The problem of optimal transport (Monge-Kantorovich problem) in Kantorovich formulation looks as follows:
Let there be a metric space $(E, D)$. And we have to transort a certain amount of mass from the initial distribution, to the final distribution of points. And there is cost function associated with the transport: $$ C(x, y) : E \times E \rightarrow R. $$ And $\mu, \nu \in P_2(E)$ are measures to be transported. So we have to find a map $\pi(x, y) \in P(E \times E)$, such that minizes the cost function: $$ \mathrm{inf} \int_{E \times E} C(x, y) \ d \pi(x, y) $$ With the constrains: $$ \int_E \pi(x, y) dy = \mu(x) \qquad \int_E \pi(x, y) dx = \nu(y) $$
And my question is, what does the $d \pi (x, y)$ denote in the present case? Is it some measure on a functional space? Integration overall all possible mappings, subject to a certain constrains?
I would be grateful for comments and suggestions