\begin{align*}
\mathsf K_c(\mu,\nu):=\inf\left\{\int_{X\times Y} c(x,y)\,d\gamma(x,y): \gamma\in\Pi(\mu,\nu) \text{ and the extended integral is well-defined}\right\}.
\end{align*}
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Define the Monge admissible class
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\begin{align*}
\mathcal M_c:=\{S:X\to Y: S \text{ is }\mathcal A/\mathcal B\text{-measurable},\ S_{\#}\mu=\nu,\ \int_X c(x,S(x))\,d\mu(x) \text{ is well-defined}\}.
\end{align*}