Cylinder convergence is weak convergence on a finite-spin product

If $\sigma \mathrm{\ne }\eta$, some coordinate differs, so $\rho (\sigma ,\eta )>0$; symmetry is immediate, and the coordinatewise triangle inequality gives the triangle inequality after summing. If two configurations agree on $x_1,\dots ,x_N$, then $\rho \le {\sum }_{k>N}{2}^{-k}=2^{-N}$. Conversely, if $\rho (\sigma ,\eta )<2^{-N}$, no coordinate among $x_1,\dots ,x_N$ differs, because a mismatch at $x_k$ contributes at least $2^{-N}$. These bounds show that metric balls and finite-coordinate basic neighborhoods generate the same topology.

If ${\mu }_n{\to }_{\mathrm{l}\mathrm{o}\mathrm{c}}\mu$, every cylinder indicator is bounded and continuous by [F2], so its integrals converge. Every bounded function depending on finitely many coordinates is a finite linear combination of cylinder indicators, and its integrals therefore converge as well.

Suppose an open cover of $\mathrm{\Omega }$ has no finite subcover. Let $C_0=\mathrm{\Omega }$. Recursively, if $C_{N-1}$ is the cylinder fixing the first $N-1$ spins and has no finite subcover, its finitely many children obtained by assigning ${\sigma }_{x_N}=s$ cannot all have finite subcovers; otherwise their finite union would cover $C_{N-1}$. Choose the least $s$ in the fixed order whose child still has no finite subcover and call that child $C_N$. These nested cylinders determine a configuration $\sigma$. A cover member containing $\sigma$ contains a basic neighborhood fixing finitely many sites; for sufficiently large $N$, $C_N$ lies inside that neighborhood and hence has a one-member subcover, a contradiction. Thus every open cover has a finite subcover and $\mathrm{\Omega }$ is compact.

Each cylinder is clopen by [F2], and cylinders form a countable basis: ${\mathbb{Z}}^d$ has countably many finite subsets, and each finite coordinate set has finitely many spin assignments. Therefore every open set is a countable union of cylinders and belongs to the cylinder sigma-algebra, while every cylinder is Borel. Hence the two sigma-algebras are equal.

Fix a bounded continuous $g:\mathrm{\Omega }\to \mathbb{R}$ and $\epsilon >0$. For each $\sigma$, continuity and the cylinder basis give a cylinder neighborhood $C_{\sigma }$ of $\sigma$ on which $\mathrm{\mid }g(\eta )-g(\sigma )\mathrm{\mid }<\epsilon \mathrm{/}3$. Compactness from step 2.1 gives a finite subcover $C_1,\dots ,C_r$. Let $F$ be the union of the finitely many coordinate sets fixed by these cylinders. If $\eta ,\zeta$ agree on $F$, choose a $C_i$ containing $\eta$; because $C_i$ depends only on a subset of $F$, it also contains $\zeta$, so $\mathrm{\mid }g(\eta )-g(\zeta )\mathrm{\mid }<2\epsilon \mathrm{/}3$. Choose $N$ with $F\subseteq \{x_1,\dots ,x_N\}$ and fix one default spin $s_0\in S$. Replace all coordinates after $x_N$ by $s_0$ to define a canonical extension $\widehat{\sigma }$ of each first-$N$ prefix, and set $g_N(\sigma )=g(\widehat{\sigma })$. Then $g_N$ depends only on the first $N$ coordinates and ${\sup }_{\sigma }\mathrm{\mid }g(\sigma )-g_N(\sigma )\mathrm{\mid }<\epsilon$.

For every $n$, $\mathrm{\mid }\int gd{\mu }_n-\int g_{N}d{\mu }_n\mathrm{\mid }\le \epsilon$ and the same bound holds for $\mu$. Step 1.2 gives $\int g_{N}d{\mu }_n\to \int g_{N}d\mu$, so the limit superior of $\mathrm{\mid }\int gd{\mu }_n-\int gd\mu \mathrm{\mid }$ is at most $2\epsilon$. Since $\epsilon >0$ is arbitrary, weak convergence follows. Conversely, if ${\mu }_n$ converges weakly, indicators of all cylinders are bounded continuous by [F2], so their probabilities converge; finite sums give every finite marginal, hence local weak convergence by [F1].