Compactness of the space of measure-preserving maps

Question

Let $X, Y$ be Polish spaces and $\mu, \nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $\mu$ is non-atomic. Denote by $\mathcal{M}(\mu,\nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $\mu(T^{-1}(B))=\nu(B)$). Is the set $\mathcal{M}(\mu,\nu)$ compact with respect to the convergence of $\mu$-measure?