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Thread: proving measurability

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

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

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

    $\displaystyle \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 $\displaystyle \psi$ depend on the measurability of $\displaystyle f_{n,k}$ (i'm thinking of dominated convergence)? Do we need to know first whether $\displaystyle f_{n,x}$ is measurable?
    Last edited by Dinkydoe; Apr 4th 2012 at 04:21 AM.
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  2. #2
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    Re: proving measurability

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

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