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Math Help - Measurable Subset Question

  1. #1
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    Measurable Subset Question

    1)Let E be a measurable subset of \mathbb{R} with m(E)=1 and let (E_n) be a sequence of measurable subsets of E. If, for any \epsilon > 0, there exists a set E_i in the sequence (E_n) with m(E_i) > 1 - \epsilon, find the value of m(\cup E_n).

    My working is as below:
    Since m(E) = 1 & m(E_i) > 1 - \epsilon,
    m(E_i) > m(E) - \epsilon, so m(E \setminus E_i) < \epsilon.
    So set \epsilon = \frac{1}{n} such that m(E \setminus E_n) < \epsilon for n \geq i
    So m(E \setminus \cup E_n) = 0.
    Hence m(\cup E_n) = 1.

    Is this correct?

    2)Let E be a measurable set with m(E) = 1. Suppose that {E_n} is a sequence of measurable subsets of E such that m(E_n) = 1 for n=1,2,3,.... Find m(\cap E_n).

    I suppose the answer is 1?
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  2. #2
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    Re: Measurable Subset Question

    Quote Originally Posted by Markeur View Post
    2)Let E be a measurable set with m(E) = 1. Suppose that {E_n} is a sequence of measurable subsets of E such that m(E_n) = 1 for n=1,2,3,.... Find m(\cap E_n).

    I suppose the answer is 1?
    Yes, because each of the complementary sets E\setminus E_n is a null set, and the union of countably many null sets is null.
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