Inner regular measure ($\mu (B)={\sup }_{K\subseteq B,K\mathrm{compact}}\mu (K)$)

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.