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Thread: Sigma-Algebra

  1. #1
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    Sigma-Algebra

    Let $\displaystyle A,B,C $ be a partition of $\displaystyle \Omega $. Write out the smallest $\displaystyle \sigma $-algebra containing the sets $\displaystyle A,B,$ and $\displaystyle 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
    $\displaystyle A \cap B = \emptyset $
    $\displaystyle A \cap C = \emptyset $
    $\displaystyle B \cap C = \emptyset $

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

    Then by definition of borel-algebra we have
    (1) $\displaystyle \Omega \ \in \ \mathcal{F} $
    (2) As $\displaystyle A,B,C \ \in \mathcal{F}, A^{c}, B^{c}, C^{c} \in \mathcal{F} $
    (3) As $\displaystyle 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 $\displaystyle \Omega$ belong to $\displaystyle \mathcal{F}$ (first+second axiom)

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

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

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

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

    And thus the third axiom is automatically checked.

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