Radon measure

A Radon measure is a Borel measure that is finite on compact sets and is inner regular, so the measure of a Borel set can be approximated from below by compact subsets. On a locally compact Hausdorff space one also usually includes outer regularity.

Every finite positive Borel measure on Euclidean space is inner regular by compact sets and outer regular by open sets. For an open set $U$, the compact sets $\{x:|x|\le n,\mathrm{dist}(x,U^c)\ge 1/n\}$ increase to $U$, so continuity from below of a measure gives inner approximation. The empty complement is handled by compact balls. The class of Borel sets admitting both approximations is closed under complements: interchange inner and outer approximations, then truncate the inner closed set by a large compact ball, using finiteness of the measure. For a countable union, approximate its components from outside with summable errors, and approximate finitely many components from inside after making the remaining union's measure small. Thus this class is a sigma-algebra containing the open sets. In particular, a finite measure carried by a Borel null set has compact subsets of that carrier capturing arbitrarily nearly all its mass.