The van den Berg--Kesten--Reimer operator and inequality for infinite spaces

TL;DR

This paper extends the BKR inequality to infinite spaces and multiple events, removing previous finiteness assumptions and addressing related set-theoretic issues.

Contribution

It generalizes the BKR operator and inequality to infinite product spaces and multiple events, broadening their applicability.

Findings

Extended BKR inequality to infinite product spaces

Removed finiteness constraints on the dimension

Addressed set-theoretic issues in the extension

Abstract

We remove the hypothesis "$S$ is finite" from the BKR inequality for product measures on $S^d$, which raises some issues related to descriptive set theory. We also discuss the extension of the BKR operator and inequality, from 2 events to 2 or more events, and we remove, in one sense, the hypothesis that $d$ be finite.