 Disjoint occurrence of increasing events

A finite open coordinate set witnesses an increasing event when prescribing those coordinates open forces the event, regardless of the remaining configuration. The disjoint occurrence of increasing events $A\square B$ means that $A$ and $B$ have disjoint finite witnesses. For finite-coordinate events this is the usual disjoint-occurrence definition. The van den Berg-Kesten inequality bounds its probability by $\mathbb{P}(A)\mathbb{P}(B)$ under an independent Bernoulli distribution product measure.