# Kantorovich duality with pseudometrics

The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that generalize the setting to pseudometric spaces? Or is this impossible? I will now specify the setting I am mainly interested in: For a Polish space $$X$$ and a metric $$d$$ on $$X$$ the cost $$T_d$$ of optimal transportation with (continuous) cost function $$c(x,y)=d(x,y)$$ is given by \begin{align*} T_d(\mu,\nu)=\inf_{\pi \in \Pi(\mu,\nu)} \int_{X \times X} d(x,y) d\pi(x,y). \end{align*}, where the $$\inf$$ is running over all joint distributions with marginals $$\mu$$ and $$\nu$$. Let $$Lip(X)$$ denote the set of all Lipschitz functions on $$X$$, and \begin{align} \| \varphi \|_{Lip}=\sup_{x\ne y} \frac{|\varphi(x)-\varphi(y)|}{d(x,y)}. \end{align} Then \begin{align} T_d(\mu,\nu)=\sup \{ \int_X \phi d(\mu-\nu) | \quad \phi \in L^1(d|\mu-\nu|), \| \varphi \|_{Lip}\le 1 \}. \end{align} My question now would be, if this conclusion also holds, if $$d$$ is a pseudometric instead of a metric. Mainly: Is it possible to draw immediate conclusions that follow from the proof in the metric case, or is this dependent on the topology induced by $$d$$

• Also the application of Prokhorov's Theorem works without any problems in this setting? Villani considers Polish spaces in his book, but in my setting, as mentioned before, the ground space $X$ is now equipped with a pseudometric. In other words: as long as the cost function is lower semicontinuous, the generalization to pseudometric spaces instead of Polish spaces is straightforward? Did I understand this correctly? In any case, thanks a lot for helping me! – Markus Holzleitner Dec 19 '18 at 14:59