Inner regular measure
= Inner regular measure
{title2=$\mu(B)=\sup_{K\subseteq B,\ K\ {\rm compact}}\mu(K)$}
= Inner regularity
{synonym}
A <measure> is inner regular on its measurable sets if their measures are suprema of the measures of 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.