Locally finite counting measures and vague topology

Fix $d\ge 1$. Let $\mathcal{B}({\mathbb{R}}^d)$ be the $\sigma$-algebra generated by the Euclidean open sets, and understand all measures below as defined on $({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d))$. A measure here is a countably additive map from $\mathcal{B}({\mathbb{R}}^d)$ to $[0,\mathrm{\infty }]$ that assigns $0$ to the empty set. A locally finite counting measure on ${\mathbb{R}}^d$ is a measure of the form $\xi ={\sum }_{r\in I}{\delta }_{x_r}$, where $I$ is finite or countable, repetitions among the points $x_r\in {\mathbb{R}}^d$ are allowed, and every compact set contains only finitely many listed points, counting repetitions. The empty sum is allowed and is the zero measure. For a Borel set $B$, $\xi (B)$ is the number of listed points in $B$, with multiplicity. Write ${\mathrm{\Gamma }}_{\mathrm{l}\mathrm{f}}({\mathbb{R}}^d)$ for the set of these measures. For $x\in {\mathbb{R}}^d$, the point mass ${\delta }_x$ assigns mass $1$ to a Borel set containing $x$ and mass $0$ otherwise.

For any continuous compactly supported $f:{\mathbb{R}}^d\to \mathbb{R}$ and $\xi ={\sum }_{r\in I}{\delta }_{x_r}$, define $\int fd\xi :={\sum }_{r:x_r\in \mathrm{supp}f}f(x_r)$, where $\mathrm{supp}f$ is the closure of $\{x:f(x)\mathrm{\ne }0\}$. This sum is finite by local finiteness, counts multiplicities, and is independent of the listing because $\xi$ fixes the multiplicity at every point. For the zero measure it is $0$.

The vague topology on ${\mathrm{\Gamma }}_{\mathrm{l}\mathrm{f}}({\mathbb{R}}^d)$ is the coarsest topology for which $\xi \mapsto \int fd\xi$ is continuous for every nonnegative continuous compactly supported function $f:{\mathbb{R}}^d\to \mathbb{R}$. Thus ${\xi }_n{\to }_{\mathrm{v}}\xi$ exactly when $\int fd{\xi }_n\to \int fd\xi$ for every such $f$.

Given a probability space $(\mathrm{\Theta },\mathcal{A},\mathbb{P})$, meaning that $\mathcal{A}$ is a $\sigma$-algebra on $\mathrm{\Theta }$ and $\mathbb{P}$ is a countably additive probability measure on $\mathcal{A}$ with $\mathbb{P}(\mathrm{\Theta })=1$, a random locally finite counting measure is an $(\mathcal{A},{\mathcal{B}}_{\mathrm{v}})$-measurable map $\mathrm{\Xi }:\mathrm{\Theta }\to {\mathrm{\Gamma }}_{\mathrm{l}\mathrm{f}}({\mathbb{R}}^d)$, where ${\mathcal{B}}_{\mathrm{v}}:=\sigma \{U:U\text{ is open in the vague topology just defined}\}$. We require that, for every bounded Borel set $B\subseteq {\mathbb{R}}^d$, the evaluation $\mathrm{\Xi }(B):\mathrm{\Theta }\to [0,\mathrm{\infty })$ given by $\mathrm{\Xi }(B)(\theta )=\mathrm{\Xi }(\theta )(B)$ is $\mathcal{A}$-measurable; it is finite by local finiteness. Write ${\mathbb{E}}_{\mathbb{P}}$ for integration with respect to $\mathbb{P}$. A sequence $({\mathrm{\Xi }}_n)$ of such measures on a common probability space uses this same domain and target for every $n$. If $K$ is compact, its local counts are uniformly integrable on $K$ when

${\lim }_{R\to \mathrm{\infty }}{\sup }_n{\mathbb{E}}_{\mathbb{P}}\text{\hspace{0.17em}⁣}[{\mathrm{\Xi }}_n(K){\mathbf{1}}_{\{{\mathrm{\Xi }}_n(K)>R\}}]=0.$

This is the count-tail condition used when an argument needs control of unbounded observables supported in $K$. The definition asserts no convergence of expectations without the exact test-function, support, and tail hypotheses of the result being applied.

Finite-spin product and local topologies

This measure-space topology is distinct from finite-spin product topology and cylinder convergence. Vague convergence tests compactly supported spatial functions and can lose mass at infinity; weak convergence of probability measures cannot be inferred from it without an additional tightness condition.

Finite-spin product and local topologies