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Math Help - Measurability

  1. #1
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    Measurability

    Let n,q \in \mathbb{N} such that q \leq n and let E_1,E_2,...,E_n be measurable subsets of [0,1]. Suppose that for each point x \in [0,1], there are at least q sets in \{E_1,E_2,...,E_n\} that contain x. Prove that there exists 1 \leq i \leq n such that m(E_i) \geq \frac{q}{n}.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Measurability

    Quote Originally Posted by Markeur View Post
    Let n,q \in \mathbb{N} such that q \leq n and let E_1,E_2,...,E_n be measurable subsets of [0,1]. Suppose that for each point x \in [0,1], there are at least q sets in \{E_1,E_2,...,E_n\} that contain x. Prove that there exists 1 \leq i \leq n such that m(E_i) \geq \frac{q}{n}.
    Here's the basic idea. Suppose that m(E_i)<\frac{q}{n} then \sum_{i=1}^{n}m(E_i)<q, but why is that stupid (hint: use the idea that rougly the E_i's ' q-fold cover' [0,1]).
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  3. #3
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    Re: Measurability

    Okay solved it. Thanks!
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