Local-Tame Topology

Let ${\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1(\mathrm{\Omega })$ be the probability laws $P$ on the configuration space of Poisson Configuration Reference that are invariant under every $T_x$ and satisfy $E_P{N}_{C_0}<\mathrm{\infty }$. Stationarity then makes every bounded-region count integrable, because a bounded set meets finitely many unit cells.

A real measurable function $f$ on $\mathrm{\Omega }$ is local if there is a bounded Borel set $A$ such that $f$ is measurable with respect to the restriction $\omega \mapsto \omega {\mathrm{\mid }}_A$. It is tame if there is a finite $b\ge 0$ such that $\mathrm{\mid }f(\omega )\mathrm{\mid }\le b(1+N_A(\omega ))$ for every $\omega$ and one such bounded $A$. Write $\mathcal{T}$ for the family of all local tame functions. Each $f\in \mathcal{T}$ is integrable under every $P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1(\mathrm{\Omega })$.

The local-tame topology ${\tau }_L$ on ${\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1(\mathrm{\Omega })$ is the initial topology generated by all maps $P\mapsto E_{P}f$, $f\in \mathcal{T}$. Equivalently, a net $(P_{\alpha })$ converges to $P$ in ${\tau }_L$ exactly when $E_{P_{\alpha }}f\to E_{P}f$ for every measurable local $f$ with at most linear growth in the count of a bounded support. Bounded local indicators are among these tests, so this topology separates distinct probability laws. The domain restriction to finite-intensity stationary laws ensures every displayed expectation is a finite real number.