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Homework Help: Probability help/sigma-algebras

  1. Oct 20, 2011 #1
    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
     
  2. jcsd
  3. Oct 21, 2011 #2
    I assume that F is supposed to be

    [itex]F:=\left\{A\cap E: E\in\epsilon\right\}[/itex].​

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


    1. [itex]\emptyset,A\in F[/itex]

    • If [itex]B\in F[/itex], then it is necessary that [itex]B^c\in F[/itex]

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

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