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Math Help - Measure

  1. #1
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    Measure

    I'm having trouble, please can somebody help me?

    Prove that for all ε>0 there exists a closed set Fc[0,1] such that F = and λ(F)>1-ε, where λ is the Lebesgue measure.

    Thank you very much!
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  2. #2
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    That sounds like a very difficult problem, if not impossible. The irrationals in the interval [0,1] are not closed, and I'm not sure I see how any subset of them can be closed. Are you sure this problem is solvable?
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  3. #3
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    I have seen this problem as an exercise in two books, then I think there is a solution.
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  4. #4
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    Quote Originally Posted by eltondelamancha View Post
    I'm having trouble, please can somebody help me?

    Prove that for all ε>0 there exists a closed set Fc[0,1] such that F = and λ(F)>1-ε, where λ is the Lebesgue measure.

    Thank you very much!
    This is an immediate consequence of the following lemma:

    A set X\subset \mathbb{R} ^n has measure 0 if and only if for all \varepsilon >0 there exists an open set X\subset \Omega with \lambda (\Omega ) <\varepsilon

    The proof of this is easy: There exists a lower semicontinous function f\geq 1_{X} with \int_{\mathbb{R} ^n} f <\frac{ \varepsilon}{2} then take \Omega := f^{-1} \left( \frac{1}{2} , \infty \right)
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  5. #5
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    Nice thinking 'Jose27'
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