real analysis - Question about the existence of a Lebesgue measurable set

Question: Does there exist any Lebesgue measurable set $E \subset [0,1]$ such that for any $x \in \mathbb{R}$, there exists a $y \in E$ satifying $x - y \in \mathbb{Q}$?

I guess there does not exist such a measurable set $E$ but I failed to prove that. Can anyone give a proof for me as a beginner on Lebesgue integrals?

Answer

Take $E=[0,1]$ and, for each $x\in\mathbb R$, take $y=x-\lfloor x\rfloor$. Then $x-y\in\mathbb Z\subset\mathbb Q$.