given polish space $(X,d)$, consider weak\* topology of probability. optimal transport of probability $u,v$ is defined by $\pi(u,v)$ such that $\pi(u,v)$ minimizes:


$\{\int d(x,y) d \pi(dx,dy): \pi \text{ is the couple of } u,v\}$


if $u\_k \to u, v\_k \to v$, can we show $\pi(u\_k, v\_k) \to \pi(u,v)$?


and how is the measurability w.r.t. the $\sigma$-alg generated by weak\* top? 



continuity/ measurablity of optimal transport
Of course not! This is hopeless if $X$ has unbounded diameter. 


As a counterexample, consider $X=\mathbb Z$ with the standard metric; $\mu\_k=(1-\frac 1k)\delta\_0+\frac 1k\delta\_k$; and $\nu\_k=\nu=\mu=\delta\_0$. Then $\nu\_k\to\nu$, $\mu\_k\to \mu$, but $\pi(\mu\_k,\nu\_k)=1$ for each $k$. 


