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

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 ?

1. $A $ is Lebesgue measurable and $m(A)=0$. ($m(A)$ is Lebesgue measure of $A $ )

2. $A $ is Lebesgue measurable and $m(A)=\infty$

3. $A $ is Lebesgue measurable but $m(A)=r$ such that $0<r <\infty $

4. The set $A$ is not necessarily Lebesgue measurable

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

Answer 1

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.