Definition:Stopped Sigma-Algebra

Definition
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

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

Let:


 * $\FF_T = \set {A \in \Sigma : A \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t \text { for all } t \in \Z_{\ge 0} }$

We call $\FF_T$ the stopped $\sigma$-algebra associated with $T$.

Also see

 * Stopped Sigma-Algebra is Sigma-Algebra

Also known as
$\FF_T$ is also known as the $\sigma$-algebra of $T$-past.