Sub-sigma-fields and Random Variables

This lecture introduces the concept of sub-sigma fields, explaining how different levels of information can be represented by sigma fields. It delves into the definition of random variables, emphasizing their role as functions associating outcomes of an experiment with real numbers. The lecture explores the measurability of random variables, highlighting the importance of being F-measurable. It also discusses Borel measurable functions, continuous functions, and the composition of F-measurable random variables with Borel measurable functions. Examples are provided to illustrate the concepts, emphasizing the relationship between sigma fields, random variables, and measurable functions.