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

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.