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Math Help - sigma-algebra generated by a r.v.

  1. #1
    Junior Member
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    sigma-algebra generated by a r.v.

    Hallo,

    How can I show that the \sigma-algebra generated by a random variable X and the \sigma-algebra generated by X and f(X) ( f is an appropriate measurble function) are equal?

    I.e. I want to show

    \sigma(X)=\sigma(X,f(X)).

    Obviously, it holds that
    \sigma(f(X))\subseteq \sigma(X) \subseteq \sigma(X,f(X)).

    Can anybody help me?
    Thanks in advance.
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  2. #2
    Super Member girdav's Avatar
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    Re: sigma-algebra generated by a r.v.

    Show that \sigma(X) contains all the elements of the form X^{-1}(B) and (f\circ X)^{-1}(B), where B is measurable.
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