A note on positive association

Abstract

We show that if $\mathcal{A},\mathcal{B},\mathcal{C}$ are increasing subsets of $\Omega :=\{0,1{\}}^n$ with $\mathcal{A}\ne \varnothing$, then with respect to any product probability measure on $\Omega$, \[ \mbox{if each of the pairs $\{\mathcal{A}\cap \mathcal{B},\mathcal{C}\}$, $\{\mathcal{A}\cap \mathcal{C},\mathcal{A}\}$ is independent, then $\mathcal{B}$ and $\mathcal{C}$ are independent.} \] This implies an answer to a motivating question of J. Steif, and is related to a basic, still open variant of that question, and to a well-known conjecture of S. Sahi.