Course syllabus
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
Exercises for oral
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).
October 6 Lecture
| Day | Sections | Content |
|---|---|---|
| 8/29 |
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 |
| 9/1 |
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 |
| 9/2 |
JS 3.9-3.12 |
Cantor function, Distribution functions, Borel-Cantelli |
| 9/5 |
|
Exercises, Review, discussion |
| 9/8 |
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. |
| 9/9 |
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 |
| 9/12 | Exercises, Review, discussion | |
| 9/15 |
JS 4.4-4.5 F 2.4 |
Modes of convergence, Some inequalities (Markov and Chebyshev) |
| 9/16 |
JS 5.1-5.5 F p 22-23 and F2.5 |
Product sigma algebras, product measures, Fubini-Tonelli Theorem, certain counterexamples |
| 9/19 | Continuation with product measures | |
| 9/22 | Exercises, Review, discussion | |
| 9/23 |
JS 6.1-6.3 JJ p. 16, pp 21-22, 24-25 |
Random variables, expectation (putting probability theory inside measure and integration theory) |
| 9/26 |
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 |
| 9/29 | Exercises, Review, discussion | |
| 9/30 |
JS 7.1 F 3.1 |
Signed measures, Jordan-Hahn decomposition theorems, mutual singularity |
| 10/3 |
JS 7.2-7.3 F 3.2 |
Absolute continuity, Radon-Nikodym theorem,Lebesgue’s decomposition theory |
| 10/6 | Exercises, Review, discussion | |
| 10/7 |
JS 8.1-8.4 F pp.95-96 |
3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem |
| 10/10 |
JS 8.5 F pp.97-100
|
Lebesgue’s differentiation theorem |
| 10/13 | Exercises, Review, discussion | |
| 10/14 |
JS 9.1-9.2 F pp. 101-107 |
Functions of bounded variation, absolute continuity, Fundamental theorem of calculus |
| 10/17 |
JS 9.1-9.2 F pp. 101-107 |
Continuation with the above |
| 10/20 | Exercises, Review, discussion | |
| 10/21 | 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 |
|---|---|---|