Process law, modification, and indistinguishability

Two processes with the same index set and corresponding target spaces have the same process law if all corresponding finite-dimensional laws agree; their source probability spaces need not be the same. Suppose in addition that $X=(X_t{)}_{t\in I}$ and $Y=(Y_t{)}_{t\in I}$ are defined on one common probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, take values in the same measurable space at each $t$, and $\{X_t=Y_t\}$ is measurable for each $t$. Then $Y$ is a modification (or version) of $X$ when $\mathbb{P}(X_t=Y_t)=1$ for every fixed $t$. They are indistinguishable when the event $\{\omega :X_t(\omega )=Y_t(\omega )\text{ for every }t\in I\}$ is measurable and has probability one. The latter has one exceptional null event; the former may have a different null event at every time.