A note on positive association

We show that if $\mathcal{A},\mathcal{B},\mathcal{C}$ are increasing subsets of $\mathrm{\Omega }:=\{0,1{\}}^n$ with $\mathcal{A}\mathrm{\ne }\mathrm{\varnothing }$, then with respect to any product probability measure on $\mathrm{\Omega }$, 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.