TITLE: Let $A \subseteq \mathbb R$ be an infinite set and for every $x,y \in A, x-y \in \mathbb{Q}$. Now which of following options is true?
QUESTION [0 upvotes]: Let $A \subseteq \mathbb R$ be an infinite set and for every $x,y \in A, x-y \in \mathbb{Q}$. Now which of following options is true ? 

$A $ is Lebesgue measurable and $m(A)=0$. ($m(A)$ is Lebesgue measure of $A $ )
$A $ is Lebesgue measurable and $m(A)=\infty$
$A $ is Lebesgue measurable but $m(A)=r$ such that $0<r <\infty $
The set $A$ is not necessarily Lebesgue measurable

I think $A$ is countable then option 1 is true.

REPLY [2 votes]: The set must be countable (and hence must have measure $0$).
To see this, fix an element $y_0\in A$.  Then every $x\in A$ can be written as $y_0+q$ for some $q\in \mathbb Q$.  As the set of such combinations is countable, so must $A$ be.
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