= Regularity of finite Borel measures on Euclidean space
{title2=$\mu(A)=\sup_{K\subset A,\ K\text{ compact}}\mu(K)$}
Every finite positive <Borel measure> on Euclidean space is inner regular by <compact sets> and outer regular by open sets. For an <open set> $U$, the <compact sets> $\{x:|x|\le n,\ \operatorname{dist}(x,U^c)\ge1/n\}$ increase to $U$, so <continuity from below of a measure> gives inner approximation. The empty complement is handled by compact balls. The class of Borel sets admitting both approximations is closed under complements: interchange inner and outer approximations, then truncate the inner closed set by a large compact ball, using finiteness of the <measure>. For a countable union, approximate its components from outside with summable errors, and approximate finitely many components from inside after making the remaining union's <measure> small. Thus this class is a <sigma-algebra> containing the open sets. In particular, a finite <measure> carried by a Borel <null set> has <compact subsets> of that carrier capturing arbitrarily nearly all its mass.
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