Outer regular measure ($\mu (E)={\inf }_{U\supseteq E,U\mathrm{open}}\mu (U)$)

A measure on a topological space is outer regular on its measurable sets when the displayed open-cover identity holds. Lebesgue measure has this property. It allows a null set to be placed in arbitrarily small open neighborhoods, and a compact small set to receive a smooth cutoff of similarly small integral.