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 and examination, are in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Lectures (the correspondence between what will be covered and the days is very approximate)
| Day | Sections | Content |
|---|---|---|
| 9/2 |
F pp 19-22 JJ 3.1,3.2,3.5 |
Why we can't measure all sets, sigma-algebras, d-systems, Dynkin's Lemma, measure spaces |
| 9/5 |
F 1.3,4 JJ 3.7 and Theorem 3.10 |
Measure spaces, outer measure, Caratheodory's Theorem, Construction of Lebesgue measure |
| 9/6 |
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 |
F Theorems 2.14-2.20 F 2.3 until p 55 |
Monotone convergence theorem, integration of complex valued functions, Fatou's lemma, Lebesgue dominated convergence) |
| 9/12 | F 2.4 | Modes of convergence |
| 9/13 | F p 22-23 and F2.5 | Product sigma algebras, product measures, Fubini-Tonelli Theorem, certain counterexamples |
| 9/16 | NO CLASS | |
| 9/19 | JJ p. 16, pp 21-22, 24-25 | Random variables, expectation (putting probability theory inside measure and integration theory) |
| 9/20 | JJ p 9-10, 26-27 (until 4.3), Theorem 8.2, 8.3 | Borel-Cantelli lemmas, weak and strong law of large numbers |
| 9/23 | Continuing with the above if lack of time | |
| 9/26 | F 3.1 | Signed measures, Jordan-Hahn decomposition theorem |
| 9/27 | F 3.2 | Radon-Nikodym theorem,Lebesgue’s decomposition theorem |
| 9/30 | NO CLASS | |
| 10/3 | F pp.95-96 | 3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem |
| 10/4 | NO CLASS | |
| 10/7 | F pp.97-100 | Lebesgue’s differentiation theorem |
| 10/10 | F pp. 101-104 | Functions of bounded variation |
| 10/14 | F pp. 105-107 | Fundamental theorem of calculus and integration by parts for Lebesgue integral |
| 10/17 | Summary/reserve | |
|
10/21 10/24 10/25 |
Presentations of Projects Presentations of Projects Presentations of Projects |
Recommended exercises
| Chapter | Exercises |
|---|---|
| Folland 1 |
1.2.4, 1.4.17,1.4.18, 1.4.24 (Added 8,9,10,30,31,33 see announcement sept 9) |
| Folland 2 |
2.1.3, 2.1.4, 2.2.13, 2.2.15, 2.4.33-2.4.39, 2.4.42, 2.5.46, 2.5.47, 2.5.48 |
| Folland 3 | 3.1.1, 3.1.2, 3.1.3, 3.1.6, 3.2.8, 3.2.10, 3.2.11, 3.2.12*, 3.2.13, 3.2.16, 3.2.17, 3.4.22-26, 3.5.30-33, 3.5.40-41 |
Project
The projects entail each group reading about some topic related to the course
and then presenting it to the class. The presentation should be 30-45 minutes
with each person presenting approximately the same amount.
A group should be between 3 and 4 people, 4 being the ideal.
The topic needs to be approved by us. Alternatively, we can suggest a topic.
Hopefully most people will have formed groups by the second week. For those who have not, we will match them.
Please send proposals by email to both of us by September 15. If you don't have a group, write to us and we find you a group.
Possible Project Topics:
Hausdorff measures and dimension
Construction of Borel sets (via ordinals)
Original construction of Lebesgue measure (via inner and outer capacities)
Finding all sigma-algebras for a finite set
Examples of sigma-algebras on infinite sets (for example finite/co-finite sets, etc.)
More covering lemmas/theorems
Project groups (formed so far):
Course summary:
| Date | Details | Due |
|---|---|---|