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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?

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

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