= Inner regular measure
{title2=$\mu(B)=\sup_{K\subseteq B,\ K\ {\rm compact}}\mu(K)$}
= Inner regularity
{synonym}
For a <measure> on a <topological space> in which <compact sets> are measurable, inner regularity means that the <measure> of each <measurable set> is the supremum of the <measures> of its contained <compact sets>. Every <Borel probability measure> on a <compact metric space> is inner regular and has <outer regularity>. Combining the two properties allows approximation of an <indicator function> by a <continuous function> in the $L^1$ norm.
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