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Approximating Borel Sets

  1. Dec 11, 2013 #1
    1. The problem statement, all variables and given/known data

    We're given the measure space [tex](X,A,μ)[/tex] with [tex]X=\bigcup_{i=1}^{\infty} X_i[/tex] where [tex] X_i⊂X_{i+1}⊂..., X_i[/tex] are open for all i and μ(X_i)<+∞ for all i. Show that for every Borel set B there exists an open set U where μ(B\U)<ϵ.

    2. Relevant equations

    measures are subadditive

    3. The attempt at a solution

    Pick closed sets [tex]c_i⊂X_i−B[/tex] with [tex]μ((X_i−c_i)−B)=μ((X_i−B)−c_i)<ϵ/{2^i}[/tex]. Let [tex]G=\bigcup_i^\infty X_i-c_i[/tex]. Then recognizing that [tex]X_i−c_i=X_i⋂c_i^c[/tex] is the intersection of open sets reveals that G must be open. Moreover, B⊂G is clear and thus we can consider G−B. If follows that μ(G−B)<ϵ by sub additivity.

    My problem is explaining why such closed sets exist. I know that [tex]X_i-B[/tex] is measurable, but why am I able to find a closed set inside of it that satisfies these requirements?
  2. jcsd
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