### Measure Theory

Wednesday 11:30-13:00, HS02 (OMP 1)
Thursday 11:30-13:00, HS02 (OMP 1)

Proseminar (optional): Monday 9:45-11:15, HS02 (OMP1), instructor Henna Koivusalo

Summary: The Lebesgue measure and integration theory is one of the pillars of modern analysis and the cornerstone of the probability theory. The lecture presents its basic concepts and results (in particular the existence, uniqueness and important examples of measures, Lebesgue integral, convergence theorems, spaces of integrable functions, product measures) and connects these results with other areas of mathematics.

Exam: The exam for the lecture is oral, lasting roughly 30min. You need to know the material covered during the lecture. For longer proofs proof idea/major proof steps are only required. The questions asked during the exam are taken from the following list.

Problems for proseminar: Oct 10 (solution for exercise 6), Oct 17, Oct 24, Nov 07, Nov 14, Nov 21, Nov 28, Dec 5, Dec 12, Jan 9, Jan 16, Jan 23, Jan 30,

Literature:

• P. Billingsley: Probability and measure. (Wiley, 1986)
• Cohn: Measure Theory. (Birkhäuser)
• Evans-Gariepy: Measure theory and fine properties of functions. (CRC Press)
• Rudin: Real and complex analysis. (McGraw-Hill)
• Tao: An introduction to measure theory. (AMS)

Lecture notes: handwritten lecture notes from WS2013 with some corrections and suplements: correction for page 15, correction for page 21, suplement for pages 23-24

### Selected topics in probability: Extremes and Gaussian fields

Tuesday 11:45-13:15, Seminarraum 12 (OMP 1)

Lecture notes: (handwritten)
Oct 4, Oct 11, Oct 18, Oct 25, Nov 8, Nov 15, Nov 22, Nov 29, Dec 6, Dec 13, Jan 10, Jan 24.

Literature: (will be completed later)

• Leadbetter, Lindgren, Rootzen: Extremes and related properties of random sequences and processes.
• Resnick: Extreme values, regular variation and point processes (Springer 1987).
• Adler, Taylor: Random fields and geometry (Springer 2007).
• Zeitouni: Lecture notes on Gaussian processes

### Seminar in Probability Theory (together with Prof. Walter Schachermayer)

Tuesday 9:45-11:15, Seminarraum 08 (OMP 1)
First session: 4 Oct 2016

This semester, the general direction of the seminar is financial mathematics. The planned topics is 'Stochastic Portfolio Theory'.
Detailed list of presentations will be given on October 4

Literature: R. Fernholz, I. Karatzas: Stochastic portfolio theory: an overview [PDF]

Programm:

Oct 11
No seminar
Oct 18
Steppacher, Maxymowicz
Oct 25
Hornakova, Kumhera
Nov 1
No seminar
Nov 8
Presentation of master theses
Nov 15
Merz, Jakic
Nov 22
Rössler, ???