- Discuss the measure- and Lebesgue integration theory that is relevant in probability theory.
- Introduce some vital concepts in probability theory, such as conditional expectations, the Radon-Nikodym theorem, martingale convergence theorems, characteristic functions and why they are characteristic, the Brownian motion.
- Provide rigorous proofs for two central convergence theorems in probability: the Strong Law of Large Numbers and the Central Limit Theorem.

**Prerequisites**

None (but it does help to have some basic understanding of (functional) analysis; e.g. convergence, (uniform) continuity, metric spaces, Banach/Hilbert spaces). Also, although measure- and Lebesgue integration theory is built up "from scratch", the course is quite tough for those who have never seen measure theory before. In that case it may help to consult René Schilling's *Measures, Integrals and Martingales*, or Jeffrey Rosenthal's *A First Look at Rigorous Probability Theory* for some extra material on this topic.** **

**Rules about Homework / Exam**

There are weekly homework assignments by which one may raise (but not lower) the final grade. More precisely, if the grade of the exam is 5 or higher, then the homework grade counts for 1/3 of your final grade, provided this results in a higher grade (otherwise the homework grade is ignored).

Contrary to previous years, this year no bonus can be earned with a presentation.

Homework may be handed in at the beginning of the lecture. Provided it is typed out in LaTeX, it may also be sent by e-mail to the responsible teaching assistant. Kirsten Wang is responsible for the homework of week 1 through 7, Madelon de Kemp is responsible for the homework of week 8 through 13.

- Docent: Sonja Cox