### 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, ???