Local lemma for an infinite index set, under continuity from above

Let $L$ be a complete lattice and $R : L \to \mathbb{R}$ a valuation (nonnegative, monotone, modular, $R(\top) = 1$, $R(\bot) = 0$). Let $(X_i)_{i \in I}$ be a family in $L$ indexed by an arbitrary set $I$, and let $\Gamma(i) \subseteq I$ be finite sets forming a dependency graph: for every $i$ and every finite $S \subseteq I$ with $i \notin S$ and $S \cap \Gamma(i) = \emptyset$, $R\big(X_i \wedge \bigwedge_{j \in S} X_j\big) = R(X_i)\,R\big(\bigwedge_{j \in S} X_j\big)$. Let $0 \le y_i < 1$ with $R(X_i) \ge 1 - y_i \prod_{j \in \Gamma(i)} (1 - y_j)$ for every $i$, and let $c$ be a real number with $c \le \prod_{j \in S}(1 - y_j)$ for every finite $S \subseteq I$. Assume $R$ is continuous from above in the following sense: whenever $b \le R\big(\bigwedge_{j \in S} X_j\big)$ for every finite $S \subseteq I$, also $b \le R\big(\bigwedge_{i \in I} X_i\big)$.

Then

This extends Theorem 14 of Ambainis, Kempe and Sattath to infinite families. The continuity hypothesis is exactly what is needed: it holds for a normal trace, but fails for relative dimension in infinite dimension, where nonzero subspaces can have all finite intersections nonzero and zero total intersection.