Let $(X,\mathcal A)$ and $(Y,\mathcal B)$ be measurable spaces. Let $\mu \in \mathcal P(X,\mathcal A)$ and $\nu \in \mathcal P(Y,\mathcal B)$ be probability measures, and let
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\begin{align*}
T:(X,\mathcal A)\to(Y,\mathcal B)
\end{align*}
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be a measurable map satisfying $T_\#\mu=\nu$. Define
and set $\pi:=S_\#\mu$. Then $\pi\in\Pi(\mu,\nu)$, meaning that the first marginal of $\pi$ is $\mu$ and the second marginal of $\pi$ is $\nu$. Moreover, for every nonnegative $\mathcal A\otimes\mathcal B$-measurable function