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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.

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    Looks like an answer rather than a question.2012-06-28

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Let $P$ be a closed nowhere dense set in $\mathbb R$ with positive measure and let $E$ be a countable subset of $P$ that is dense in $P.$ For example, $E$ could be the set of endpoints of the complementary open intervals for $P.$ Then $m(\overline{E}-E)=m(P)$ is positive.

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    I suppose you could construct a fat cantor set "inside" so that the complement is dense in $E$.2012-06-28