Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$. Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then
$$
 \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ \tau \le n \} \in \mathcal A_n \mbox{ for all } n \right\}.
$$
My textbook wrote:

an event is contained in $\mathcal A_{\tau}$ if it is determined by the course of the "process" until time $\tau$.

What should this mean? And what is the intuitive Meaning of $\mathcal A_{\tau}$? I cannot make sense of this construction...
 A: Otherwise stated $\mathcal{A}_{\tau}$ is formed of events that can be defined by questions that can be answered (positively or negatively) only if you know whether $\tau$ as happened at time any time $n$. 
Notice that involving the knowledge of $X$ in the informal definition is not necessary IMO unless the filtration is the natural filtration generated by the process $X$. It is easy to set an example where the knowledge of $\mathcal{A}_{\tau}$ is independent of an adapted process $X$.
Best regards
Edited to answer to stefan's comment :
No it's not like that. To determine if $A$ is in  $\mathcal{A}_{\tau}$ (for a stopping time $\tau$), you need as a first ingredient to get all the sets $B_n=\{\tau\leq n\}$. This tells you for each time $n$ if the stopping time has occured or not because $B_n\subset \mathcal{F}_n$ by definition of stopping times. 
Then you check if $B_n\cap A \subset \mathcal{F}_n$. So this means that you know $A$ has occured for every $n$ on the condition that you know if $\tau$ has occured. 
What you said in your comment is wrong (because it has to b for every $n$). Moreover the formulation is ambiguous as the fact that a set is in a $\sigma$-algebra doesn't mean that is has occured. Here the thing to "grasp" is the conditionning on all the sets $B_n$. You can turn this more properly into the following, $\mathcal{A}_\tau$ is generated by the sets for which you can say if they have happend prior to time $\tau$.
