Measure Theory for Probabilists (WS 2026/27)
Prof. Dr. Peter Pfaffelhuber
Assistants: Félix Barrez Tambe Ndonfack , Carola Heinzel
This is a hybrid course which is given at the University of Freiburg. Lectures will be given with videos (see below). Tutorials take place in person, Tuesdays, 14:15–15:45, in seminar room 232 (Ernst-Zermelo-Str. 1), starting 20th of October, 2026. There is no tutorial on 22nd of December, 2026.
Requirements for the Studienleistung: Every exercise is worth a number of points, which is stated on the sheet. You collect these points by ticking exercises: on every sheet, you tick those exercises which you are able to present in the tutorial, and every exercise you tick gives you its points.
Handing in your solutions is voluntary – do so if you would like feedback from Félix. In this case, you may set your ticks after you have received his feedback, so that you know whether your solution is correct.
In the tutorial, Félix picks at random who presents a ticked exercise, preferably among those who have not presented yet. If you cannot properly explain a solution you have ticked, you lose all points you collected in that session. If this happens three times, the Studienleistung is not awarded.
In order to obtain the Studienleistung, you need at least 50% of the points that can be achieved for the exercises.
Requirements for the Prüfungsleistung: You must pass the oral exam.
Content: The course covers the material which is usually needed in order to dive into measure theoretic aspects of probability theory. Usually, in basic lectures on probability (e.g. taught in Freiburg as Stochastik 1 in the 3rd semester), no measure theory is needed. Rather, the lecture in Analysis 3 covers measure theory. However, in case you want to specialize in probability theory and did not attend Analysis 3, this course is for you.
Prerequisites: You need to know fundamentals of mathematics, such as sets, functions etc. The course requires some knowledge in topology, but tries to recall all explanations in this field.
Manuscript: The current version is here.
Videos: Below you find links to all Videos, or you look at the playlist . A collection of all slides is here.
Exercise sheets: The exercise sheets will be uploaded to Ilias .
The exercise sheets and other course materials will be uploaded to Ilias .
The password will be announced in the first tutorial. Students who are registered for the course will receive the password by email.
1. Introduction
Some introductory remarks
2. Semi-rings and σ-fields
Covers Section 1.1
3. Generators and extensions
Covers Section 1.2, and includes some repetition in topology (Appendix A)
4. Dynkin systems and compact systems
Covers Sections 1.3 and 1.4
5. Set functions and outer measures
Covers Section 2.1
6. σ-additivity
Covers Section 1.2, and includes some repetition in topology (Appendix A)
7. Uniqueness and extension of set functions
Covers Section 2.3
8. Measures on ℝ and image measures
Covers Sections 2.4 and 2.5
11. Convergence results
Covers Section 3.4
12. Basics of L^p-spaces
Covers Sections 4.1, 4.2
13. The space L^2
Covers Section 4.3
14. Theorem of Radon-Nikodým
Covers Section 4.4
15. Set systems on product spaces
Covers Sections 5.1, 5.2
16. Measures on product spaces
Covers Section 5.3, 5.4