Let $\{X_t\}_{t\in[0,T]}$ be a stochastic process and $(\mathcal{F}_t)_{t\in[0,T]}$ its natural filtration. Show that for any $\mathcal{F}_1$-measurable r.v. $Y$, there exists and countable $S\subset[0,1]$ such that $Y(\omega)=Y(\omega')$ as soon as $X_t(\omega)=X_t(\omega')$, $\forall t\in S$.
I am really stumped and confused by this question, so any help is most appreciated. Thank you in advance.
By this thread, there exists a countable subset $S$ of $[0,1]$ such that $\sigma(Y)\subset \sigma\left(X_s,s\in S\right)$. Write $S=\{s_n, n\in\mathbb N\}$. By the Doob-Dynkin lemma, there exists a function $f\colon\mathbb R^{\mathbb N}\to\mathbb R$ such that $Y=f\left(X_{s_n}\right)$. If $X_{s_n}(\omega) =X_{s_n}(\omega')$ then $Y(\omega)=Y'(\omega')$.