Suppose we have two stochastic processes $X$ and $Y$ such that for each $t\ge0$, $P(X(t)=Y(t))=1$
I read often something like: by "countable additivity", we can find a countable set $T \subset \mathbb R^+$ such that $P(X(t)=Y(t), \forall t \in T)=1$
How do we construct such a $T$ ? Where does the countable additivity of $P$ play a role ?
The result holds for any countable set $T$. For $t_i \in T$, let $A_i$ be the event $X(t_i)=Y(t_i)$. This event occurs almost surely. By countable additivity, the intersection of a countable number of almost sure events is almost sure. Your event, $X(t)=Y(t), \forall t \in T$ is the intersection of the $A_i$, so has probability one.