Definition:Complete 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 say that $\sequence {\FF_t}_{t \ge 0}$ is a complete filtration if and only if:

$\NN \subseteq \FF_0$