**Van Den Berg-Kesten-Reimer inequality**


Let $n$ be a positive integer. For $i\in[n]$, let $\Omega\_i$ be a finite set and $\mu\_i$ a probability measure on it. Set $\Omega=\Omega\_1\!\times\!\ldots\!\times\!\Omega\_n$ and $\mu=\mu\_1\!\times\!\ldots\!\times\!\mu\_n$.
For $A\!\subset\!\Omega$ and $\sigma\!\subset\![n]$, let
$$
A\_\sigma=\lbrace\omega\!\in\!A\!:\,\forall\upsilon\!\in\!\Omega,\,(\forall i\!\in\!\sigma,\,\upsilon\_i\!=\!\omega\_i)\!\implies\!\upsilon\!\in\!A\rbrace
$$
In other terms, *the occurrence of the event $A\_\sigma$ is solely controlled by $\sigma$*.
Given two events $A,B\subset\Omega$, the notion of *disjoint occurrence of $A$ and $B$* is defined by the following event  

$$
A\!\circ\!B=\lbrace\omega\!\in\!A\!\cap\!B\!:\,\exists\,\sigma,\tau\!\subset\![n],\,\sigma\!\cap\!\tau\!=\!\emptyset \wedge\omega\!\in\!A\_\sigma\!\cap\!B\_\tau\rbrace
$$ 


*Van Den Berg-Kesten-Reimer* inequality states that 


$$
\forall A,B\!\subset\!\Omega,\,\,\mu(A\circ B)\le\mu(A)\cdot\mu(B)
$$


**Question** 


Are there non-trivial events that turn the inequality stated above into an equality?



Van Den Berg-Kesten-Reimer inequality
In the case of *increasing* events, the standard proof of the BK inequality works like this. Start with your set $\Omega$, and successively replace each $\Omega\_i$ by a disjoint union of two copies $\Omega\_i^1$ and $\Omega\_i^2$. After step $i$, the event ${(A\circ{B})}(i)$ is defined by saying that $A$ and $B$ need to occur disjointly on $\Omega\_{i+1}$ up to $\Omega\_n$, and $A$ (resp. $B$) is allowed to use the $\Omega\_k^1$ (resp. $\Omega\_k^2$).


Then, $(A\circ B)(0)$ is just disjoint occurrence, and $(A\circ B)(n)$ is the occurrence of two independent events, its probability is $P(A)P(B)$. Besides, for every $i$, we relax the restrictions by allowing the events to be less and less disjoint - hence the BK inequality.


If there exists an $i$ that is pivotal for both $A$ and $B$ with positive probability, then when you split the corresponding $\Omega\_i$, the inequality between the probabilities of $(A\circ B)(i-1)$ and $(A\circ B)(i)$ is strict, and so is BK. In other words, BK is an equality if and only if you can partition $[n]$ into two subsets, $A$ depending only on the first one and $B$ on the second one. 


In yet other words, in the case of increasing events, there is no non-trivial (in the sense, where disjoint occurrence is not automatic) pair of events realizing the equality.


Without monotonicity, the argument does not work directly ...


