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.
Definition (π-system):
Let be a set. A -system is a collection of sets such that whenever , then also .
Definition (Dynkin system):
Let be a set. A Dynkin system or -system is a collection of sets such that the following three axioms hold:
- .
Definition (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.
Definition (σ-algebra generated by a collection of sets):
Let be a set, and let . Then we define
- .
Definition (λ-system generated by a collection of sets):
Let be a set, and let . Then we define
- .
Theorem (Dynkin's λ-π theorem):
Let be a set, and let be a -system on . Then
- .
Proof: 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
[edit | edit source]- 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 .