Borel measure

A Borel measure on a topological space is a measure defined on the sigma-algebra generated by the open sets, equivalently on the Borel sets.

For a positive Borel measure on a topological space, the support consists of the points whose every open neighborhood has positive measure. For a finite regular complex measure use its variation measure. Regularity makes the complement of the support null: compact subsets of that open complement have finite covers by null neighborhoods. A regular probability measure on a compact Hausdorff space whose support is a single point is therefore a Dirac measure.

For a Borel probability measure on a Polish space and a continuous map to another Polish space, the image measure has the displayed support. Every neighborhood of an image of a support point has an open inverse image of positive measure. Conversely, the closed image closure has full image measure because the original support has full measure. This transfers a support description for a driving rough path to the solution of a rough differential equation.

A Borel probability measure is a Borel measure with total mass one. On a compact metric space it is determined by its integrals against continuous functions. On that compact metric space, the family of Borel probability measures is compact for weak convergence of probability measures.

For a Borel probability measure on a compact metric space, inner regularity and outer regularity give $K\subseteq B\subseteq U$ with $K$ compact, $U$ open and $\mu (U\setminus K)<ϵ$. The Urysohn lemma supplies a continuous function $\phi$ equal to one on $K$ and zero outside $U$, with values in $[0,1]$. Its difference from the indicator function of $B$ is supported in $U\setminus K$ and bounded by one, proving the displayed estimate. This transfers identities established for continuous test functions to measurable sets.

A finite positive Borel measure on a compact Hausdorff space is regular when its values on Borel sets can be approximated from inside by compact sets and from outside by open sets. For signed or complex measures require this property of the variation measure. These are the measures representing the continuous dual space of $C(K)$ in the Riesz-Markov-Kakutani representation theorem.

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.