Set Theory/Systems of sets

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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:

  1. .

Definition (σ-algebra):

Let be a set. A -algebra on is a collection of subsets of , say , such that the following axioms are satisfied:

  1. 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]

  1. Let be a set, and let . Prove that is a -system if and only if
    1. .
  2. Let be a set, and let . Prove that is a -algebra if and only if
    1. for all implies .