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Math Help - set of positive outer measure

  1. #1
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    set of positive outer measure

    Show that if a set X in R has positive outer measure, then there is a bounded subset of X also having positive outer measure.
    Here's my unoriginal attempt:

    Let X be any set in R, X measurable, 0 < m*(X) < \infty, so X is bounded
    Let Y = X \cap Q and Y' = X \sim Y
    So, Y, Y' bounded
    m*(Y) = 0 (countable set)
    m*(X) = m*(Y \cup Y') = m*(Y) + m*(Y') = 0 + m*(Y')
    So, m*(Y') = m*(X) > 0

    Does this make sense?

    Another way I wanted to show this was by representing X as a disjoint union of (finite number of) measurable, bounded sets. Then one of these sets would satisfy the criteria, I think.
    If X open, I know that X is the disjoint union of a countable collection of open intervals.
    If X closed, or X neither open or closed - what is the disjoint union?

    Thanks for any tips.
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  2. #2
    Junior Member
    Joined
    May 2010
    Posts
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    Re: set of positive outer measure

    I figured out the answer to this question. I'm all set.
    And the silly thing with the Y,Y' subsets is not quite correct but it doesn't matter.
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