[See this link](https://mathoverflow.net/questions/97574/van-den-berg-kesten-reimer-inequality) 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](http://www.jstor.org/stable/3213860?seq=1#page_scan_tab_contents), 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?



Berg-Kesten-Reimer inequality on infinite spaces?
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>


