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Measure space, null set

  1. May 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,\mathcal{A},\mu)[/tex] be a fixed measure space.

    Let [tex]A_k \in \mathcal{A}[/tex] such that [tex]\displaystyle
    \sum^\infty_{k=1} \mu(A_k) < \infty[/tex]. Prove that
    \{ x \in X | x \in A_k \text{ for infinitely many k} \}
    is a null set.

    2. Relevant equations

    3. The attempt at a solution

    Let [tex]S = \{ x \in X | x \in A_k \text{ for infinitely many k} \}[/tex].
    Suppose [tex]\mu (S) > 0[/tex]. Then [tex]\displaystyle \mu(\bigcap A_k) > 0, A_k
    \ni x, x \in S[/tex]. Then [tex]\mu (A_k) > 0, A_k \ni x, x \in S[/tex] and hence
    [tex]\displaystyle \sum^\infty_{k=1} \mu(A_k) = \infty[/tex], which
    contradicts to assumption.

    Is this correct?
  2. jcsd
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