Definition:Limit of Filtration of Sigma-Algebra

Definition

Discrete Time

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.

Continuous Time

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.