# Question on Full Measure

1. Oct 21, 2009

### e(ho0n3

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. Oct 21, 2009

### foxjwill

As an example of such a subset X, take the irrational numbers.

3. Oct 21, 2009

### e(ho0n3

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. Oct 21, 2009

### Dick

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
5. Oct 21, 2009

### e(ho0n3

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. Oct 21, 2009

### e(ho0n3

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. Oct 21, 2009

### Dick

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. Oct 21, 2009

### e(ho0n3

Oh, I see it now. Duh! Thanks.