Existence of a Constant c in Subset X with Complement Measure 0

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In summary, the conversation discusses the existence of a subset X of real numbers, whose complement has Lebesgue measure 0. The goal is to show that there exists a constant c such that for all integers n, c + n is in X. The conversation explores different strategies and eventually concludes that the intersection of all sets (X-i) for i an integer cannot be empty, leading to the desired result.
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Homework Statement
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?
 
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  • #2
As an example of such a subset X, take the irrational numbers.
 
  • #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?
 
  • #4
Think about the intersection of all of the sets (X-i) for i an integer. Could it possibly be empty?
 
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  • #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.
 
  • #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?
 
  • #7
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?
 
  • #8
Oh, I see it now. Duh! Thanks.
 

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