Let $X$ and $Y$ be Polish spaces, let $\mu \in \mathcal{P}(X)$ and $\nu \in \mathcal{P}(Y)$, and let $c: X \times Y \to (-\infty,\infty]$ be lower semicontinuous and bounded below. Define the feasible set of transport plans by $\Pi(\mu,\nu) := \{\pi \in \mathcal{P}(X \times Y) : (\operatorname{pr}_X)_{\#}\pi = \mu \text{ and } (\operatorname{pr}_Y)_{\#}\pi = \nu\}$. If

\begin{align*} \inf_{\gamma \in \Pi(\mu,\nu)} \int_{X \times Y} c(x,y)\, d\gamma(x,y) < \infty, \end{align*}

then there exists $\pi \in \Pi(\mu,\nu)$ such that

\begin{align*} \int_{X \times Y} c(x,y)\, d\pi(x,y) = \inf_{\gamma \in \Pi(\mu,\nu)} \int_{X \times Y} c(x,y)\, d\gamma(x,y). \end{align*}