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.

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.

Exams:

• Quizz 1: Course material of weeks 6-8 (see above) plus associated exercises over the chapters 3-6: March 25, 15:15-16:45 in BBL 001.
• Quizz 2: May 6, 15:15-16:45 in BBL 001; subject matter to be announced.
• Final exam: date to be announced.

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.).