In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and inner-regular if \begin{equation} \mu(E) = \sup_{U \textrm{ compact}} \{\mu(U): U \subseteq E\}. \end{equation}
What's the motivation for the asymmetry in the definition? I.e. why for outer regularity do we require $U$ to be open and for inner regularity we require $U$ to be compact?