Consider $\Omega = C(0,\infty;\mathbb{R})$ with the topology of compact convergence, defining the $\sigma$-algebra $\mathcal{N}$.
Define $$\pi_t(w) = w(t)\,,$$ and define the $\sigma$-algebras, for $s \geq 0$, $$\mathcal{F}_s = \sigma(\{\pi_t : t \leq s \})\,,$$ and $$\mathcal{F}^s = \sigma(\{\pi_t : t \geq s \})\,.$$
The family $(\mathcal{F}_s)_{s\geq 0}$ is a filtration.
Let $\tau$ be a stopping time with respect to $(\mathcal{F}_s)_{s\geq 0}$. There is a filtration usually associated with $\tau$ : $$\mathcal{F}_\tau = \{A \in \mathcal{N} : A \cap \{\tau \leq s \} \in \mathcal{F}_s\,\forall s \geq 0 \}\,.$$
Is there a way to define $\mathcal{F}^\tau$, similarly to $\mathcal{F}^s$?
I was thinking in use $$\mathcal{F}^\tau = \{A \in \mathcal{N} : A \cap \{\tau \geq s \} \in \mathcal{F}^s\,\forall s \geq 0 \}\,.$$
Addendum 03/27: I now think this definition is not useful, I want to consider sets of the form $\varTheta_\tau^{-1} (A)$ for $A \in \mathcal{N}$, and $\varTheta_\tau (w) = w(\tau(w) + \cdot)$ defined on $\{w : \tau(w) < \infty \}$, I guess this still depends heavily in the information before $\tau$, I can't find a way to describe this set succinctly.
The intuition I have for $A \in \mathcal{F}_\tau$, is that in any realization of the process if I observe the process until $\tau$ occurs, then I'm able to decide if the event $A$ has occurred or not.
For $A \in \mathcal{F}^s$, if I observe until time $s$, then up to time $s$ I'm fully ignorant if the event $A$ occurs or not. The conditions defining the event are in terms of moments of time equal or after $s$.
For $A \in \mathcal{F}^\tau$ (I expect to have that, though I still have no clear intuition for this) if I observe the process until any time $s$, if $\tau$ has not occur, then I'm fully ignorant if the event $A$ occurs or not.
In certain way, for $A \in \mathcal{F}_\tau$, $\tau$ signals if I have gathered enough information to decide whether $A$ as occurred or not, anything after that is useless.
For $A \in \mathcal{F}^\tau$, $\tau$ signals when I am just starting to gather the information to decide if $A$ occurs or not, anything before that is useless.
Do you agree with me on this? I can only find information about $\mathcal{F}_\tau$ but nothing about a construction similar to $\mathcal{F}^\tau,$ do you know of any reference?