Probability Theory: Lecture 3

This lecture covers the basic properties of random variables, sub-sigma algebras, measurability, independence, and shift-invariant probability measures. It discusses the need to restrict the sigma algebra when defining a probability measure for models of infinitely many independent coin tosses. The lecture also explains the concept of cylinder sets and algebras, emphasizing the closure under finite unions and taking complements. It concludes with the extension of measures to sigma-algebras and the uniqueness of measures. The content is presented in a rigorous and systematic manner, building a solid foundation in probability theory.