Let $\Omega_1, \Omega_2, \ldots, \Omega_n$ be probability spaces, each of finitely many elements. The inequality applies to spaces of the form $\Omega = \Omega_1 \times \Omega_2 \times \cdots \times \Omega_n$, equipped with the product measure, so that each element $x = (x_1, \ldots, x_n) \in \Omega$ is given the probability
$$\mathbb P({x}) = \mathbb P_1({x_1}) \cdots \mathbb P_n({x_n})$$
For two events $A, B\subseteq \Omega$, their ''disjoint occurrence'' $A \mathbin{\square} B$ is defined as the event consisting of configurations $x$ whose memberships in $A$ and in $B$ can be verified on disjoint subsets of indices. Formally, $x \in A \mathbin{\square} B$ if there exist subsets $I, J \subseteq [n]$ such that:
-
$I \cap J = \emptyset$,
- for all $y$ that agrees with $x$ on $I$ (in other words, $y_i = x_i\ \forall i \in I$), $y$ is also in $A,$ and
- similarly every $z$ that agrees with $x$ on $J$ is in $B$.
The inequality asserts that $\mathbb P (A \mathbin{\square} B) \le \mathbb P (A) \mathbb P (B)$ for every pair of events $A$ and $B$.
We only need it in the case where $A$ and $B$ are upper events, which is the Van den Berg-Kesten inequality, but it should be possible to go straight for the general case following Reimer's original paper.
