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Math Help - Showing a function is measurable

  1. #1
    Senior Member Dinkydoe's Avatar
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    Showing a function is measurable

    Let X=(S,d) a banach space. I want to show the following: For any borel measure \mu, the map \psi: X\to \mathbb{R} given by x\mapsto \mu(\overline{B}_x(r)) is measurable.

    I believe this can be done in a number of ways, one of which is proving that the set U_c:=\left\{x:\psi(x)\leq c\right\} for any c is contained in \mathcal{B}(X) (borel sigma-algebra of X)

    I think proving this, can not be done without using the fact that d is induced by some norm \left\|\cdot \right\|. If we take for example X= (\mathbb{R}, d_E) with Euclidian metric, ( \mu lebesgue measure) it is obvious U_c is contained in \mathcal{B}(X) as U_c = X or U_c = \emptyset for any c. But for any random metric d, this is not obvious...

    A norm-induced metric has (as I recall) 2 extra properties. 1. translation invariance d(x+a,y+a)=d(x,y). and 2. d(\alpha x,\alpha y) = |\alpha|d(x,y)

    I hope what I think is correct here...so suppose \psi(x)= \alpha for some \alpha >0 then for any y\in X we have \psi(y)=\alpha as well...

    It's just a hunch, i have no clue actually...as I don't know what properties \mu might have. (except for the standard properties)

    Any idea what I'm missing here?

    Thank you

    (good to see u guys back online though...btw: can it be implemented that $ x^2 $ is valid latex-code here, instead of wraps..it's sometimes
    quite bothersome to write tags everywhere :P)
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  2. #2
    Super Member girdav's Avatar
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    Re: Showing a function is measurable

    Can yo use Fubini's theorem?
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