Prove $A+A=\mathbb{R}$ in which the measure of the complement of $A$ is zero

Suppose not. Then there exists $r\in \mathbb{R}$ s.t. $r-a\notin A$ for any $a\in A$. Let $B:=\{r-a: a\in A\}$. By translation invariance of the Lebesgue measure $m(B)>0$. However, $B\subset \mathbb{R}\setminus A$ and so $m(B)=0$.