Finite-spin product and local topologies

Fix an integer $d\ge 1$, a finite nonempty alphabet $S$ with the discrete topology, and a bijective enumeration $x_1,x_2,\dots$ of ${\mathbb{Z}}^d$. Let $\mathrm{\Omega }=S^{{\mathbb{Z}}^d}$ be the set of all configurations $\sigma :{\mathbb{Z}}^d\to S$ with the product topology, the smallest topology that makes each coordinate map $\sigma \mapsto {\sigma }_x$ continuous. For a finite set $F\subseteq {\mathbb{Z}}^d$ and assignment $a\in S^F$, the cylinder set determined by $(F,a)$ is $[F,a]=\{\sigma \in \mathrm{\Omega }:\sigma {\mathrm{\mid }}_F=a\}$. The cylinder sigma-algebra $\mathcal{C}$ is generated by these sets.

For each finite $F$, let ${\pi }_F:\mathrm{\Omega }\to S^F$ be restriction to $F$. A probability measure $\rho$ on $(\mathrm{\Omega },\mathcal{C})$ is a countably additive map $\mathcal{C}\to [0,1]$ with $\rho (\mathrm{\varnothing })=0$ and $\rho (\mathrm{\Omega })=1$. Its finite-coordinate pushforward is the measure on all subsets of $S^F$ defined by $({\pi }_F{)}_{\ast }\rho (B):=\rho ({\pi }_F^{-1}(B))$; the preimage is a finite union of cylinders and belongs to $\mathcal{C}$. For probability measures ${\mu }_n,\mu$ on this space, local weak convergence ${\mu }_n{\to }_{\mathrm{l}\mathrm{o}\mathrm{c}}\mu$ means that $({\pi }_F{)}_{\ast }{\mu }_n(B)\to ({\pi }_F{)}_{\ast }\mu (B)$ for every finite $F\subseteq {\mathbb{Z}}^d$ and every $B\subseteq S^F$. Equivalently, all finite-coordinate cylinder probabilities converge. The cylinder sigma-algebra is shown below in a local result to equal the Borel sigma-algebra, with local weak convergence equal to ordinary weak convergence on this space.

Finite-range lattice interaction and conditional energy

Cylinder convergence is weak convergence on a finite-spin product

This finite-spin product topology and cylinder convergence are distinct from the vague topology on locally finite counting measures. Vague convergence tests compactly supported spatial functions and may lose mass at infinity; it does not by itself imply weak convergence of probability measures.

Locally finite counting measures and vague topology