YouTube)
66. [10.4 - Lebesgue spaces](https://invisiblenetwork.ai/learn/nptel/111106161/db38f335fee5) (YouTube)
67. [10.5 - Examples. Inclusion questions](https://invisiblenetwork.ai/learn/nptel/111106161/7a452ac3a74e) (YouTube)
68. [10.6 - Convergence in L^p](https://invisiblenetwork.ai/learn/nptel/111106161/0e0d9a46ca18) (YouTube)
69. [11.1 - Approximation](https://invisiblenetwork.ai/learn/nptel/111106161/865c7506a4e3) (YouTube)
70. [11.2 - Applications](https://invisiblenetwork.ai/learn/nptel/111106161/81980793faed) (YouTube)
71. [11.3 - Duality](https://invisiblenetwork.ai/learn/nptel/111106161/d71177c12628) (YouTube)
72. [11.4 - Duality](https://invisiblenetwork.ai/learn/nptel/111106161/18b7d9b0b300) (YouTube)
73. [11.5 - Convolutions](https://invisiblenetwork.ai/learn/nptel/111106161/263f2897208c) (YouTube)
74. [11.6 - Convolutions](https://invisiblenetwork.ai/learn/nptel/111106161/ae9eef95fc5a) (YouTube)
75. [12.1 - Convolutions](https://invisiblenetwork.ai/learn/nptel/111106161/009dcf257381) (YouTube)
76. [12.2 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/8f7a1678f9b6) (YouTube)
77. [12.3 - Exercises](https://invisiblenetwork.ai/learn/nptel/111106161/8d2fd44a8ecc) (YouTube)
78. [12.4 - Change of variable](https://invisiblenetwork.ai/learn/nptel/111106161/6fe428254456) (YouTube)
79. [12.5 - Change of variable](https://invisiblenetwork.ai/learn/nptel/111106161/1c279567cb2c) (YouTube)