Here is the question:
Let $(\Omega, \mathcal{F}, \mathrm{P})$ be a probability space. Let $T$ be an arbitrary (possibly uncountable) index set, and $\forall t\in T$, let $X_t:(\Omega,\mathcal{F})\rightarrow (E_t, \mathcal{E}_t)$ be a random variable with state space $(E_t, \mathcal{E}_t)$. Then a function $F:\Omega\rightarrow \mathrm{R}$ is measurable with respect to $\sigma({\{X_t\mid t \in T\}})$ if and only if there is a sequence $(t_n)_{n\in N}\subset T$ and a function $g:\times_{n\in \mathrm{N}} E_{t_n} \rightarrow \mathrm{R} $ that is $(\bigotimes_{n\in\mathrm{N}}\mathcal{E}_{t_n})$- measurable (i.e., whith respect to the product $\sigma$-algebra of $(E_{t_n}\mid n\in \mathrm{N})$) such that $F=g(X_{t_1}, X_{t_2},\dots)$.
Any thoughts on how to attack this (one direction is straightforward, but the other???)?. Any help would be much appreciated! Thanks in advance...