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Show that E and the closure of E have the same Jordan outer measure

  1. Sep 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Let E be a bounded set in [itex] R^n [/itex]
    Show that E and the closure of E have the same Jordan outer measure

    2. Relevant equations
    Jordan outer measure is defined as [itex] m^* J(E)=inf(m(b)) [/itex]
    where [itex] B \supset E [/itex] B is elementary.
    3. The attempt at a solution

    If E and the closure of E are the same there is nothing to prove.
    other case cl(E) contains limit points of E that are not in E.
    Lets first look at the measure of E. BY the definition of Jordan outer measure, The closure of E will contain the inf(B) and E is a subset of B and the closure. Let B, be the closure of E, The inf(B) will be the measure of E.
    Now this is where Im a little stuck. The Close of E, has all its limit points, so the inf(E) is part of E, its like a closed interval. There is no set B such that inf(m(B)). Unless i pick B to be the set such that
    [itex] B=E+ \epsilon [/itex] so the inf of the measure of B is E.
     
  2. jcsd
  3. Sep 27, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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