Lemma 5.27.387F112BE621FD

Proof. Choose relatively compact open sets $U_\ell $ exhausting $\Omega $, with $\overline U_\ell \subset U_{\ell +1}$. For each $\ell $, choose a countable uniformly dense subset of $C_0(U_{\ell +1};\mathbb R)$ consisting of smooth compactly supported functions, extended by zero to $\Omega $; such families exist because $C_0(U_{\ell +1};\mathbb R)$ is separable and its elements can be approximated uniformly by smooth functions. Add a nonnegative smooth compactly supported cutoff function equal to one near $\overline U_{\ell +1}$, and enumerate the union of all these functions as $(f_ i)$.

Call the maps $\alpha \mapsto \int _\Omega f_ i\, \mathrm d\alpha $ the coordinates. If positive measures converge to a fixed measure in every coordinate, the cutoff coordinates give common mass bounds on each $\overline U_{\ell +1}$, and uniform approximation then gives convergence against every $g\in C_ c(\Omega )$. The same argument, applied near a fixed measure, shows that $\alpha \mapsto \int _\Omega g\, \mathrm d\alpha $ is continuous for the topology generated by the coordinates: bound the relevant cutoff coordinate on a neighborhood, choose one smooth test that is uniformly close to $g$, and shrink the neighborhood using the coordinate of that test. Thus the coordinates generate exactly the vague topology, and they separate measures.

The function $s\mapsto s/(1+s)$ is increasing and subadditive on $[0,\infty )$. Applying it to the triangle inequalities for the coordinates gives the triangle inequality for $d_ v$, and $d_ v$ is positive on distinct measures because the coordinates separate them. Each coordinate is continuous for this metric, and the tails of the series are uniformly small, so the metric induces exactly the topology generated by the coordinates.

Finally, the finite atomic measures with nonnegative rational weights at points of a fixed countable dense subset of $\Omega $ form a countable dense family. To approximate a measure in finitely many coordinates, partition a compact neighborhood of the supports of the corresponding tests into sets on which all these tests have small oscillation, place an atom in each set of nonzero mass, and then perturb its location and weight. Only the finite mass of that compact neighborhood is used, and the remaining tail of the metric is arbitrarily small. $\square$