Let $X = (X_t,t\in[0,T])$ be a bounded stochastic process, which is stochastically continuous: for any $t_0\in[0,T]$, $X_t\overset{\mathrm{P}}{\longrightarrow} X_{t_0}$, $t\to t_0$. Does the process $X$ have a Riemann integrable modification?
I tried to the following. It is known that $X$ has both a modification, which is both measurable and separable with respect to the family of closed sets. This raises the following purely analytical question:
Let a bounded measurable $f\colon [0,T] \to \mathbb R$ be "separable" in the following sense. There exists a countable dense set $S\subset [0,T]$ such that for any closed $F\subset \mathbb R$ and any $(a,b) \subset [0,T]$, $$\big (\forall t\in(a,b)\cap S \ f(t)\in F\big) \Rightarrow (\forall t\in(a,b)\ f(t)\in F\big).$$ Can we say that $f$ is Riemann inntegrable?