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

For a given positive integer $n$ and for every $i\in[n]$, denote by $\mu_i$ a probability measure on a finite set $\Omega_i$. Call $\mu$ and $\Omega$ the products $\mu_1\!\times\! \ldots\!\times\!\mu_n$ and $\Omega_1\!\times\!\ldots\!\times\!\Omega_n$ respectively.

For an event $A\!\subset\!\Omega$ and an index subset $\sigma\!\subset\![n]$, let
$$
A_\sigma=\lbrace\omega\!\in\!A\!:\,\forall\psi\!\in\!\Omega,\,(\forall i\!\in\!\sigma,\,\psi_i\!=\!\omega_i)\!\implies\!\psi\!\in\!A\rbrace
$$
Hence, *the occurrence of $A_\sigma$ is solely controlled by $\sigma$*.
For all events $A$ and $B$ of $\Omega$, *the disjoint occurrence of $A$ and $B$* is described by

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

The *Van Den Berg-Kesten-Reimer* inequality states that for all events $A$ and $B$ of $\Omega$,

$$ \boxed{\mu(A\circ B)\le\mu(A)\cdot\mu(B)} $$

**Question**

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