In this MO question it is said that:
On a complete, non necessarily separable metric space $E$, the set $P_r(E)$ of all Radon probability measures with the Wasserstein-Kantorovich metric $W_d$ is complete;
and that the above is a known fact. However I can't seem to find a published reference. Could someone please point me to one? I need in particular a reference of the result above where separability is not required.