Measure, Integration and Probability Theory

When/Where

From 2016.09.14 to 2016.12.30

Wednesday: 8:50-11:50
Place: Chen Ruiqiu Building, room 304

Office hour: Wednesday from 13:00 to 15:30
Office: Math building, room 2.105

Course Description

This lecture is about measure, integration and probability theory. Calculus is a requirement, and it is rather targeted at graduate students as well as PhD students. Though, senior undergraduate students are welcome to attend.

Practical organization

Lecture, homework sheet, and exam are in English.
Every week a lecture note following what is taught in class is provided on this webpage, section Lecture Material
Every week, a homework sheet has to be solved (by group of 2 or 3 students) and will be graded. The homework sheets are available on this webpage, section Lecture Material. The homework sheet should be handed electronically to the teaching assistant on Wednesday before the next lecture.
Final exam in January
Final grade is a pondered average between homework sheet and final exam

Content of the lecture

We will handle the following topics. Due to time constraints, the points in italic may be left apart but will be provided in the lecture notes.

Theory of sets and topology, measurable functions, continuous functions. (2 weeks)
- Set theory (properties of the Boolean algebra of sets)
- semi-ring, ring, algebra, sigma-algebra Dynkin theorem
- Topology, convergence, metric and normed spaces.
- Measurable and Continuous functions.
- Generated systems of sets: initial sigma-algebra/topology, product algebra/topology.
Set Functions, measures, probability measures, construction of measures. Product measure, stochastic kernel (2 weeks)
- Content, premeasure, measure, and property thereof.
- Generated probability measure
- Caratheordory theorem, Kolmogorov extension of measure
- Existence and properties of product measures and stochastic kernels.
Definition and properties of the integral and $L^p$ spaces, Fubini-Tonelli, Radon-Nykodym (2 weeks)
- Definition of integral, $L^p$ spaces, identification in the almost sure sense.
- Monotone convergence, Fatou’s lemma, dominated convergence.
- Fubini-Tonelli.
- Jensen, Hölder and Minkowsky inequalities.
- Radon-Nykodym theorem
Convergence in measure and in $L^p$. Uniform integrability. (2 weeks)
- Different notions of convergence
- Uniform integrability
- De la Vallee-Poussin theorem
- Convergence in $L^1$. eventually Dunford Schwartz weak compactness characterization of UI sets
Probability distribution, Independence, Borel-Cantelli, weak convergence, laws of large numbers (2 weeks)
- Probability distribution and independence, Borel Cantelli
- Strong law of large numbers
- Weak convergence, portemanteau theorem
- Characteristic function
- Weak law of large numbers
Conditional expectation. (2 weeks)
- Existence, properties of Conditional Expectation
- Regular conditional probability measure
- Conditional distributions
- Independence
- Disintegration theorem
Discrete time martingales, stopping times, Doob’s theorems, convergence of martingales. (3 weeks)
- Stochastic processes, filtration, stopping time
- Martingale
- Doob’s optional sampling theorem
- Doob’s upcrossing’s lemma, Doob’s inequalities
- Convergence of martingales in the almost sure sense, in the $L^p$ sense.
Markov Processes (1 week)
- Definition of Markov chain, time homogeneity
- Construction of Markov Probability measures
- Markov property, Chapman-Kolmogorov theorem
- Recurrence and transience
- Stationarity, irreducibility

Literature

The course won’t follow strictly speaking any textbook, however you might find some interesting/alternative approaches in

For measure and integration theory:

[1] “Introduction to Measure and Integration”, S.J. Taylor, Cambridge university press 1973

[2] “Foundation of Modern Probability”, O. Kallenberg, 2nd Edition 2002 (very general but complete)