Compactness of Finite-Specific-Entropy Levels in the Local-Tame Topology

Let $W=a+[0,m{)}^d$ be any integer cube, with $a\in {\mathbb{Z}}^d$ and integer $m\ge 1$. Translation invariance and [F1] give $D(P_W\mathrm{\parallel }{Q}_W)\le c\mathrm{\mid }W\mathrm{\mid }$. If $A$ is any bounded Borel set, choose one such cube $W$ containing it; by restriction monotonicity [F2], every $P\in {\mathcal{H}}_c$ satisfies $D(P_A\mathrm{\parallel }{Q}_A)\le C_A:=c\mathrm{\mid }W\mathrm{\mid }$.

For any probability density $h$ relative to a probability $Q$ with $D(hQ\mathrm{\parallel }Q)\le C$, and any $K>1$, ${\int }_{\{h>K\}}hdQ\le \frac{C+1\mathrm{/}e}{\log K}\mathrm{.}$ Indeed, $-u\log u\le 1\mathrm{/}e$ on $[0,1]$, while $h\log h\ge h\log K$ on $\{h>K\}$. Split the entropy integral at $h=1$ and combine these bounds. Thus, for each fixed cube $W$, the densities $d{P}_W\mathrm{/}d{Q}_W$ with $P\in {\mathcal{H}}_c$ have $L^1(Q_W)$ norm one and are uniformly integrable.

Fix $W$. By [F3], the weak closure $K_W$ of these densities in real $L^1(Q_W)$ is weakly compact. It is Hausdorff by [F6]. Every $h\in K_W$ remains nonnegative and has integral one: testing against each indicator and against $1$ gives weakly closed conditions by [F6]. Also $D(h{Q}_W\mathrm{\parallel }{Q}_W)\le c\mathrm{\mid }W\mathrm{\mid }$, because [F2] expresses $D$ as the supremum of weakly continuous affine functions of $h$, with continuity given by [F6]. Hence all entropy sublevels used below are closed in their local weak topologies.

Index all integer cubes $W=a+[0,m{)}^d$ with $a\in {\mathbb{Z}}^d$ and integer $m\ge 1$ by their countable set of pairs $(a,m)$, and form the product $K_{\ast }:={\prod }_W{K}_W$. It is compact by [A2] and Hausdorff because each weak $L^1$ topology is Hausdorff by [F6]. Impose the following closed conditions on its tuples: whenever $W\subseteq V$, the $W$-marginal of $h_V{Q}_V$ equals $h_W{Q}_W$; and, for every real translation $x$ and bounded local measurable test $f$, the two marginals containing the supports of $f$ and $f\circ T_x$ satisfy $E_{P}f=E_P(f\circ T_x)$. Each equality is closed because it tests weak convergence against bounded measurable functions on a containing cube, which are continuous by [F6]. The intersection $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$ of these closed conditions is compact.

A tuple in $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$ defines consistent finite-dimensional laws on the countable standard-Borel cell spaces: for a finite set of cells, choose any integer cube containing it and take that marginal. If two cubes contain the same finite set, a larger integer cube contains both; the imposed nested consistency makes their induced laws agree. By [F4] there is a unique law $P$ on $\mathrm{\Omega }$ with those marginals. The imposed equalities make $P$ stationary. For each integer cube $W$, consistency identifies $P_W=h_W{Q}_W$ for its coordinate $h_W\in K_W$, so step 2.1 gives $D(P_W\mathrm{\parallel }{Q}_W)\le c\mathrm{\mid }W\mathrm{\mid }$. In particular, these local entropy bounds hold for every law constructed from $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$ before its finite intensity has been established.

The same variational inequality controls particle-count tails uniformly on the whole family of laws constructed from $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$. Fix bounded $A$ and an integer cube $W\supseteq A$. For every such law $P$, its $W$ marginal is $h_W{Q}_W$ with $h_W\in K_W$, so step 2.1 gives $D(P_W\mathrm{\parallel }{Q}_W)\le C_A:=c\mathrm{\mid }W\mathrm{\mid }$. Let $K\ge 0$, $a>0$, and $M\in \mathbb{N}$. The bounded local test ${\varphi }_M=a\min (N_A{\mathbf{1}}_{\{N_A>K\}},M)$ in [F2] gives $a{E}_P\min (N_A{\mathbf{1}}_{\{N_A>K\}},M)\le C_A+\log E_Q\exp ({\varphi }_M)\mathrm{.}$ As $M\to \mathrm{\infty }$, monotone convergence yields $a{E}_P[N_A{\mathbf{1}}_{\{N_A>K\}}]\le C_A+\log E_Q{e}^{a{N}_A{\mathbf{1}}_{\{N_A>K\}}}\mathrm{.}$ By [F5] and dominated convergence the logarithmic term tends to zero as $K\to \mathrm{\infty }$. First choose $a$ large and then $K$ large; this proves uniform integrability of the counts $N_A$ over all laws constructed from $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$. Taking $A=C_0$ gives finite cell intensity for each such $P$. Since step 4.1 gives stationarity, $P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1(\mathrm{\Omega })$; its origin-anchored cube bounds from step 4.1 and [F1] now give $I(P)\le c$.

The product compactness in step 3.1 first gives compactness for all bounded local tests. For any tame $f$ supported in $A$ with $\mathrm{\mid }f\mathrm{\mid }\le b(1+N_A)$, set $f_K=f{\mathbf{1}}_{\{N_A\le K\}}$. Then $f_K$ is bounded and local, and its expectation is weakly continuous by [F6], while ${\sup }_{P\in {\mathcal{H}}_c}\mathrm{\mid }{E}_P(f-f_K)\mathrm{\mid }\le b{\sup }_{P\in {\mathcal{H}}_c}{E}_P[(1+N_A){\mathbf{1}}_{\{N_A>K\}}]⟶0$ by step 5.1. Thus on the entropy level the tame expectation maps are uniform limits of bounded-local continuous maps. Every $P\in {\mathcal{H}}_c$ supplies its cube densities and gives a tuple in $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$, so the extension map from $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$ is onto; the local expectations just shown make it continuous. Its image ${\mathcal{H}}_c$ is compact in ${\tau }_L$.

Each $K_W$ is weakly compact in real $L^1(Q_W)$ and hence weakly sequentially compact by [F3]. Given a sequence in $K_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$, successively take subsequences converging in the first, second, and later cube coordinates and diagonalize; AC supplies the countable choices. The limit tuple preserves all closed consistency, stationarity, and entropy constraints from step 3.1, so steps 4.1–5.1 give a law in ${\mathcal{H}}_c$. Count-tail uniformity from step 5.1 upgrades coordinate convergence on bounded local tests to convergence for every tame test by the same truncation estimate in step 6.1. Hence ${\mathcal{H}}_c$ is sequentially compact in ${\tau }_L$.