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

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

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

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


 * $\ds \FF_\infty = \map \sigma {\bigcup_{n \mathop = 0}^\infty \FF_n}$

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