TMV100 / MMA110 Integration theory
Course PM
This page contains the program of the course: lectures and suggested exercises. Other information, such as learning outcomes, teachers, literature, lecture notes and examination information, are in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Information for the exam
Lectures (the correspondence between what will be covered and the days is very approximate: in particular, the days with planned exercises might not be the days we do the exercises).
Day | Sections | Content |
---|---|---|
8/28 |
JS 3.1-3.3 F pp 19-22 JJ 3.1,3.2,3.5 |
Algebras, sigma-algebras, d-systems, Dynkin's Lemma, measure spaces |
8/31 |
JS 3.3-3.7 F 1.3,4 JJ 3.7 and Theorem 3.10 |
Measure spaces, outer measure, Caratheodory's Theorem, Construction of Lebesgue measure, uniqueness, nononmeasurable sets |
9/1 |
JS 3.9-3.12 |
Cantor function, Distribution functions, Borel-Cantelli |
9/4 |
|
Exercises, Review, discussion |
9/7 |
JS 4.1-4.3 F 2.1,2 up to (but not including) Theorem 2.14. JJ 5 until Theorem 5.6 |
Measurable functions, integration of non-negative functions. |
9/8 |
JS 4.2-4.3 F Theorems 2.14-2.20 F 2.3 until p 55 |
Monotone convergence theorem, integration of real valued functions, Fatou's lemma, Lebesgue dominated convergence |
9/11 | Exercises, Review, discussion | |
9/14 |
JS 4.4-4.5 F 2.4 |
Modes of convergence, Some inequalities (Markov and Chebyshev) |
9/15 |
JS 5.1-5.5 F p 22-23 and F2.5 |
Product sigma algebras, product measures, Fubini-Tonelli Theorem, certain counterexamples |
9/18 | Continuation with product measures | |
9/21 | Exercises, Review, discussion | |
9/22 |
JS 6.1-6.3 JJ p. 16, pp 21-22, 24-25 |
Random variables, expectation (putting probability theory inside measure and integration theory) |
9/25 |
JS 6.1-6.3 JJ p 9-10, 26-27 (until 4.3), Theorem 8.2, 8.3 |
Borel-Cantelli lemmas, weak and strong law of large numbers |
9/28 | Exercises, Review, discussion | |
9/29 |
JS 7.1 F 3.1 |
Signed measures, Jordan-Hahn decomposition theorems, mutual singularity |
10/2 |
JS 7.2-7.3 F 3.2 |
Absolute continuity, Radon-Nikodym theorem,Lebesgue’s decomposition theory |
10/5 | Exercises, Review, discussion | |
10/6 |
JS 8.1-8.4 F pp.95-96 |
3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem |
10/9 |
JS 8.5 F pp.97-100
|
Lebesgue’s differentiation theorem |
10/12 | Exercises, Review, discussion | |
10/13 |
JS 9.1-9.2 F pp. 101-107 |
Functions of bounded variation, absolute continuity, Fundamental theorem of calculus |
10/16 |
JS 9.1-9.2 F pp. 101-107 |
Continuation with the above |
10/19 | Exercises, Review, discussion | |
10/20 | Possible Review |
Recommended exercises: (1) Those contained in the lecture notes, (2) exercises
from Folland below and (3) exercises from the supplementary exercise sheet (if you have time).
You should prioritise (1) and (2).
Chapter | Exercises |
---|---|
Folland 1 |
2, 3, 4, 8, 9, 10, 18(a,b), 19,23, 28, 30, 31, 33 |
Folland 2 |
1, 3, 4, 5, 6, 8, 9, 13, 14, 16, 19, 21, 22, 25, 28(a,b), 30, 33, 37, 38, 47, 48 |
Folland 3 |
4, 5, 6, 9, 11, 17 (but only if you are interested in probability), 22, 25, 30, 31, 32, 39, 40, 41, 42(d) |
supplementary exercise sheet (these might or might not be updated as the course proceeds)
Course summary:
Date | Details | Due |
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