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Real Analysis! HELP!

  1. Nov 6, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that there exists measurable functions f_n defined on some measure subspace, st f_n-> f a.e. but such that f is not measurable.

    2. Relevant equations

    Converges a.e. means that converges everywhere except on a set of measure zero.

    3. The attempt at a solution
    Need to construct a measure space in which some subset of a measurable set of measure zero is not measurable. However, such measure space is not compelte.
     
  2. jcsd
  3. Nov 7, 2008 #2
    I know that I need to construct a measure space in which some subset of a measurable set of measure zero is not measurable. However, such measure space is not complete.
    It seems that there is a subset of the Cantor set that is not borel measurable...so, if you choose the Borel measure, then you know it is not complete and that m(Cantor set)=0...
    I am not sure how to choose the function though...maybe choose the Cantor function?
     
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