Foundations of Probability and Measure Theory
The course is designed for third-semester students at MIPT and consists of two parts. The first part is a classical introduction to probability theory (probability space, random variables, independence, expectation and variance). The second part is a rigorous construction of Lebesgue measure and the Lebesgue integral, on which all further, βgenuineβ probability theory rests: sigma-algebras, CarathΓ©odoryβs extension theorem, measurable functions and their modes of convergence, the Lebesgue integral and its properties, products of measures, signed measures, and the Radon-Nikodym theorem.
For the full program broken down by session, see the program page.
Prerequisites
The course does not assume any prior acquaintance with probability theory or measure theory: mathematical analysis from the first two semesters (limits, series, integrals) is enough.
Assessment format
Assessment follows the current rules of the semester (tests, colloquium, exam); see the departmentβs current schedule.
Contents
I. Foundations of Probability
II. Measure Theory
- Systems of sets. The Borel sigma-algebra
- Set functions on semirings and sigma-algebras. Measure
- Outer measure, CarathΓ©odoryβs extension theorem
- Lebesgue measure, Lebesgue-Stieltjes measure
- Measurable functions and their modes of convergence
- The Lebesgue integral: integration with respect to a measure
- Products of measures
- Lebesgue spaces \(L^p(\mu)\)
- Signed measures, densities
- Differentiation of the Lebesgue integral. Absolutely continuous functions
About this text
This text was prepared based on notes from seminar sessions.
Author: M.A. Nedoshivin
Email: nedoshivin.ma@phystech.edu
Found a typo or an error? Email the address above.
The materials are distributed under the CC BY-NC-SA 4.0 license; see the βAbout the projectβ page for details.