Borel measures live on the topology-generated sigma-algebra. Radon measures are Borel measures that can be approximated from compact and open sets.
Borel sigma-algebra
The Borel sets are generated by open sets and therefore encode topology.
Definition. If \(X\) is a topological space, \(\mathcal B(X)\) is the sigma-algebra generated by open subsets of \(X\). A Borel measure is a measure on \(\mathcal B(X)\).
Proposition. On a metric space, \(\mathcal B(X)\) is generated by open balls, and if \(X\) is separable it is generated by balls with centers in a countable dense set and rational radii.
Proof. Open sets are unions of open balls. In a separable metric space, every open ball is a countable union of rational-radius balls with centers in a fixed countable dense subset contained slightly inside it. Thus the countable collection generates all open sets.
Regularity
Regularity means measurable sets can be approximated by topologically nice sets.
Definition. A Borel measure \(\mu\) is outer regular if \[ \mu(A)=\inf\{\mu(U):A\subset U,\ U\text{ open}\}, \] and inner regular on a class of sets if \[ \mu(A)=\sup\{\mu(K):K\subset A,\ K\text{ compact}\}. \] A finite Borel measure is regular when both identities hold for every Borel \(A\).
Theorem. Every finite Borel measure on \(\mathbb R^d\) is regular.
Proof. First, every open \(U\subset\mathbb R^d\) is a countable union of closed dyadic cubes contained in \(U\). Finite measure continuity from below therefore gives \[ \mu(U)=\sup\{\mu(K):K\subset U,\ K\text{ compact}\}. \] Let \(\mathcal C\) be the sets that are both inner and outer regular. The preceding argument puts all open sets in \(\mathcal C\). If \(A\in\mathcal C\), finite total mass turns inner approximation of \(A\) into outer approximation of \(A^c\), and conversely, so \(\mathcal C\) is closed under complements. Disjoint countable unions are handled by approximating finitely many pieces from within and controlling the remaining measure tail. Thus \(\mathcal C\) is a sigma-algebra containing the open sets, hence all Borel sets.
Radon measures
Radon measures are the right measures for locally compact spaces.
Definition. A Radon measure on a locally compact Hausdorff space is a Borel measure finite on compact sets, inner regular on open sets, and outer regular on Borel sets.
Proposition. Every locally finite Borel measure on \(\mathbb R^d\) is Radon.
Proof. Let \(Q_n=[-n,n]^d\). Local finiteness gives \(\mu(Q_n)<\infty\), so the restriction of \(\mu\) to \(Q_n\) is regular by the finite theorem. For open \(U\), \[ \mu(U)=\lim_n\mu(U\cap Q_n), \] and each term can be approximated from below by a compact subset of \(U\cap Q_n\); this proves inner regularity. For a Borel \(A\), choose \(n\) so that the portion of \(A\) relevant to any prescribed finite truncation lies in \(Q_n\), apply outer regularity there, and then exhaust. This yields global outer regularity, while compact finiteness is assumed.
Support
The support is the closed set where the measure cannot be locally ignored.
Definition. For a Borel measure on a second-countable space, \[ \operatorname{supp}\mu =\{x:\mu(U)>0\text{ for every open neighborhood }U\ni x\}. \] Equivalently, it is the complement of the largest open \(\mu\)-null set.
Proof. Let \(\{B_n\}\) be a countable base and let \[ N=\bigcup_{\mu(B_n)=0}B_n. \] Then \(N\) is open and null. Every open null set is a union of basis elements, each of which is null, so it lies in \(N\). Hence \(N\) is the largest open null set, and its complement consists exactly of points whose every neighborhood has positive measure.
Example. If \[ \mu=\sum_{n=1}^{\infty}2^{-n}\delta_{q_n}, \] where \(\{q_n\}\) enumerates \(\mathbb Q\), then \(\operatorname{supp}\mu=\mathbb R\): every open interval contains some \(q_n\) and therefore has positive mass. The support may be much larger than the set of atoms.
Continuous approximation of indicators
Regularity makes measures visible to continuous test functions.
Remark. If \(K\subset U\) with \(K\) compact and \(U\) open in a metric space, then \[ \varphi(x)=\frac{d(x,U^c)}{d(x,U^c)+d(x,K)} \] is continuous, equals \(1\) on \(K\), and vanishes on \(U^c\). Regularity lets \(K\) and \(U\) approximate a measurable set in measure, so continuous functions approximate its indicator. This is the mechanism behind the portmanteau theorem in Weak convergence of measures.