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Thread: sigma-algebra generated by a r.v.

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

    Hallo,

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

    I.e. I want to show

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

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

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