Almost-sure normalized log posterior odds

Let $S$ specify a positive finite number of parties, a positive finite number of settings for each party, and a positive finite number of outcomes for every party and setting. Write $A$ for the set of joint settings, $X_a$ for the joint outcomes at setting $a$, $V$ for the set of assignments $v$ prescribing an outcome for every party at every one of its settings, and $v|_a\in X_a$ for the outcomes prescribed at $a$. Let $Q(a,x)$ be a real table satisfying these hypotheses: each $Q(a,\cdot)$ is nonnegative and sums to $1$; for every subset $J$ of parties, settings $a,b$ agreeing on $J$, and $v\in V$, the sums of $Q(a,x)$ over outcomes satisfying $x_i=v_i(a_i)$ for $i\in J$ and of $Q(b,x)$ over outcomes satisfying $x_i=v_i(b_i)$ for $i\in J$ are equal; and there exist positive local dimensions $d_i$, a positive semidefinite matrix $\rho$ of trace $1$ on the joint index set $\prod_i\{0,\ldots,d_i-1\}$, and positive semidefinite matrices $M_{i,a_i,x_i}$, summing to the identity over $x_i$ for each $i,a_i$, such that $Q(a,x)=\operatorname{Re}\operatorname{tr}(E_{a,x}\rho)$, where $E_{a,x}(u,v)=\prod_i M_{i,a_i,x_i}(u_i,v_i)$. Let $s$ be any nonnegative distribution of total mass $1$ on $A$, allowing zero coordinates. For a distribution $p$ on $V$, set $P_p(a,x)=\sum_{v:\,v|_a=x}p(v)$, and on the trial set $B=\{(a,x):a\in A,\ x\in X_a\}$ set $q(a,x)=s(a)Q(a,x)$ and $r_p(a,x)=s(a)P_p(a,x)$. Define $D(q\|r)=+\infty$ if some trial has $q>0$ and $r=0$, and otherwise define it as $\max\{0,\sum_{b:q(b)\ne0}q(b)\log_2(q(b)/r(b))\}$, viewed in $[0,+\infty]$, and put $\Delta=\inf_{p\in\mathcal P(V)}D(q\|r_p)$, where $\mathcal P(V)$ is the simplex of distributions on $V$. Let $\nu$ be Hausdorff measure of dimension $|V|-1$, restricted to this simplex, and let $w:\mathbb R^V\to\mathbb R$ be continuous on the simplex, satisfy $c<w(p)$ there for some real $c>0$, and satisfy $\int w\,d\nu=1$. Finally let $h\in\mathbb R$ satisfy $0<h<1$. For a finite trial sample $t$, define $L_q(t)=\prod_iq(t_i)$, $m(t)=\int w(p)\prod_i r_p(t_i)\,d\nu(p)$, and $O_h(t)=\frac{h}{1-h}L_q(t)/m(t)$; let $\mathbb P_q$ be the infinite product of the trial measure $\sum_{b\in B}\max\{q(b),0\}\delta_b$. The conclusion is that $w$ is integrable with respect to $\nu$, $\mathbb P_q$ has total mass $1$, $\Delta<+\infty$, and for $\mathbb P_q$-almost every infinite trial sequence $z$,

where $z_{<n}=(z_0,\ldots,z_{n-1})$ and $\Delta_{\mathbb R}$ is the real value of the finite extended nonnegative number $\Delta$. The quantified sample lengths include $n=0$, where both likelihoods are $1$; real division by zero makes the displayed sequence's term at $n=0$ equal to $0$.