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Math Help - Sequence of leesgue measurable sets

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    Sequence of leesgue measurable sets

    Let \{E_n\} be a sequence of lebesgue measurable subsets of R such that

     \sum\limits_{n=1}^{\infty} m(E_n) < \infty . If E= \lim_{n} \sup E_{n} , then show that m(E)=0
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    Hello,

    Look at the Borel-Cantelli theorem for measure theory.
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    Quote Originally Posted by Chandru1 View Post
    Let \{E_n\} be a sequence of lebesgue measurable subsets of R such that

     \sum\limits_{n=1}^{\infty} m(E_n) < \infty . If E= \lim_{n} \sup E_{n} , then show that m(E)=0
    Since  \sum\limits_{n=1}^{\infty} m(E_n) < \infty , then for all  \epsilon > 0, there exists an N such that if k > N, then  \sum\limits_{n=k}^{\infty} m(E_n) < \epsilon . Thus, it must be the case that for each k > N we have  m(E_k) < \epsilon . And finally, we have that sup{ m(E_k)} <  \epsilon for all k > N.
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