Suppose $E$ is a nowhere dense set. For simplicity, assume it is in $R$. Is it true that the Lebesgue measure of $\overline{E}-E$ is zero? I.e., $m(\overline{E}-E)=0$.
The statement is not true in general. If $E$ is allowed to be open then take the complement of a fat cantor set.
This is also not the same set as what is often defined to be the boundary. If $\overline{E}-E^o$ is the boundary of a set, then fat cantor sets have positive measure boundaries.