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Thread: Showing that a specific outer measure is a measure

  1. #1
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    Showing that a specific outer measure is a measure

    Problems:

    a. Let X be an in infinite set and let mu be a fnite outer measure on the
    subsets of X such that every set {x}; x in X is mu-measurable and there
    exists a countable subset A of X with mu(X\A) = 0. Show that mu is a
    measure on the sigma-algebra of all subsets of X.

    b) Let mu, vu be finite outer measures on X with the property in a), that is,
    there exist countable sets A, B in X with mu(X\A) = vu(X\B) = 0.
    Find the Lebesgue decomposition of vu with respect to mu.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Showing that a specific outer measure is a measure

    Quote Originally Posted by mgarson View Post
    Problems:

    a. Let X be an in infinite set and let mu be a fnite outer measure on the
    subsets of X such that every set {x}; x in X is mu-measurable and there
    exists a countable subset A of X with mu(X\A) = 0. Show that mu is a
    measure on the sigma-algebra of all subsets of X.

    b) Let mu, vu be finite outer measures on X with the property in a), that is,
    there exist countable sets A, B in X with mu(X\A) = vu(X\B) = 0.
    Find the Lebesgue decomposition of vu with respect to mu.
    We need to see some work friend. The first problem really is not that hard (and I don't mean that in a condescending way), have you given it a try? Can you show us something? Maybe then we'll help with the second part.
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  3. #3
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    Re: Showing that a specific outer measure is a measure

    Sure! Here it goes (I didn't want to post any of my stuff before because I didn't want to confuse anybody with my attempt). Well, here it is,

    a. I realize that our $\displaystyle \mu:P(X)\rightarrow[0,\infty]$ has to satisfy the following properties:
    (i) $\displaystyle \mu(\emptyset)=0$,
    (ii) $\displaystyle \mu(\cup Ej)=\sum_{1}^{\infty}\mu(E_j)$ where $\displaystyle E_i\cap E_j=\emptyset $ if $\displaystyle i\neq j $.
    ----------------
    Now, (i) and $\displaystyle \mu(\cup Ej)\le\sum_{1}^{\infty}\mu(E_j)$ follow from the definition of outer measure. So all we have to show is:
    $\displaystyle \mu(\cup Ej)\ge\sum_{1}^{\infty}\mu(E_j)$.

    And then I tried to use the following but it didn't do me any good (I'm not sure if it is true):
    $\displaystyle \mu(X\setminus A_{\mu})=0 \Rightarrow X$ is countable $\displaystyle \Rightarrow X=\cup_{1}^{\infty}{x_i}$.

    And that's pretty much it...
    -------------------------------------------------------------------
    b. As for this one I really have no clue. I mean I know that I'm trying to find $\displaystyle \nu_{a}$ and $\displaystyle \nu_{s}$ such that: $\displaystyle \nu=\nu_{a}+\nu_{s}$ where $\displaystyle \nu_{a}<<\mu$ and $\displaystyle \nu_{s}\bot\mu$. Obviously $\displaystyle \nu(X\setminusA_{\mu})=\nu_{s}(X\setminus A_{\mu}))$ and $\displaystyle \nu_{s}(A_{\mu})=0$ since $\displaystyle \mu(X\setminus A_{\mu})$.
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