I have a question about the following:
Given two Polish spaces $X$ and $Y$ with probability measures $\mu$ and $\nu$ defined on their Borel-$\sigma$-algebras, respectively.
Why is there no possibility to endow the set $T(\mu,\nu):=\{T:X\rightarrow Y\text{ measurable }; T_\#\mu = \nu\}$ with a topology such that $T(\mu,\nu)$ is sequentially closed and compact, where $T_\#\mu(B) := \mu(T^{-1}(B))$ with $B\subseteq Y$ measurable?
Thanks for every idea, hint or explanation!