There is this problem on my sheet:
Let $X = (X)_{n≥0}$ be a martingale and $T$ be a finite stopping time. Suppose $X_T$ is integrable. Show that $$E [X_T] = E [X_0]$$ if and only if $$\lim_{n→∞} E[X_n : T > n] = 0$$
I have a proof of this which boils down to using DCT and the fact the $X_{T ∧n}$ tends to $X_T$ as $n → ∞$
However, we have that a consequence from Doob's optional stopping Theorem that if $S,T$ are $2$ bounded stopping times, then $E[X_S]=E[X_T]$.
Doesn't this imply that $\lim_{n→∞} E[X_n : T > n] = 0$ for every martingale which is basically uniform integrability?
I don't think it implies that, because you're given just that $T$ is a finite stopping time, not that $T$ is a bounded stopping time.
If $T$ is a bounded stopping time then for large enough $n$ the event $T>n$ is empty. I have no idea what the definition of $E[X:\emptyset]$ might be.