Independence in Probability Theory

This lecture covers the concept of independence in probability theory, starting with the independence of two events and moving on to independence of random variables. The instructor explains how to check for independence using joint probabilities and sigma-fields, emphasizing the difference between proving independence and dependence. The lecture also delves into independence of sub-sigma fields, highlighting the relationship between information and independence. Specific examples with discrete and continuous random variables are provided to illustrate the concept, along with practical criteria for checking independence. The importance of understanding independence in various contexts, such as measure theory and probability spaces, is emphasized.