Let $X$ and $Y$ be Polish spaces equipped with their Borel $\sigma$-algebras, let $\mu \in \mathcal{P}(X)$ and $\nu \in \mathcal{P}(Y)$ be Borel probability measures, and let $c: X \times Y \to [0,\infty)$ be finite and lower semicontinuous. Define the Kantorovich transport value
where $\Pi(\mu,\nu)$ is the set of Borel probability measures on $X \times Y$ with first marginal $\mu$ and second marginal $\nu$.
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Assume $C_c(\mu,\nu)<\infty$ and assume there exist Borel functions $a: X \to \mathbb{R}$ and $b: Y \to \mathbb{R}$ such that $a \in L^1(X,\mu)$, $b \in L^1(Y,\nu)$, and
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\begin{align*}
c(x,y) \leq a(x)+b(y)
\end{align*}
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for every $(x,y) \in X \times Y$.
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Define $\mathcal{A}_c(\mu,\nu)$ to be the set of all pairs $(f,g)$ such that $f: X \to \mathbb{R}$ and $g: Y \to \mathbb{R}$ are finite real-valued Borel functions, $f \in L^1(X,\mu)$, $g \in L^1(Y,\nu)$, and
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\begin{align*}
f(x)+g(y) \leq c(x,y)
\end{align*}
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for every $(x,y) \in X \times Y$. Define the Kantorovich dual value over this pointwise Borel integrable admissible class by
Then there exist finite real-valued Borel functions $\varphi: X \to \mathbb{R}$ and $\psi: Y \to \mathbb{R}$ with $\varphi \in L^1(X,\mu)$ and $\psi \in L^1(Y,\nu)$ such that $(\varphi,\psi)\in\mathcal{A}_c(\mu,\nu)$ and
Equivalently, the Kantorovich dual supremum over finite real-valued Borel integrable potentials satisfying the admissibility inequality pointwise everywhere is attained.