Stochastic Experiments and Probability Spaces
Uffe Høgsbro Thygesen · 2023
This chapter is a an introduction to measure-theoretic probability. The study of stochastic differential equations require a firm basis of probability theory, and this theory relies in turn on measure theory. In this chapter, we discuss stochastic experiments, systems of events organized in a sigma-algebra, probability measures, and random variables. We show how expectation can be viewed as integration, how information can be modeled using systems of events. We define conditional expectations and show how random variables can be organized in linear spaces, which allows us, for example, to view estimation as an orthogonal projection.