Let $(X,d)$ be a Polish space. Let $$ \mu = \sum_{k=1}^\infty a_n \delta_{x_n}, $$ where $x_n \in X, a_n \in [0,1]$ and $$ \sum_{k=1}^\infty a_n =1. $$ Let $$ \nu = \sum_{k=1}^N b_k \delta_{x_k}, $$ with $b_n \in [0,1]$ and $$ \sum_{k=1}^N b_n =1. $$
Does there exits a map $T: X \to X $ such that $$ \mu \circ T^{-1} = \nu. $$
This claim comes up in the article https://www.lpsm.paris/pageperso/bolley/wasserstein.pdf, and there is an explanation there, but I am not too familiar with optimal transport language.