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

This category contains pages concerning Event of Stopping Time Equal to Infinity is Measurable in Limit of Filtration:

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:

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