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Math Help - Measurable Set Question (2)

  1. #1
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    Measurable Set Question (2)

    Let E be a measurable subset of [a,b]. Let \{ I_k \} be a sequence of open intervals in [a,b] such that
    m(I_k \cap E) \geq \frac{2}{3}m(I_k), k=1,2,3,....
    Prove that m((\cup_{k=1}^{\infty} I_k) \cap E) \geq \frac{1}{3}m(\cup_{k=1}^{\infty} I_k).

    I've managed to prove below:
    Suppose that m((\cup_{k=1}^{\infty} I_k) \cap E) < \frac{1}{3}m(\cup_{k=1}^{\infty} I_k).
    Then m((\cup_{k=1}^{\infty} I_k) \cap E) < \frac{1}{3}m(\cup_{k=1}^{\infty} I_k)
    \leq \frac{1}{3} \sum_{k=1}^{\infty} m(I_k)
    \leq \frac{1}{3} \sum_{k=1}^{\infty} \frac{3}{2} m(I_k \cap E)
    = \frac{1}{2} \sum_{k=1}^{\infty} m(I_k \cap E)
    Hence m((\cup_{k=1}^{\infty} I_k) \cap E) < \frac{1}{2} \sum_{k=1}^{\infty} m(I_k \cap E).

    I'm stuck here. How could I continue from here? Or is there another approach?

    Thank you in advance.
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  2. #2
    Junior Member
    Joined
    Jan 2009
    Posts
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    Re: Measurable Set Question (2)

    Deleted, I went wrong there.
    Last edited by InvisibleMan; September 28th 2011 at 04:28 AM.
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