= 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 $X_t=\mathbf 1_{\{U=t\}}$, each fixed $X_t$ is zero <almost surely>, yet $\sup_tX_t=1$ 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.<Measurability Woes by David Pollard>[Section 10.4, separable versions and the random-spike counterexample]{ref}
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