Using the following definition I want to proof that a stopping time $\tau$ is measureable w.r.t. its associated $\sigma$-Algebra $\mathcal{F}_\tau$.
Definition. Let $T$ be a totally ordered set, $t^*:= \sup T$ and $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in T},\mathbb{P})$ a filtered probability space. A map $\tau :\Omega \to \overline{T}$ is called $(\mathcal{F}_t)_{t \in T}$-stopping time, if $\{\tau \leq t\} \in \mathcal{F}_t$ for all $t \in T$.
Furthermore, the $\sigma$-Algebra $\mathcal{F}_\tau$ associated with $\tau$ is defined by $\mathcal{F}_\tau = \{F \in \mathcal{F}_{t^*} | F \cap \{\tau \leq t\} \in \mathcal{F}_t \text{ for all } t \in T\}$.
I think I can show the statement for real-valued stopping times. $\tau$ is $\mathcal{F}_\tau$-$\mathscr{B}_{\mathbb{R}}$-measurable if $\tau^{-1}(A) \in \mathcal{F}_\tau$ for all $A \in \mathscr{B}_{\mathbb{R}}$. To show this I use two statements I remember from Probability Theory:
- It suffices to show $\tau^{-1} (A) \in \mathcal{F}_\tau$ for all $A \in \mathcal{A}$ if $\sigma(\mathcal{A}) = \mathscr{B}_{\mathbb{R}}$ and
- The set of all intervals $(-\infty,b]$ for $b \in \mathbb{R}$ generates $\mathscr{B}_{\mathbb{R}}$.
Because I got $\tau^{-1}((-\infty,b]) = \{ \tau \leq b\} \in \mathcal{F}_b \subseteq \mathcal{F}_{t^*}$ and $\tau^{-1}((-\infty,b]) \cap \{\tau \leq t\} = \begin{cases} \{\tau \leq b\} & \text{for } b \leq t & \in \mathcal{F}_b \subseteq \mathcal{F}_t,\\ \{\tau \leq t\} & \text{for } b > t & \in \mathcal{F}_t \end{cases}$ for all $b \in \mathbb{R}$ it follows with the two statements above that every $\mathbb{R}$-valued stopping time is $\mathcal{F}_\tau$-$\mathscr{B}_{\mathbb{R}}$-measurable.
I think my problems with this proof are because of some gaps in the fundamentals concerning measurability.
My questions would be: First of all, is my proof for real-valued stopping times correct? Why can I "assume" that $\mathcal{F}_\tau$-measurable means $\mathcal{F}_\tau$-$\mathscr{B}_\mathbb{R}$-measurable? Do the statements 1. and 2. hold for a totally ordered $T$ (and therefore can the proof be done similiarly in the more general setting)?
Thank you in advance for your help!
When we talk about measurability of a real valued function without specifying any sigma algebra on $\mathbb R$ it is understood that we are using the Borel sigma algebra on $\mathbb R$.
Your proof for real valued $\tau$ is fine. The proof is same if the parameter set is replaced by a totally ordered set.
Hint for the general case. Show that if $\tau$ is a stopping time then $\min \{\tau, n\}$ is a real valued stopping time. Now use the fact that the event $(\min \{\tau, n\}\leq t)$ decreases to the event $(\tau \leq t)$.