Dense transfer of a set with positive lebesgue measure: is it conull? I'm facing a problem in measure theory and I need to prove the following conjecture to move on.
Attention: I'm not sure the following statement is true.
Let $A \subset \mathbb{R}$ be a measurable set such that $m(A)>0$ and $H$ be a countable, dense subset of $\mathbb{R}$. If $A+H=\{a+h: a \in A, h \in H\}$, prove that $m((A+H)^c)=0$.
I'm totally stuck. It's easy to see that $A+H=\displaystyle{\bigcup_{h \in H} A+h}$, so it's definitely a measurable set, but that's the only progress I've been able to make. Any help would be greatly appreciated!

Use Lebesgue density theorem (LDT) which has an elementary proof.
Towards a contradiction, suppose $B = \mathbb{R} \setminus (A + H)$ has positive measure. Using LDT, choose open intervals $I, J$ of same length such that $B \cap I$ has $\geq 99$ percent measure of $I$ and $A \cap J$ has $\geq 99$ percent measure of $J$. Choose $h \in H$ (using the density of $H$) such that $J + h$ meets $I$ on a set of measure $\geq 99$ percent of $I$ (which has same length as $J$). Do you see a problem now?
