Convergence of triangular transformations of measures

We prove that if a Borel probability measure $\mu$ on a countable product of Souslin spaces satisfies a certain condition of atomlessness, then for every Borel probability measure $\nu$ on this product, there exists a triangular mapping $T_{\mu,\nu}$ that takes $\mu$ to $\nu$. It is shown that in the case of metrizable spaces one can choose triangular mappings in such a way that convergence in variation of measures $\mu_n$ to $\mu$ and of measures $\nu_n$ to $\nu$ implies convergence of the mappings $T_{\mu_n,\nu_n}$ to $T_{\mu,\nu}$ in measure $\mu$.