Definition of $\sigma$-algebra $\mathcal{F}_\tau$ with $\tau$ a stopping time If $\tau$ is a stopping time and $(\mathcal{F_t})_{t\in I}\subset \mathcal{F}$ is a filtration, then the $\sigma$-algebra of the $\tau$-past is defined as
$$\mathcal{F}_\tau := \{A\in\mathcal{F} : A\cap \{\tau \leq t\} \in \mathcal{F}_t \text{ for any } t \in I\}\, .$$
What is the reasoning behind this definition? How can one characterize the sets which are in $\mathcal{F}_\tau$? Since $ A\cap \{\tau \leq t\} \in \mathcal{F}_t$ but not necessarily $ A\subseteq \{\tau \leq t\}$, might it not be the case that we can't decide if $A$ happened at time $\tau$? And can one understand $\mathcal{F}_\tau$ similar to $\mathcal{F}_t$ but with the index (somewhat) randomly chosen? If yes, how?
 A: $A \in \mathcal{F}_t$ means: Given the information up to time $t$, we can decide for some fixed $\omega \in \Omega$ whether $\omega \in A$ (or $\omega \in A^c$), i.e. whether the event $A$ happens (or not).
The interpretation for $A \in \mathcal{F}_{\tau}$ is very similar: Given the information up to time $t$ and given that the stopping time $\tau(\omega)$ already occured before time $t$, we want to decide whether $\omega \in A$ (or $\omega \in A^c$).
Example: Let $(B_t)_{t \geq 0}$ be a stochastic process with continuous sample paths and $B_0=0$. Define
$$\tau := \inf\{t>0; B_t \notin (-1,1)\}.$$


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*$A:=\{B_{\tau}=1\} \in \mathcal{F}_{\tau}$: Indeed, given the path up to time $t$ and given that the stopping time occured before time $t$, we can easily decide whether $B_{\tau}=1$ or $B_{\tau}=-1$ (since $(B_t)_{t \geq 0}$ has continuous sample paths, these are the only two possibilities).

*$A := \{B_{\tau+1}=0\} \notin \mathcal{F}_{\tau}$: Obviously, this event requires information about the future (compare this with $\{B_{t+1}=0\} \notin \mathcal{F}_t$) and therefore $A \notin \mathcal{F}_{\tau}$.

