Definition:Limit of Filtration of Sigma-Algebra/Continuous Time

Definition

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

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

We define the limit $\FF_\infty$ by:

$\ds \FF_\infty = \map \sigma {\bigcup_{t \in \hointr 0 \infty} \FF_t}$

where $\map \sigma \cdot$ denotes the $\sigma$-algebra generated by a collection of subsets.