**Statement of Van Den Berg-Kesten-Reimer inequality:**

Let $n$ be a positive integer. For $i\in[n]$, let $\mu_i$ be a probability measure on a finite set $\Omega_i$. Let $\Omega=\Omega_1\times\ldots\times\Omega_n$ and $\mu=\mu_1\times \ldots\times\mu_n$.

For an event $A\subset\Omega$, and an index subset $\Sigma\subset[n]$, let $A_{|\Sigma}=\lbrace\omega\in A:\forall\psi\in\Omega,(\forall i\in\Sigma,\psi_i=\omega_i)\implies\psi\in A\rbrace$. This means that, as an event, the occurrence of $A_{|\Sigma}$ is *controlled by the random variables indexed by* $\Sigma$ *only*.

For any two events $A,B\subset \Omega$, let $A\circ B=\lbrace\omega\in A\cap B:\exists\Sigma,\Lambda\subset[n],(\Sigma\cap\Lambda=\emptyset)\wedge(\omega\in A_{|\Sigma}\cap B_{|\Lambda})\rbrace$. $A\circ B$ describes the *disjoint occurrence* of $A$ and $B$.

The Van Den Berg-Kesten-Reimer inequality states that for all events $A,B\subset\Omega$, $\mu(A\circ B)\le\mu(A)\cdot\mu(B)$.

**Question:** What (non-trivial) events turn the inequality into an equality?