Approximation of a unbounded stopping time and convergence of respective $\sigma$-algebras

Question

Fixed a filtration $(\mathscr{F_n})_n$ in a probability space and given a stopping time $\tau$ w.r.t $(\mathscr{F_n})_n$ that is finite almost surely we can construct a non-decreasing sequence of bounded stopping times $\tau_n=min\{\tau,n\}$ that converges to $\tau$. How to prove that $\sigma(\cup_n\mathscr{F_{\tau_n}})=\mathscr{F_{\tau}}$?

Answer 1

Hints:

1. It holds for any two stopping times $\sigma \leq \varrho$ that $\mathcal{F}_{\sigma} \subseteq \mathcal{F}_{\varrho}$. Conclude from $\mathcal{F}_{\tau_n} \subseteq \mathcal{F}_{\tau}$ that $$\sigma \left(\bigcup_n \mathcal{F}_{\tau_n} \right) \subseteq \mathcal{F}_{\tau}.$$
2. Fix $A \in \mathcal{F}_{\tau}$ and write $$A = \bigcup_{k \geq 1} A \cap [\tau \leq k] =: \bigcup_{k \geq 1} A_k.$$ Prove that $A_k \in \mathcal{F}_{\tau_{k+1}}$ and conclude $\mathcal{F}_{\tau} \subseteq \sigma\left( \bigcup_n \mathcal{F}_{\tau_n} \right).$