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!

This follows from the following Steinhaus theorem: if $A,B$ are subsets of the real line of positive measure then $A-B$ contains an interval.
