The Radon Measure of a Positive Linear Functional on $C_c(X)$: the Outer Measure, Compact Sets, Borel Measurability and Regularity

In the setting of The Dual of C_c(X): Standing Notation, let $(X,d)$ be a locally compact metric space and let $\Lambda$ be a positive linear functional on $C_{c}(X)$. For open $V\subseteq X$ put

$$\mu(V)=\sup\{\Lambda f:f\in C_{c}(X),\ 0\le f\le1,\ \operatorname{supp}f\subseteq V\},$$

with $\operatorname{supp}f$ the support of $f$, and for $E\subseteq X$ put

$$\mu^{*}(E)=\inf\{\mu(V):V\subseteq X\text{ open},\ E\subseteq V\}.$$

(i) Outer measure. $\mu^{*}$ is an outer measure on $X$, and $\mu^{*}(V)=\mu(V)$ for every open $V\subseteq X$.

(ii) Compact sets. For every compact $K\subseteq X$,

$$\mu^{*}(K)=\inf\{\Lambda f:f\in C_{c}(X),\ f\ge\mathbf{1}_{K}\},$$

and $\mu^{*}(K)<\infty$.

(iii) Borel measure. Every Borel subset of $X$ is $\mu^{*}$-measurable, and the restriction $\bar{\mu}=\mu^{*}|_{\mathcal{B}(X)}$ of $\mu^{*}$ to $\mathcal{B}(X)$ is a Borel measure on $X$.

(iv) Radon measure. $\bar{\mu}$ is a Radon measure on $X$.