Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration/Continuous Time

Theorem

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\sequence {\FF_t}_{t \ge 0}$ be a continuous-time filtration of $\Sigma$.

Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Let $\FF_\infty$ be the limit of the filtration $\sequence {\FF_t}_{t \ge 0}$.

Then:

$\set {\omega \in \Omega : \map T \omega = \infty} \in \FF_\infty$

Note that we have $\map T \omega < \infty$ if and only if $\map T \omega \le N$ for some $N \in \N$.

So, we have:

$\ds \set {\omega \in \Omega : \map T \omega < \infty} = \bigcup_{N \in \N} \set {\omega \in \Omega : \map T \omega \le N}$

Since $T$ is a stopping time, we have:

$\set {\omega \in \Omega : \map T \omega \le N} \in \FF_N$

For each $N \in \N$, we have:

$\ds \FF_N \subseteq \bigcup_{t \in \hointl 0 \infty} \FF_t$

$\FF_N \subseteq \FF_\infty$

$\ds \bigcup_{N \in \N} \set {\omega \in \Omega : \map T \omega \le N} \in \FF_\infty$

So:

$\set {\omega \in \Omega : \map T \omega < \infty} \in \FF_\infty$

Then since $\FF_\infty$ is closed under relative complement, we have:

$\set {\omega \in \Omega : \map T \omega = \infty} \in \FF_\infty$

$\blacksquare$