# Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if $$\mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\}$$ and inner-regular if $$\mu(E) = \sup_{U \textrm{ compact}} \{\mu(U): U \subseteq E\}.$$

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?

• Do you think it would be more natural with closed instead of compact? – Ulrik Feb 18 '16 at 13:16