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Math Help - Probability measure on Borel-sigma

  1. #1
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    Probability measure on Borel-sigma

    Let  (\Omega , \mathcal{F} , \mathbb{P} ) be a probabilty space and  X: \ \Omega \rightarrow \ \bar{\mathbb{R}} be a random variable with distribution  \mu . Show that  \mu is a probabilty measure on  \bar{\mathcal{B}} . That is, verify the three axioms of probabilty for  \mu , using the collection of "events" on  \bar{\mathcal{B}} .
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  2. #2
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    Hello,

    But how is \mathcal{\overline B} defined ?
    Is it the Borel algebra over \mathbb{\overline R} ?
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  3. #3
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    Hi Moo,
    Yes,  \mathcal{\overline B} is the borel algebra over  \mathbb{\overline R}
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    I've become stuck with the question. My teacher said that I need to use the fact that  \mathbb{P} is a probability measure and that  X is a random variable. I've racked my brain but can't work it out Whats more annoying is that the answer is staring me right in the face.
    I know that the 3 axioms of probability are

    (1)  0 \leq \mathbb{P} \leq 1 for some event A

    (2)  \mathbb{P} (\Sigma) = 1

    (3) For any sequence  A_1, A_2, ... which are disjoint
     \mathbb{P} ( \bigcup_i A_i ) = \sum_i \mathbb{P} (A_i).

    I just can't get my head around this and any help would be greatly appreciated.
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  5. #5
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    I'm just learning about probability and measure theory myself, and I am not a mathematician. So here's my best shot at it:

    We have that X: (\Omega,\mathcal{F}) \rightarrow (\mathbb{\bar{R}},\mathcal{\bar{B}}), and I suppose that makes X an extended random variable. Now, X induces a probability measure on \bar{\mathcal{B}}, namely \mathbb{P}_X(B) = \mathbb{P}\{\omega: X(\omega) \in B\},~~B \in \bar{\mathcal{B}}. The distribution function of X is then \mu(x) = \mathbb{P}\{\omega: X(\omega) \leq x\}. Now, because for  a < b, \mu(b) - \mu(a) = \mathbb{P}\{\omega: a < X(\omega) \leq b\} = \mathbb{P}_X(a,b], the distribution \mu corresponds to the measure \mathbb{P}_X. We should have \mu(x) = 1 as x \rightarrow \infty and \mu = 0 as x \rightarrow -\infty. Is that of any help?

    By the way, which book do you use to study measure theory?
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  6. #6
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    If X is a random variable then you can think of X^{-1}:\mathcal{\overline{B}}\rightarrow \mathcal{F}. Now  \mu=\mathbb{P}\circ X^{-1}.

    Quote Originally Posted by funnyinga View Post
    (1)  0 \leq \mathbb{P} \leq 1 for some event A
    This is immediately obvious.
    (2)  \mathbb{P} (\Sigma) = 1
    We have that X^{-1}(\overline{\mathbb{R}})=\Omega, and so this is obvious as well.
    (3) For any sequence  A_1, A_2, ... which are disjoint
     \mathbb{P} ( \bigcup_i A_i ) = \sum_i \mathbb{P} (A_i).
    Take A_1,A_2,... \in \overline{\mathcal{B}} disjoint, i.e.  \bigcap_{i \in \mathbb{N}}A_i=\emptyset. This implies that X^{-1}\left(\bigcap_{i \in \mathbb{N}}A_i\right)=\bigcap_{i \in \mathbb{N}}X^{-1}(A_i)=\emptyset. As for each i, we have X^{-1}(A_i) \in \mathcal{F} the result follows.
    Last edited by Focus; August 12th 2009 at 10:53 AM. Reason: Nonsensical assumption removed
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