Geodesics and generalized geodesics

Statement

Theorem.(Benamow-Brenier) Let $\mu _{n}$ be a sequence of tight probability measures on Polish space $X$. Then, there exists $\mu \in P(X)$ and convergent subsequence $\mu _{n_{k}}$ such that $\mu _{n_{k}}\rightarrow \mu$ in the dual of $C_{b}(X)$. Conversely, every sequence $\mu _{n}\rightarrow \mu$ is tight.