Maat en integratie (WISB 312)

Intended audience: third-year students

Book: Measures, Integrals and Martingales.

Outline: sigma-algebras and measures, Dynkin classes, Lebesgue measure, integration, modes of convergence for measurable and integrable functions, inequalities, image measures and integrals, uniform integrability, martingales, Radon-Nikodym theorem.

Prerequisites: It is expected that you have knowledge of analysis at the level of the course Functies en Reeksen. Some knowledge of functional analysis and/or probability theory can also be helpful.

Treated in the course:

On 11-2: Chapter 3.
On 18-2: Chapter 4.
On 25-2: Chapters 5 and 6. The result of Theorem 6.1 was stated precisely (as such it is exam material), but its proof was only sketched and will not be exam material.
On 4-3: Chapter 7 up to and including Example 7.8 (p.52).
On 11-3: Chapter 8 up to and including page 63.
On 18-3: no course given.
On 25-3: From Chapter 8, page 63 up to and including Theorem 9.6.
On 1-4: Finished Chapter 9. Chapter 10 up to and including Theorem 10.9.
On 8-4: Finished Chapter 10. Chapter 11 up to and including Theorem 11.2
On 15-4: Reduced course. Continued chapter 11 up to sketch of Corollary 11.9.
On 22-4: no course given.
On 29-4: Finished chapter 11. Chapter 12 until page 112.
On 6-5: Finished chapter 12 and treated first two pages of Chapter 13.
On 13-5: no course given.
On 20-5: pages 122 until 127. Skipped the subsection "More on measurable functions". Page 128: only Thm. 13.11 and its proof.
On 27-5: on p. 129 only Lemma 13.12 and its proof. Skipped page 130. Chapter 14: only Thm. 14.1 and its proof. Skipped entire chapter 15. Started with Ch. 16 until p. 166, where we shall start on line 1 next time.
Finished proof of Theorem 16.6 and Remark 16.7. Rest of Chapter 16 was skipped. Skipped also Chapters 17, 18. Presented Chapter 19 only up to Radon Nikodym Theorem 19.2. After this the extensive handout distributed on 3-6 takes over (including exercises -- see below), with subjects not treated in the book like Hahn-Jordan decomposition, Lebesgue decomposition, etc. These subjects are exam material.

Exercises:

Week 6: 3.2, 3.4iii, 3.5, 3.9, 3.10 and 3.11
Week 7: 3.12*, 4.3, 4.5, 4.6, 4.10, 4.13
Week 8: 5.2, 5.3, 5.4 5.6, 5.8*, 5.10 and 6.5
Week 9: 6.3, 6.4i,ii,6.5(repeated),6.7,6.9,7.3,7.8; no starred exercises.
Week 10: 7.4,7.5,7.6,7.9 parts (v),(vi),(vii),(viii),7.10,7.11; no starred exercises.
Week 11: no course given.
Week 12: 8.3, 8.10, 8.12 and the following starred exercise.
Week 13: 9.2, 9.4, 9.5, 9.7, 9.8, 9.9.
Week 14: 9.10, 9.11, 9.12, 10.5(iii),(vi), 10.7, 10.8 and the following starred exercise.
Week 15: 10.9, 10.10, 10.11, 10.12, and 10.15.
Week 16: no course given.
Week 17: 11.2, 11.4, 11.6, 11.7, 11.11, 11.12.
Week 18: 11.13, 11.15, 12.1, 12.6, 12.8, 12.9, 12.10 and the following starred exercise.
Week 19: no course given.
Week 20: Some extra homework will be announced in weeks 20-22. You are supposed to work on this at home and can discuss it during the four hour exercise session on June 10. First batch: 12.12, 12.13, 12.14, 12.15, 12.17, 12.18, 12.20.
Week 21: 13.4, 13.5, 13.6, 13.7, 13.8, 13.9 and the following starred exercise.
Week 22: 14.1 (cf. also problem 9.5), 16.1, 16.3, 16.4, 16.8, 16.9, the following starred exercise> and problems RN1, RN2, SM1, SM2, SM3 and CE1, CE2 of the handout distributed on 3-6.
Weeks 23, 24: only exercise sessions (resp. whole afternoon and 15:15-17 h.).

Exams:

Quizz 1: Course material of weeks 6-8 (see above) plus associated exercises over the chapters 3-6: March 25.
Quizz 2: Course material of chapters 7, 8, 9, 10 and 11 plus associated exercises: May 20.
Final exam: on Thursday, 24-6. A complete solution is given here. If you happen to participate on 26-8 (resit): do NOT use this solution in a passive way.
Resit (=herkansing): on 26-8, 14:00-17:00 in BBL 000. This covers ALL the material treated in the course (including the handout distributed on 3-6). See the above weekly logbook for a detailed description.

Grades:

Each quizz counts for 20% towards the final grade, the final exam itself counts for 50% and the homework grade (see below) counts for 10%.

Homework:

Every two weeks (as an approximate rule) an exercise is assigned that can be turned in exactly one week later, at the beginning of the course, i.e., at 13:15. If so desired, they can also be sent by email, and composed in LaTeX (i.e., not scanned!) to the teaching assistant (Egbert Rijke: e.m.rijke@gmail.com), but only by that same time.
As a rule, such homework will be graded and returned to the student after a week, at the beginning of the practice session. If it is not picked up by then, the teaching assistant will keep the work for two more weeks, after which it will be discarded.
Turning in homework can be skipped at most once before 22 April and at most once after that date. In case of omissions because of illness, long absence, etc. please contact the teacher at once. As a rule, such omissions are only acceptable if they can be substantially motivated (doctor's letter, etc.).