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  1. #1
    Member mabruka's Avatar
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    Finite measure property ?

    Hi!
    Can any one make this a little bit more clear please?


    Let $\displaystyle ( E, p ) $ be a metric space, and $\displaystyle P:\mathcal B(E)\longrightarrow \mathbb{R}_+$ a probability measure on borel sets of E.

    I do not understand the next implication: $\displaystyle x \in E, \epsilon >0$


    Since $\displaystyle \delta (B_\epsilon (x)) \subset \{y \in E : p(x,y)=\epsilon \} $
    then

    $\displaystyle P( B_\epsilon (x))= 0$ for (at most) a numerable number of $\displaystyle \epsilon$


    $\displaystyle \delta (B_\epsilon (x))$ is the topological boundary

    I think i may be missing a "core" property of finite measures but i dont remember!

    Thanx in advance

    EDIT: Bad title i can't fix this sorry
    Last edited by mabruka; Feb 2nd 2010 at 05:37 PM. Reason: Bad title
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  2. #2
    Member mabruka's Avatar
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    Turns out the conclusion is wrong! No wonder why i couldnt figure it out!



    it should say

    $\displaystyle P(B_\epsilon (x))= 0 $ EXCEPT for at most a numerable number of $\displaystyle \epsilon$ .
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