Measure Theory and Integration (MA40042)
Semester I, 2024-25
This site to be used in conjunction with the course Moodle page.
Content:
Systems of measurable sets: sigma-algebras, pi-systems, d-systems,
Dynkin's Lemma, Borel sigma-algebras. Measure in the abstract: convergence properties, Uniqueness Lemma, Caratheodory's Theorem (statement). Lebesgue outer measure and measure on Rn. Measurable functions. Monotone-Class Theorem. Probability. Random variables. Independence. Integration of non-negative and signed functions. Monotone-Convergence Theorem. Fatou's Lemma. Dominated-Convergence Theorem. Expectation. Product measures. Tonelli's and Fubini's Theorem. Radon-Nikodym Theorem (statement). Inequalities of Jensen, Holder, Minkowski. Completeness of Lp.
Books
Office Hour
Mondays 4.15 to 5.10, Room 4W4.11
News
- In the class on 8 November, as well as going over the prolem sheet we covered Theorem 11.6 of the notes (Monotone Convergence) so we'll be
proceding from after that in the next lecture.
Consolidated lecture notes
- Consolidated notes for LAST year (2023-24).
pdf file.
- Consolidated notes for THIS year (2024-25),
Sections 1 to 10
(these consolidated notes contain a few minor corrections/improvements compared to the notes that have been posted as lectures went along.)
pdf file.
Lecture notes
- Pages 1 to 10 (Sec. 1 to 3):
pdf file.
- Pages 11 to 16 (Sec. 4 to 5, updated 9 October 11.14 AM): pdf file.
- Pages 13 to 16 (Sec. 5, updated 16 October): pdf file. (compared to the version from October 9, the only change is to remove the word `probability' from Lenna 5,6)
- Pages 17 to 21 (Sec. 6, updated version 16 October):
pdf file.
- Pages 22 to 23 (Sec. 7, 17 October):
pdf file.
- Pages 24 to 27 (Sec. 8, first version 23 October):
pdf file.
- Pages 24 to 27 (Sec. 8, updated version 24 October):
pdf file.
- Pages 28 to 29 (Sec. 9, posted 28 October):
pdf file.
- Pages 30 to 33 (Sec. 10, preliminary version 31 October 3PM):
pdf file.
- Pages 30 to 33 (Sec. 10, updated version 31 October 5PM):
pdf file.
- Pages 34 to 36 (Sec. 11): pdf file.
- Pages 37 to 39 (Sec. 12): pdf file.
- Pages 40 to 43 (Sec. 13): pdf file.
Problem sheets
Solutions