Let $(X,\mathcal A,\mu)$ and $(Y,\mathcal B,\nu)$ be probability spaces, and let $T:(X,\mathcal A)\to (Y,\mathcal B)$ be a measurable map. Then $T$ is a transport map from $\mu$ to $\nu$, meaning $T_{\#}\mu=\nu$, if and only if, for every measurable set $B\in\mathcal B$,