 Stochastic domination of probability measures

On a finite partially ordered state space, $\mu ⪯\nu$ means $\int fd\mu \le \int fd\nu$ for every real-valued order-preserving function $f$. Equivalently, every increasing subset has no larger probability under $\mu$ than under $\nu$. A coupling supported on pairs $(X,Y)$ with $X\le Y$ proves this relation. This concerns order of probability laws, rather than stochastic asymptotic order notation.