Probability help/sigma-algebras

[FTaylor244]

Let (S, ε, P) be a probability space and let A be an element of ε with P(A)>0. Let F={AπE :E is an element of }
-Prove that F is a sigma-algebra on A.


Not sure even where to go with this really. I know that to be a sigma-algebra has to be closed under complementation and countable unions. I'm not very good with proofs, and just a push in the right direction would help me out a ton

 

[canis89]

I assume that F is supposed to be

$F:=\left\{A\cap E: E\in\epsilon\right\}$.​

Now, for F to be a sigma-algebra on A, you have to first show that $F\subseteq 2^A$. Then, show that the following 3 conditions are satisfied by F.

1. 
   $\emptyset,A\in F$

• 
  If $B\in F$, then it is necessary that $B^c\in F$

• 
  For every sequence of sets $(B_n)$, where everyone of them is a member of F, it is necessary that $\bigcup_{n\in\mathbb{N}}B_n\in F$.

Try to prove the conditions one-by-one. Also use the fact that $\epsilon$ is a sigma-algebra. If you have some more questions, feel free to ask.