Given a set Ω with a σ-field F defined on it. Let B be a...
Given a set with a -field F defined on it. Let B be a subset of and define
. Call this .
The claim is that this collection of sets is a -field and hence (B, ) is a measurable space.
I don't see why is a -field just because it is a subset of one. Any insight appreciated
Answer & Explanation
Suggested proof. (Axioms are -algebra axioms.) Notation different from question.
Let () be
To prove the claim it is a -algebra on we need to show:
(ii) in .
(iii) satisfies -additivity.
Proof of (i): We know that is in by Axiom (1).
Therefore, by (), .
But , hence .
Proof of (ii): (de Morgan).
Now and are in by Axiom (2) and their union is in by Axiom (3).
Hence, by () .
But , and which is in by (i).
Proof of (iii): Let .
We want to show that
by Axiom (3).