Set Theory/Systems of sets

In this chapter, we would like to study, for a given set , subsets of the power set . We consider in particular those subsets of that are closed under certain operations. π-system Let be a set. A -system is a collection of sets such that whenever , then also . Dynkin system Let be a set. A Dynkin system or -system is a collection of sets such that the following three axioms hold: . σ-algebra Let be a set. A -algebra on is a collection of subsets of , say , such that the following axioms are satisfied: for all implies . Note that being a -algebra is a stronger requirement than being a Dynkin system: A -algebra is closed under all countable intersections, whereas a Dynkin system is only closed under intersections of countable ascending chains. σ-algebra generated by a collection of sets Let be a set, and let . Then we define . λ-system generated by a collection of sets Let be a set, and let . Then we define . Dynkin's λ-π theorem Let be a set, and let be a -system on . Then . The direction "" is clear, so that we only have to prove "". To do so, we prove that is in fact a -algebra that contains , using the definition of as the intersection of all -algebrae that contain . Exercises Let be a set, and let . Prove that is a -system if and only if . Let be a set, and let . Prove that is a -algebra if and only if for all implies .