Let $A\subseteq \mathbb{R}^2$ be a measurable set s.t $m(A)>0$. Show that exists a sequence $\{x_n\}_\mathbb{N}\subset\mathbb{R}^2$ s.t$$m(\mathbb{R}^2-(\bigcup_{n=1}^\infty(x_n+A))=0\quad(x+A=\{x+a\mid a\in A\}$$ hint:You can use Lebesgue density theorem
I thought taking squares $E_n$ which satisfy the condition $$m(E_n\cap A)>(1-\epsilon_n)m(E_n)$$ and to tile $\mathbb{R}^2$ by translation of $E_n$ but from there I don't know how to use the density theorem. How cna I continue using this theorem?
Cover the ball $B(0,n)$ with finitely many translates $x_{n,k}+E_n$ of $E_n$, $1\le k\le m_n$. Then the corresponding translates $x_{n,k}+A$ cover at least a share approximately (why only approximately? How can this be made precise?) $1-\epsilon_n$ of $B(0,n)$. Taking the union over all $n$, you get countably many translates of $A$ that cover an arbitrarily large share of arbitrarily large disks. This is not compatible witrh the existenc of any square $E$ and $\epsilon>0$ with $m(E\cap(\mathbb R^2-\bigcup_{n,k}(x_{n,k}+A)))>(1-\epsilon)m(E)$ (at least if you pick $\epsilon_n$ carefully enough and $E_n$ smallenough).