Measure Theory and Integration (MA40042)
Semester I, 2023-24
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
Wednesdays 1.15 to 2.10 (10 January - otherwise by appointment).
Revision Week
I am planning to have revision week lectures on:
-
Wednesday 10 January 10.15
-
Thursday 11 January 3.15
News
This site to be used in conjunction with the course Moodle page.
Consolidated lecture notes
- Consolidated notes for LAST year (2022-23):
pdf file.
- Consolidated notes for THIS year (2023-24),
Sections 1-9 (Pages 1-32)
- Complete consolidated notes for THIS year (2023-24).
pdf file.
Lecture notes
- Pages 1 to 5 (Sec.1, start of Sec.2):
pdf file.
- Pages 6 to 10 (end of Sec. 2, Sec 3):
pdf file.
- Pages 11 to 13 (Sec. 4): pdf file.
- Pages 14 to 16 (Sec. 5): pdf file.
The proof of Theorem 5.5 is non-examinable this year.
- Pages 17 to 21 (Sec. 6, first version posted): pdf file.
- Pages 17 to 21 (Sec. 6, First updated version, posted 19 October): pdf file.
- Pages 17 to 21 (Sec. 6, Second updated version, posted 21 December - added 2 lines to proof of Corollary 6.3): pdf file.
- Pages 22 to 28 (Sec. 7-8): pdf file.
- Pages 29 to 32 (Sec. 9): pdf file.
- Pages 33 to 35 (Sec. 10) (first version posted): pdf file.
- Pages 33 to 35 (Sec. 10) (updated 16 November):
pdf file (updated the proof of Lemma 10.7(b) to better match the proof in lectures).
- Pages 36 to 38 (Sec. 11): pdf file.
- Pages 39 to 42 (Sec. 12): pdf file.
- Pages 43 to 46 (Sec. 13): Preliminary version pdf file.
- Pages 43 to 47 (Sec. 13): Updated version pdf file.
- Pages 48 to 53 (Sec. 14): pdf file.
- Pages 54 to 61 (Sec. 15 and Sec. 16): pdf file.
Problem sheets