Theorem 2.7.3DF30CEF15D17

Proof. We embed $\mathcal M$ homeomorphically onto a closed subset of a product of compact intervals. For $f\in C_ c(\Omega ,\mathbb R)$, choose $r(f)$ with $\operatorname{Supp}f\subset K_{r(f)}$; this is possible because $\operatorname{Supp}f$ is compact and the open sets $\operatorname{Int}K_ r$ increase to $\Omega $. For $\mu \in \mathcal M$ we have $|f|\leq \| f\| _\infty \chi _{r(f)}$, so $\int f\, \mathrm d\mu $ lies in $I_ f=[-M_{r(f)}\| f\| _\infty ,M_{r(f)}\| f\| _\infty ]$. The product $P=\prod _ fI_ f$ over all $f\in C_ c(\Omega ,\mathbb R)$ is compact by lemma 2.7.2. The map $\Phi \colon \mathcal M\to P$, $\Phi (\mu )=(\int f\, \mathrm d\mu )_ f$, is injective by the uniqueness in theorem 2.3.3, and it is a homeomorphism onto its image. Indeed, since $\int f\, \mathrm d\mu =\int \operatorname{Re}f\, \mathrm d\mu +\mathrm i\int \operatorname{Im}f\, \mathrm d\mu $, the vague topology on $\mathcal M$ is the weakest topology for which the maps $\mu \mapsto \int f\, \mathrm d\mu $ with real $f$ are continuous. The subspace topology of $\Phi (\mathcal M)$ is the weakest topology for which the restrictions of the coordinate projections are continuous, and $\Phi $ intertwines these two families of maps.

The image consists of the points $(a_ f)\in P$ such that $a_{sf+tg}=sa_ f+ta_ g$ for all $f,g$ and real $s,t$, $a_ f\geq 0$ whenever $f\geq 0$, and $a_{\chi _ r}\leq M_ r$ for every $r$: a point satisfying these conditions defines a positive linear functional on $C_ c(\Omega ,\mathbb R)$, which by theorem 2.3.3 is integration against a Radon measure belonging to $\mathcal M$. Each condition involves finitely many coordinates and is closed, so $\Phi (\mathcal M)$ is closed in $P$ and hence compact. Therefore $\mathcal M$ is vaguely compact, and the assertion about nets follows from proposition 2.7.1. $\square$