Assume you have a probability space $(\Omega,\mathcal{F},P)$, and you have a filtration $\{\mathcal{F}_t\}$ and a stopping time $\tau$. Then all the books I have seen define the stopping time sigma-algebra as:
$\mathcal{F}_\tau=\{A \in \mathcal{F}: A\cap\{\tau\le t\}\in \mathcal{F}_t \forall {t}\ge 0\}$. But stopping-times can also take the value $\infty$. Does this mean that it is also assumed that If $A\in \mathcal{F}_\tau$, then $A\cap\{\tau\le \infty\}=A\in \mathcal{F}_\infty$. Where $\mathcal{F}_\infty=\sigma(\cup\mathcal{F}_t)$?
The reason I am asking is that a proof I am reading for the optional sampling theorem for uniformly integrable martingales, seems to rely on this, but very subtly. But earlier the author defines the stopping time sigma-algebra as I did above, and then it is a little unclear. Do we need to have this to hold for the optional sampling theorem for uniformly integrable martingales to hold?, or does the theorem also hold if we do not require what I wrote above to hold for the case $\infty$?