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Math Help - lebesgue outermeasure

  1. #1
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    lebesgue outermeasure

    Hi,
    I have this problem that i don't know how to start with:
    If E is a set on the real line with lebesgue outermeasure zero then its complement is dense in R
    Thank you
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  2. #2
    Senior Member Tinyboss's Avatar
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    Try the contrapositive: if the complement of E is NOT dense in R, then what can you say about E?
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  3. #3
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    if it's not dense then its closure is a subset of R and from there we prove that E has a measure different than zero?
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  4. #4
    Senior Member Tinyboss's Avatar
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    If the complement of E is not dense, then E contains an open set.
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  5. #5
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    ok so if i go roughly like this would it be true:
    Suppose E^c is not dense then there exist an interval I in R such that I is contained in E^c.
    the lebesque outermeasure is denoterd by *. so *(E)=*(EI)+*(EI^c)= 0 + *(I^c) which is differenty than zero since I is an interval.
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