On the book Probability and Potentials of Paul Mayer page 69 one reads:
I'm not sure how to use (1) and (2) to prove that the collection of processes is closed under sequential pointwise convergence
Attempt: Let $Y_n$ be a sequence of such processes, such that $Y_n(t,\omega) \to Z(t, \omega)$ for every $(t, \omega)\in \Bbb{R}_+ \times \Omega$. By (2) $$Z^{-1}(A) = \bigcup_{n \geq 0} \bigcap_{m>n} Y_n^{-1}(A)$$ we conclude that $Z$ is measurable and, by (1), takes values in the closure of the separable subspace of $M$ wich is separable.
