I am confused about the definition of $\mathcal{F}_T$, where $T$ is a stopping time. From three different books we find two different definitions:
(Karatzas and Shreve; Protter) Events $A \in \mathcal{F}$ for which $A \cap \{T \leq t\} \in \mathcal{F}_t$ for all $t \geq 0$.
(Revuz and Yor) Events $A \in \mathcal{F}_\infty$ such that $A \cap \{T \leq t\} \in \mathcal{F}_t$ for all $t$.
All the books claim that if $T=t$ is a constant stopping time then $\mathcal{F}_T = \mathcal{F}_t$, and if $S \leq T$ then $\mathcal{F}_S \subseteq \mathcal{F}_T$.
From these facts one should have $\mathcal{F}_T \subseteq \mathcal{F}_\infty$ for any stopping time $T$. This makes me wonder why the first two books make a point to use $\mathcal{F}$ instead of $\mathcal{F}_\infty$ (none of the books assume $\mathcal{F} = \mathcal{F}_\infty$). They also write $t \geq 0$ and I wonder if this includes $t=\infty$. If it did not include $t=\infty$ then the most we could say is that $A \cap \{T < \infty\} \in \mathcal{F}_\infty$, not $A \in \mathcal{F}_\infty$.
I conclude that the top definition must include $t=\infty$, so the two definitions agree. I just wanted to make sure I'm not missing something. Is my reasoning correct or have I missed something?
Let me write, for a stopping time $T$, $\mathcal F_T^*$ for "the history up to time $T$" in the Karatzas-Shreve-Protter sense, and $\mathcal F_T^\#$ for "the history up to time $T$" in the Revuz-Yor sense. Clearly $\mathcal F^*_t=\mathcal F^\#_t$ for each fixed time $t\in[0,\infty)$. But $\mathcal F^*_\infty=\mathcal F$ while $\mathcal F^\#_\infty=\mathcal F_\infty$ ($=\vee_t\mathcal F_t$), and here is where the difference lies. More generally, you can check that $\mathcal F^*_T\cap\{T<\infty\}=\mathcal F^\#_T\cap\{T<\infty\}$, but $\mathcal F^*_T\cap\{T=\infty\}=\mathcal F\cap\{T=\infty\}$ while $\mathcal F^\#_T\cap\{T=\infty\}=\mathcal F_\infty\cap\{T=\infty\}$.
In particular, $\mathcal F^*_T\subset\mathcal F^*_\infty$ is true, but $\mathcal F^*_T\subset\mathcal F_\infty$ may fail to hold.