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Prove f is measurable on any closed set

  1. Dec 9, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove if $f$ is measurable on R and C is any closed set, f^{-1}(C) is measurable.

    2. Relevant equations

    Definition of measurability, closed sets etc.

    3. The attempt at a solution

    I've been trying for a while to get this proof, but I seem to just end up stuck at the beginning. I think I want to point out that the complement of a closed set is an open set, and open sets are countable unions of open intervals, which are themselves measurable. But I'm not too sure, and I'd sure appreciate a gentle push in the right direction.

  2. jcsd
  3. Dec 9, 2011 #2
    Yes, that is correct. So what is bothering you??
  4. Dec 10, 2011 #3
    I guess I need help formalizing the argument. Would I just assume the hypotheses, point out by definition C' is an open set, and then just mention "open sets are countable unions of open intervals, which are measurable?" Doesn't seem too rigorous..
  5. Dec 10, 2011 #4
    It's rigorous enough for me. (assuming you proved the things like any open set is the countable union of intervals).
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