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Construct an open subset of [0,1] of arbitrary measure dense therein

  1. Dec 5, 2005 #1


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    Q: Construct an open subset E of [0,1] having Lebesgue measure [itex]m(E)=\epsilon[/itex] such that [itex]0<\epsilon<1[/itex] which is dense [0,1].

    A: The fat Cantor set. I need help proving it is dense in [0,1]. The usual Ternary expansion arguement stuff won't work as the sets used are of length [tex]\frac{\epsilon}{3^k}[/tex] at the kth iteration. Ideas?

    The details are in the attached PDF:

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  3. Dec 5, 2005 #2
    Do you mean proving that [0,1]/C is dense?
  4. Dec 5, 2005 #3


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    Yes, sorry. The set at large, namely E, is the complement of the so-called fat Cantor set C in [0,1], i.e. [0,1]\C.
  5. Dec 5, 2005 #4


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    Can you find a (not necessarily open) subset of [0,1] that is dense and has measure zero?
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