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Question on Full Measure

  1. Oct 21, 2009 #1
    The problem statement, all variables and given/known data
    Suppose X is a subset of R such that its complement has Lebesgue measure 0. Show that there exists a c such that for all integers n, c + n is in X.

    The attempt at a solution
    I've been thinking about this for a while and I just don't see how such a c could exists. Any tips?
     
  2. jcsd
  3. Oct 21, 2009 #2
    As an example of such a subset X, take the irrational numbers.
     
  4. Oct 21, 2009 #3
    I thought about that already. I know that X must contain some irrational c, but how do I know it will contain c + n for all integers n?
     
  5. Oct 21, 2009 #4

    Dick

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    Think about the intersection of all of the sets (X-i) for i an integer. Could it possibly be empty?
     
    Last edited: Oct 21, 2009
  6. Oct 21, 2009 #5
    I had thought of the interesection of the sets X + n for all integers n but that thought didn't develop further. But now that you wrote X - i, I now see how it works. Thanks.
     
  7. Oct 21, 2009 #6
    Hmm...maybe I wrote to soon. If the intersection is empty, then the intersection of X - 1, X - 2, etc. is a subset of the complement of X and so has measure 0. But where is the contradiction?
     
  8. Oct 21, 2009 #7

    Dick

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    If the intesection is empty, then the complement of the intersection is R. Write down an expression for the complement of the intersection expressed as a union of complements. Do you see a contradiction now?
     
  9. Oct 21, 2009 #8
    Oh, I see it now. Duh! Thanks.
     
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