Suppose $T$ is a a stopping time and $X$ is a martingale. We know that this means the stopped martingale $X_{T \wedge n}$ is also a martingale.
Furthermore this means that: $$E[X_{T \wedge n}] = E[X_{0}] $$ for all $n$.
My question is: is there e a difference between $T$ being finite i.e $Pr (T< \infty)$$=1$ and $T$ being bounded above i.e $T$ ($ω$)$ < c$ for any $ω \in\Omega$ in context of this equality: $$E[X_{T}] = E[X_{0}] $$
If so, can you provide some intuition/examples?
