Holley inequality

On a finite distributive configuration lattice, probability weights satisfying ${\mu }_1(\omega \vee \eta ){\mu }_2(\omega \wedge \eta )\ge {\mu }_1(\omega ){\mu }_2(\eta )$ for all pairs give stochastic domination ${\mu }_1\ge {\mu }_2$. The result holds for nonnegative weights as well: apply the four functions theorem to ${\mu }_1{1}_{A^c},{\mu }_2{1}_A,{\mu }_1{1}_A,{\mu }_2{1}_{A^c}$ for an increasing event $A$, obtaining ${\mu }_1(A^c){\mu }_2(A)\le {\mu }_1(A){\mu }_2(A^c)$.

The cross-lattice sufficient condition for stochastic comparison of two measures:The join and meet are coordinatewise maximum and minimum on a Boolean configuration cube. Normalization factors cancel, so the condition can be checked using unnormalized positive weights.