Remarks on convergence of random processes in non-separable metric spaces and on the non-existence of a Borel measure for processes in $C(0,\infty)$

Let a random element $\xi$ and random elements $\xi_n$ take values in a metric space $X$. Let $f$ be a measure and continuous functional on $X$. We discuss pecularities connected with convergence of the distributions of $f(\xi_n)$ to the distribution of $f(\xi)$ when the space $X$ is a non-separable one.