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Math Help - proving measurability

  1. #1
    Senior Member Dinkydoe's Avatar
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    proving measurability

    Hoi, I want to show that \psi(x)=\mu(\overline{B}_x(r)) is measurable. Here \overline{B}_x(r) is the closed ball with center x and radius r in some banach space X=(S,d), and \mu is a "finite" borel-measure.

    In the exercise we had to prove \lim_{n\to\infty}f_{n,x} = \mathbb{I}_{\overline{B}_x(r)} for some function f_{n,x}, point-wise convergence (I shall not define f here since I believe only its properties are relevant). and also for any sequence x_k\to x we have f_{x_k,n}\to f_{x,n}
    ( \mathbb{I} is indicated as the indicator-function ;p)

    I'm asked to prove the measurability using these results (not even sure if all the results are necessary). I'm somewhat stuck here. The only thing I can think of is writing the following:

     \psi(x) = \mu(\overline{B}_x(r)) = \int_X \mathbb{I}_{\overline{B}_x(r)}d\mu = \int_X \lim_{n\to\infty}f_{n,x} d\mu

    and using the fact we may interchange limit and integral. I don't quite see yet how I might prove measurability...

    Am i missing some heavy machinery here? Also, does the measurability of \psi depend on the measurability of f_{n,k} (i'm thinking of dominated convergence)? Do we need to know first whether f_{n,x} is measurable?
    Last edited by Dinkydoe; April 4th 2012 at 05:21 AM.
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  2. #2
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    Re: proving measurability

    You want to show that no matter what element A in \mathcal{B}([0,\infty)) you choose, the object \psi^{-1}(A) will always be an element in \mathcal{B}(S)\ , where \mathcal{B}(M) denotes the Borel sigma-algebra on the metric space M.

    To do this you might need the facts that \mathcal{B}([0,\infty)) is generated by half-open intervals and that \mathcal{B}(S) is generated by the collection of all open balls in S.
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