Sum of a countable dense set and a set of positive measure Assume $A$ is a countable dense set in $\mathbb{R}$, and set $B$ has positive (Lebesgue) measure. Prove that $A+B=\{a+b:a\in A, b\in B\}=\mathbb{R}\backslash N$, where $N$ is a set of measure zero. 
I haven't come up with a good idea. 
Thanks in advance!

Some more details for the answer by Kena Bel P: $A+B$ is measurable because it's the union of countably many translates of the measurable set $B$.  Let $N$ be its complement and suppose, toward a contradiction, that $N$ had positive measure.  Then by the cited theorem of Steinhaus, there would be an interval included in $\{n-a:a\in A, n\in N\}$.  As $B$ is dense, this interval contains some element $b$ of $B$.  But then we have, for this $b$, that $b=n-a$ for some $a\in A$ and $n\in N$.  That is, $n=a+b$, contradicting the fact that $n\in N=\mathbb R\setminus(A+B)$.
