Separable stochastic process

A separable stochastic process admits a fixed countable set of indices that captures the relevant sample path behavior outside one common null set. In the usual real-valued setting on a metric space $T$, one formulation requires a countable set that is a dense subset $D\subseteq T$ and a null set $N$ such that, whenever $\omega \notin N$, each $(t,X_t(\omega ))$ lies in the closure of $\{(s,X_s(\omega )):s\in D\}$. Thus $t$ and the corresponding process value can be approximated together from $D$.

In particular, ${\sup }_{t\in T}{X}_t={\sup }_{s\in D}{X}_s$ outside $N$. The right side is a supremum over a countable set of random variables, so its measurability is accessible through the usual sigma-algebra operations. Continuous sample paths on $[0,1]$ have this property using the rational numbers as indices and endpoints.

The common null set matters. If $U$ has a uniform distribution on $[0,1]$ and $X_t=1_{\{U=t\}}$, each fixed $X_t$ is zero almost surely, yet ${\sup }_t{X}_t=1$ always. For any fixed countable $D$, ${\sup }_{t\in D}{X}_t=0$ almost surely. Agreement at every fixed time therefore does not imply agreement of pathwise suprema.^[ref]