Let $A\subset \mathbb{R}$ such that $m(\mathbb{R}\smallsetminus A)=0$. Show that $A+A=\mathbb{R} $, where $$A+A=\{a+b\mid a,b\in A\}$$.
This is a question in the past qualifying exam in my university. I do not know where to approach. I encountered a similar problem that if $A$ is measurable and $m(A)>0$, then $A-A$ contains an interval, but I used $m(A)$ finite to do this problem. Can you help?