Measure Theory and Probability 1
- Introducing Measure space and Probability space and its componrnts, event spacre, sigma fields and probability set function. Also the property of measure or probability set function as finite additivity, sigma additivity are tto be discussed. Property of continuity from below and from above are to shown for the probability measure of monotone sequence of events.
- Introducing mathematically the concepts of limit sets including lim inf and lim sup, almost always and infinitely often events, concepts of convergence in probability and convergence with probability one. Mathematical discription of Fatos Lemma and Borel Canteli Lemmas
- Introducing Pi-system, Lambda-system and theorems of existense and uniqueness of extension of measure or probability from fields to the generated sigma-fields.Measurable functions, random variables,
independent, density function of one measure with respect to another measure. Convergence theorems regarding sequence of random variables as Fatos lemma and the DCT, MCT and BCT theorems
- Mathematical description of product space and corresponding components as product sample space, product sigma field and product measure function. Also the measurable rectangle and which the measure defined on it has a unique extension to the generated siga field. The relation between the joint distribution of independent random variables and the product measure are to be discribed. Finally the Fubini theorem and its conditions are to be discussed.