# All questions with tag [math: almost-everywhere]

1

## When $X_s<Y_t$ almost surely, for every $s<t$, implies that $X_s<Y_t$ for every $s<t$, almost surely?

When I have shown, for $s\le t$ and for two continuous stochastic process an inequality:
$$ X_s \le Y_t$$ P-a.s.
How can I deduce that this P-a.s. simultaneously for all rational $s\le t$ ?
Thank you for your help
EDIT: According to Ilya's answer, I see that we have $$P(X_s\le Y_t\text{ simultaneously for all rationals }s\le t) = 1.$$
How could...

2022-07-25 12:53:42