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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. – Ilya Jun 28 '12 at 16:35
up vote 2 down vote accepted

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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Thanks! Is it possible for $E$ to have positive measure? – nayrb Jun 28 '12 at 16:56
I suppose you could construct a fat cantor set "inside" so that the complement is dense in $E$. – nayrb Jun 28 '12 at 17:05

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