Measure Theory and Integration (MA40042)
Semester I, 2020-21
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.
This site to be used in conjunction with the course Moodle page.
- Pages 1 to 10 (Intro and Sections 1-3): pdf file.
- Pages 11 to 16 (Sections 4-5): pdf file.
- Pages 17 to 22 (Sections 6-7): pdf file.
- Pages 23 to 27 (Section 8): pdf file.
- Pages 28 to 31 (Section 9): pdf file.
- Pages 32 to 34 (Section 10): pdf file.
- Pages 35 to 37 (Section 11): pdf file.
- Pages 38 to 41 (Section 12): pdf file.
- Pages 42 to 45 (Section 13): pdf file.
- Pages 46 to 49 (start of Section 14): pdf file.
- Pages 50 to 53 (End of Section 14, and Section 15):
- Pages 54 to 61 (Sections 16-17):
A mock online paper
This mock exam
is adapted from last year's exam (i.e. the exam 2019-20 Semester 1).
Last year's exam was closed-book. The new Mock exam is designed for this year's open-book format.
Parts of questions that could be answered by copying material
from the typed notes or solutions to exercises have been replaced
by unseen material of a similar level of difficulty.
Both the `new' Mock exam questions, and the `Old'
questions as they appeared in last year's exam,
have been included for comparison purpuses.
NB the mock exam paper has not been through a checking process in the way that actual papers are quality-controlled.