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Thread: Measurability of of Random Variables w.r.t. Sigma Fields

  1. #1
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    Measurability of of Random Variables w.r.t. Sigma Fields

    Hello everyone,

    I'm having a little trouble with a probability problem with three parts; I think I'm having trouble wrapping my head around just what's going on here. If anyone could give me a starting point, I'd appreciate it.

    Here's the problem (Billingsley 5.1) (X a random variable)

    a. Show that X is measurable w.r.t. the sigma field J iff sigma(X) is a subset of J. Show that X is a measurable w.r.t. sigma(Y) iff sigma(x) is a subset of sigma(Y)

    b. Show that if J = {empty set, omega}, then X is measurable w.r.t. J iff X is constant.

    c. Suppose that P(A) is 0 or 1 for every A in J. This holds, for example, if J is the tail field of an independent sequence, or if J consists of the countable and cocountable sets on the unit interval with Lebesgue measure. Show that if X is measurable w.r.t. J, then P[X=c] = 1 for some constant c.

    Thanks for any and all help!

    Best regards
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  2. #2
    Super Member girdav's Avatar
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    Re: Measurability of of Random Variables w.r.t. Sigma Fields

    For the first question, use the fact that $\displaystyle \sigma (X)$ is a $\displaystyle \sigma$-algebra which consists of the elements of the form $\displaystyle X^{-1}(B)$ where $\displaystyle B$ is in the Borel $\displaystyle \sigma$-algebra.
    For the second question, if $\displaystyle X$ is measurable w.r.t. $\displaystyle J$ then for each $\displaystyle r\in\mathbb R$ we have $\displaystyle X^{-1}(\{r\})=\emptyset$ or $\displaystyle X$. There is only one $\displaystyle r$ such that $\displaystyle X^{-1}(\{r\})=X$.
    For the third question, what about $\displaystyle \inf\{t\in\mathbb R: P(X\leq t)=1\}$?
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  3. #3
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    Re: Measurability of of Random Variables w.r.t. Sigma Fields

    Thank you so much! You were a big help.
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