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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 ε }
    -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. Thanks!
  2. jcsd
  3. Oct 20, 2011 #2


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    So you will need to show that:

    - A is in F
    - The complement of A cap E is in F
    - A countable union U_i A cap E_i is in F.

    All you need to know are the properties of sets and that epsilon is a sigma algebra. Try it and show what you have tried.
  4. Oct 20, 2011 #3
    Oh, I was under the impression I had to do it the other way around, such that F is in A, etc..maybe that's why I'm having such a hard time here.

    Ok so...

    A is in F because F=A intersect E, and since an intersect of A is in F, then all of A has to be in F (not sure how to write this properly)
    F=AnE is the same as F=(A^c U E^c)^c (^c=complement)
    not really sure how to go about the last one
  5. Oct 20, 2011 #4


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    F is a set of subsets of A, namely the set of A cap E for E in the sigma algebra. Check the definition of a sigma algebra. Try again keeping that in mind.
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