Interpretation of stopping time sigma algebra (as explained in Durret)

This is definitely a tricky thing to wrap one's mind around. In what follows, $n$ is supposed to be an integer, and $\tau$ an arbitrary stopping time.

How I think of it is as follows: for each $n$, we have a sigma algebra $F_n$. Thus we have a "decomposition" of the sigma algebra $F$, with each element of the decomposition being its own sigma algebra $F_n$.

Our goal is the following: to form a new decomposition which is "compatible" with the stopping time $\tau$, and use this decomposition to define a sigma algebra compatible with $\tau$.

Note that the notion of being compatible with $\tau$ will depend on the time, hence why we split up the filtrations by each $n$ and consider how to make each $F_n$ individually compatible with $\{ \tau =n\}$.

Thus, for each $n$, in order to form this new compatible decomposition, we form a new sigma algebra collection of sets $\tilde{F}_n = " \{\tau =n \}\cap F_n"$. By this I mean $\tilde{F}_n = \{ A \cap \{ \tau = n \} |\ A \in F_n \}$.

Why would we do this? Because we only want to consider events for which $\tau$ is measurable, otherwise our sigma algebra would not be compatible with $\tau$. Thus we only consider events for which $\tau$ is measurable and which are contained in the information provided by the original decomposition. The logical "and" here means that we have to take set intersection, hence why intersection is used to form a decomposition compatible with $\tau$.

As one final step, we also have to create the "sigma algebra at infinity" $\tilde{F}_{\infty}$ formed by the limit of the filtration $F$ as well as the event $\{ \tau = \infty\}$. (EDIT 1.5 years later: I think this means $$\tilde{F}_{\infty} := "\{\tau = \infty\} \cap \bigcup\limits_{n \ge 1} F_n " := \left\{ A \cap \{\tau = \infty\} \left| A \in \bigcup\limits_{n \ge 1} F_n \right. \right\} $$

Then $F_{\tau}$ is the smallest sigma algebra formed by combining all of the collections of sets $\tilde{F}_n$ ($\forall n$) and $\tilde{F}_{\infty}$. We take the smallest sigma algebra containing all of the compatible sigma algebras collections of sets because we don't want any redundant information.

In other words, $F_{\tau}$ is formed by the "projections" or "traces" of each event $\{\tau=n\}$ onto each sigma algebra $F_n$ via intersection.