Definition:Completion of Filtration of Sigma-Algebra

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

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

Let $\FF_\infty$ be the limit of $\sequence {\FF_t}_{t \ge 0}$.

Let $\NN$ be the set of $A \in \Sigma$ such that:


 * there exists $A' \in \FF_\infty$ with $A \subseteq A'$ such that $\map \mu {A'} = 0$.

We define the completion $\sequence {\FF'_t}_{t \ge 0}$ of $\sequence {\FF_t}_{t \ge 0}$ by:


 * $\FF'_t = \map \sigma {\NN, \FF_t}$

for each $t \in \hointr 0 \infty$ where $\map \sigma {\NN, \FF_t}$ is the $\sigma$-algebra generated by $\FF_t$ and $\NN$.