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

See lecture notes

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.
The last class before Christmas (5.15 December 13th) is replaced by an office hour. Please let me know by email before 4PM if you would like to see me then.

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.
Consolidated notes for THIS year (2024-25) (complete notes, posted 12 December: 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.
There is a typo in the statement of Lemma 8.4. The sets R,R_1, etc. should be in curly R, not curly M.
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.
Pages 44 to 47 (Sec. 14, first version released): pdf file.
Pages 44 to 47 (Sec. 14, updated 29 November to better reflect the lectures): pdf file.
Pages 48 to 53 (Sec. 15, preliminary version): pdf file.
Pages 48 to 53 (Sec. 15, updated version 5 December 14.48): pdf file.
Pages 54 to 57 (Sec. 16): pdf file.
Pages 58 to 61 (Sec. 17): pdf file.

Problem sheets

Sheet 1 pdf file.
Sheet 2 pdf file.
Sheet 3 pdf file.
Sheet 4 pdf file.
Sheet 5 pdf file.
Sheet 6 pdf file.
Sheet 7 pdf file.
Sheet 8 pdf file.
Sheet 9 pdf file (Posted 29 Nov, updated later that day. The only change is that the update refers back to Definition 12.3 rather than ahead to Definition 15.3 in Question 54). (Further updated 6 December to correct an error in Question 52(e))
Sheet 10 pdf file.

Solutions

Sheet 1 pdf file.
Sheet 2 pdf file.
Sheet 3 pdf file.
Sheet 4 pdf file.
Sheet 5 pdf file.
Sheet 6 pdf file.
Sheet 7 pdf file.
Sheet 8 pdf file.
Sheet 9 pdf file.
Sheet 10 pdf file.