Berg-Kesten-Reimer inequality on infinite spaces?

Question

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?

Answer 1

Check this out, I hope it helps.

The van den Berg--Kesten--Reimer operator and inequality for infinite spaces, by Arratia-Garibaldi-Hales https://arxiv.org/abs/1508.05337