 Exponential-tilt proof of positive association

For a strictly positive law $\mu$ satisfying the FKG lattice condition on a finite Boolean lattice, and an increasing function $g$, tilt to ${\mu }_t(\omega )\propto e^{tg(\omega )}\mu (\omega )$. For $t\ge 0$, the lattice inequality and $g(\omega \vee \eta )\ge g(\omega )$ give the Holley condition for ${\mu }_t{\ge }_{\mathrm{st}}\mu$. For any increasing $f$, differentiate ${\mathbb{E}}_{{\mu }_t}f\ge {\mathbb{E}}_{\mu }f$ at $t=0$ from the right. The derivative is ${\mathrm{Cov}}_{\mu }(f,g)$, proving positive association of random variables.