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Math Help - Sigma-Algebra

  1. #1
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    Sigma-Algebra

    Let  A,B,C be a partition of  \Omega . Write out the smallest  \sigma -algebra containing the sets  A,B, and C
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  2. #2
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    I have no idea if this is correct, but this is my best estimation of what the question is asking.

    A partition is such that
     A \cap B = \emptyset
     A \cap C = \emptyset
     B \cap C = \emptyset

    and  A \cup B \cup C = \Omega i.e. is exhaustive

    Then by definition of borel-algebra we have
    (1)  \Omega \ \in \ \mathcal{F}
    (2) As  A,B,C \ \in \mathcal{F}, A^{c}, B^{c}, C^{c} \in \mathcal{F}
    (3) As  A,B,C \in \mathcal{F}, A \cup B \cup C \in \mathcal{F}

    Is this right?
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  3. #3
    Moo
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    Hello,

    (you're asked for a sigma algebra, not a Borel algebra )

    So yup, the empty set and \Omega belong to \mathcal{F} (first+second axiom)

    A,B,C belong to \mathcal{F}

    Their complement belong to \mathcal{F}.
    But... you can have a more precise formula for these complements :

    A^c=B\cup C
    B^c=A\cup C
    C^c=A\cup B

    Do you agree ?
    We don't need to include A\cup B\cup C because it's \Omega...

    And thus the third axiom is automatically checked.

    So the smallest \sigma-algebra containing A,B,C (also called the generated \sigma-algebra by A,B,C) is :
    \mathcal{F}=\{\emptyset,A,B,C,A\cup B,A\cup C,B\cup C,\Omega\}
    Last edited by Moo; August 8th 2009 at 12:01 AM. Reason: vocabulary problems
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