A Lebesgue measure question involving a dense subset of R, translates of a measurable set, etc. Let $\{b_n\}_{n=1}^\infty$ be a dense subset of $\mathbb{R}$ and let $D \subseteq \mathbb{R}$ be a measurable set such that $m(D \triangle (D + b_n))=0$ for all $n \in \mathbb{N}$ (here, the $\triangle$ denotes the symmetric difference of the two sets, $D+ b_n = \{d + b_n : d \in D\}$, and $m$ stands for the Lebesgue measure). Prove that $m(D)=0$ or $m(D^c)=0$ (here $D^c$ is the complement of $D$ in  $\mathbb{R}$). 
I am having trouble getting a proof off the ground! In particular, it is not clear to me how and where the dense hypothesis would come in. Any help would be greatly appreciated.  

Hints: 
1) $m(D \Delta (D + x))$ is a continuous function of $x$. 
2) If $x$ and $y$ are Lebesgue points of $D$ and $D^c$ respectively, what can you say about $m(D \Delta (D + x - y))$?
