See this link for a description of the van den Berg-Kesten-Reimer inequality. How important is the assumption that $\Omega_i$ are finite spaces?

When Berg-Kesten state the inequality in their 1985 paper, they state it for product measures on $\mathbb{R}$, but only for increasing events.

Does Reimer's general proof work for all events work without too much effort on infinite spaces with product measure?