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 $d$



Kantorovich duality with pseudometrics
Yes the Kantorovich Duality holds for continuous cost functions by following the proof in Villani's book without any change. The proof for general cost functions needs compactness of the set of couplings between the marginals which follows from Prokhorov's Theorem but might be true in more general settings.


Being Polish (=complete separable metrizable space) is usual used for the existence of optimal couplings, e.g. Prokhorov's Theorem and the Portmanteau Lemma are the foundations. So you might want check whether the topological properties of your space allow you to obtain the existence of optimal couplings. Note a pseudometric space can be Polish as well.


