It is well-known that in optimal transport, Kantorovich problem is a relaxation of the Monge problem, which always admits a solution. Let's write $$ M(\mu,\nu)=\inf_{T: T_\#\mu=\nu} \int c(x,T(x))d\mu $$ $$K(\mu,\nu)=\inf_{\pi: \pi \; \text{has marginals}\;\mu\;\text{and}\;\nu} \int c(x,y)d\pi $$
I am interested in understanding what happens when the optimal value functions of these two problem are the same: $M(\mu,\nu)=K(\mu,\nu)<\infty$. In this case, does the Monge problem admit a solution? If not, what would be a counterexample? In addition, when the Monge problem has a solution, is it necessarily true that $M(\mu,\nu)=K(\mu,\nu)<\infty$?
Thanks in advance!
I have found a paper with the example I am looking for: On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation by Aldo Pratelli