= 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 $t\ge0$, 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 $\operatorname{Cov}_\mu(f,g)$, proving <positive association of random variables>.
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