Take R^n with its associated Lebesgue measure. Let A be a set of positive measure. Do there exist points x and x' in A such that x'>>x, or x' is strictly larger in each component than x? Something tells me this must be true, and I've been working on it all day. Any help/suggestions?



This should be true concerning Lebesgue measure, no?
If I understand the relation $\gg$ correctly, then the answer is yes. This follows from the following proposition.



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> Let $A \subset \mathbb R^n$ be a Lebesgue-measurable set of positive measure. Then the difference set $D(A) = \{ x-y \mid x,y \in A \}$ contains an open ball about the origin.
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This is a standard exercise in measure theory (at least in $\mathbb{R}$). You can find several proofs of it on math.stackexchange: <https://math.stackexchange.com/questions/38902/the-set-of-differences-for-a-set-of-positive-lebesgue-measure> 


In particular, $D(A)$ contains points whose coordinates are entirely positive. Therefore there exist points $x,y \in A$ such that $x \gg y$.


