Rational translates of Fat Cantor set as positive but not full measure for each interval

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I was wondering how whether this works: Choose $A=cup_r in Qr+F$ where F denotes the Fat Cantor Set.



To be precise, if μ denotes Lebesgue measure, how would one show that this A as a Borel set A⊂R such that
$0<μ(A∩I)<μ(I)$
for every interval I in R?



A is not R by Baire Category theorem. But I haven't made much progress thereafter.










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    up vote
    0
    down vote

    favorite
    1












    I was wondering how whether this works: Choose $A=cup_r in Qr+F$ where F denotes the Fat Cantor Set.



    To be precise, if μ denotes Lebesgue measure, how would one show that this A as a Borel set A⊂R such that
    $0<μ(A∩I)<μ(I)$
    for every interval I in R?



    A is not R by Baire Category theorem. But I haven't made much progress thereafter.










    share|cite|improve this question























      up vote
      0
      down vote

      favorite
      1









      up vote
      0
      down vote

      favorite
      1






      1





      I was wondering how whether this works: Choose $A=cup_r in Qr+F$ where F denotes the Fat Cantor Set.



      To be precise, if μ denotes Lebesgue measure, how would one show that this A as a Borel set A⊂R such that
      $0<μ(A∩I)<μ(I)$
      for every interval I in R?



      A is not R by Baire Category theorem. But I haven't made much progress thereafter.










      share|cite|improve this question













      I was wondering how whether this works: Choose $A=cup_r in Qr+F$ where F denotes the Fat Cantor Set.



      To be precise, if μ denotes Lebesgue measure, how would one show that this A as a Borel set A⊂R such that
      $0<μ(A∩I)<μ(I)$
      for every interval I in R?



      A is not R by Baire Category theorem. But I haven't made much progress thereafter.







      real-analysis measure-theory






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      share|cite|improve this question











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      share|cite|improve this question










      asked Sep 4 at 9:15









      Ronit Debnath

      531115




      531115

























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