Category:Definitions/Filtrations of Sigma-Algebras

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This category contains definitions related to Filtrations of Sigma-Algebras.
Related results can be found in Category:Filtrations of Sigma-Algebras.

Discrete Time

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

Let $\sequence {\FF_n}_{n \mathop \in \N}$ be an sequence of sub-$\sigma$-algebras of $\Sigma$.

That is:

$\FF_i \subseteq \FF_j$ whenever $i \le j$.

We say that $\sequence {\FF_n}_{n \mathop \in \N}$ is a filtration of $\Sigma$.

Continuous Time

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

Let $\sequence {\FF_t}_{t \ge 0}$ be an $\hointr 0 \infty$-indexed family of sub-$\sigma$-algebras of $\Sigma$ such that:

$\FF_t \subseteq \FF_s$ whenever $t, s \in \hointr 0 \infty$ are such that $t \le s$.

We say that $\sequence {\FF_t}_{t \ge 0}$ is a filtration of $\Sigma$.