Lecture notes: Probability with Measure

TL;DR

This lecture note provides an introduction to measure theory and Lebesgue integration, establishing a foundation for probability theory through measure-theoretic concepts and theorems.

Contribution

It systematically develops measure-theoretic probability concepts, including product measures, Kolmogorov's extension theorem, and key integration theorems, in a comprehensive lecture format.

Findings

Introduces measurable spaces, measures, and Lebesgue integration fundamentals.

Explains probability concepts like independence and conditioning in measure-theoretic terms.

Details product measures and Kolmogorov's extension theorem for constructing complex probability spaces.

Abstract

Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of Lebesgue integration are developed: convergence theorems, product spaces and Tonelli-Fubini, indefinite integration and absolute continuity, L-spaces and integral inequalities. Everything is set up so that in the second part the fundamental concepts of probability (such as those of random elements and their laws, independence, conditioning) can be cast swiftly in the measure-theoretic setting. Some emphasis is placed on monotone class and Dynkin's lemma type arguments. Products of arbitrary families of probabilities and Kolmogorov's extension theorem are treated.