Stopping time clarification May I please seek someone's help to clear my understanding about stopping time.
According to the Wikipedia definition:
random variable $\tau:\Omega \rightarrow I$ is called stopping time if
$\{\omega \in \Omega : \tau(\omega) \leq t\} \in F_t$ where $F_t$ is the filtration up to time t.
I am trying to understand this definition:
1> first I choose a time "t"
2> Given my choice of "t" I will have a sigma algebra $F_t$, which essentially is a collection of sets with closure properties. 
After this I am confused
3> Should I take each element $\omega$ from $\Omega$ and check if my stopping time event has happened or not. And if it has happened, then is it in the $F_t$. 
And the above will give what ? a collection of elements ? how will it give a real number which I can call stopping time. 
I understand I have confused myself big time but not clear where I am making the mistake in my interpretation. If someone can comment to make my understanding clear I would be very thankful.
 A: The idea is that $F_t$ consists of all events that depend on your stochastic process only up to time $t$.  So to say that $\{\omega\in\Omega:\tau(\omega)\leq t\}\in F_t$ means that given any particular outcome $\omega$, to determine whether $\tau(\omega)$ is at most $t$, you only have to look at what happens in $\omega$ up to time $t$.
The idea behind the term "stopping time" is that you are running some stochastic process, and deciding to stop it at some time based on what has happened so far.  (For instance, maybe you choose to stop at the first time that your stochastic process reaches some particular value.)  An element $\omega\in\Omega$ represents one particular outcome of running the stochastic process, and the time at which you choose to stop for that outcome is $\tau(\omega)$.  When deciding whether to stop, you only know what has happened so far (not what will happen later).  So, to determine whether you will stop at time $t$ or earlier, you can only look at the outcome of your stochastic process prior to time $t$.  In other words, the event $\{\omega\in\Omega:\tau(\omega)\leq t\}$ should be in $F_t$.
