I need a check on the following reasoning, that happens everytime I do an exercise about the stopping theorem.
Let $\tau$ be a stopping time w.r.t some filtration
Usual in the exercises about the optional stopping theorem I read:
$$ t \wedge \tau \uparrow \tau$$ And that's true if $\tau$ is a.s. finite.
BUT if $\tau$ not a.s. finite, can I still say $t \wedge \tau \uparrow \tau$?
I can note that if $\tau = \infty$ a.s., then for every $ t \in \mathbb{R}$ I have $t \wedge \tau =t$ and hence taking the limit I just can say that $$\lim_t t \wedge \tau = \infty$$ (which is a.s. the value of $\tau$)
All in all, I would say that it's true that $t \wedge \tau \uparrow \tau$ also if $\tau$ is NOT a.s. finite
Is it correct or am I missing something?