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

Theorem

Discrete Time

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

Let $\sequence {\FF_n}_{n \ge 0}$ be a discrete-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 $\sequence {\FF_n}_{n \ge 0}$.

Then:

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

Continuous Time

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:

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