If $m$ is the Lebesgue measure on $\mathbb{R}^n$, for any $E \subseteq \mathbb{R}^n$ we can define the inner and outer Lebesgue measure as: $$ m^*(E) = \inf \{ m(U) : E \subseteq U, \text{ $U$ is open } \} $$
$$ m_*(E) = \sup \{ m(F) : F \subseteq E, \text{ $F$ is closed } \} $$
I know that if $E$ is measurable then $m^*(E) = m_*(E)$, but is it true for all subsets of $\mathbb{R}^n$? Could someone help me providing an example when the equality does not hold?