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Math Help - measure theory question

  1. #1
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    measure theory question

    prove that an arbitrary intersection of σ-algebras is a σ-algebra, i.e. let {A_α} where α is in I (indexing set) be a collection of σ-algebras on a given set Ω. Prove that A=intersection {A_α}(where α is in I)

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  2. #2
    Lord of certain Rings
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    Courtesy Wiki:
    A subset Σ of the power set of a set X is a σ-algebra if and only if it has the following properties:
    1. Σ is nonempty
    2. If E is in Σ then so is the complement (X \ E) of E.
    3. The union of countably many sets in Σ is also in Σ.
    I dont know topology but I think proving 2 and 3 is easy. Since they follow directly from set properties...
    Which axiom are you having trouble with?
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  3. #3
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    i think the first condition is pretty trivial, but i'm having trouble with 2nd and 3rd ones. maybe i'm not grasping the definition of sigma algebra that well..
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  4. #4
    Lord of certain Rings
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    For 2) \forall \alpha \in I, E \in A_{\alpha}\Rightarrow \Omega - E \in A_{\alpha} since A_{\alpha} is a sigma-algebra.

    And how is (1) trivial? It is easily possible that intersection of a few sets can easily be empty
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  5. #5
    Junior Member frenzy's Avatar
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    Quote Originally Posted by Isomorphism View Post
    For 2) \forall \alpha \in I, E \in A_{\alpha}\Rightarrow \Omega - E \in A_{\alpha} since A_{\alpha} is a sigma-algebra.

    And how is (1) trivial? It is easily possible that intersection of a few sets can easily be empty
    1 is trivial...


    for any \alpha \in I we know that A_{\alpha} is nonempty since it is a sigma algebra. Thus E \in A_{\alpha}

    and therefore E^c \in A_{\alpha}



    also \Omega=E\cup E^c \in A_{\alpha}

    thus \Omega\in A_{\alpha} for all \alpha \in I

    Hence

    \Omega\in \displaystyle\bigcap_{\alpha \in I}{A_{\alpha}}
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