Category:Filtrations of Sigma-Algebras

From ProofWiki
Jump to navigation Jump to search

This category contains results about 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 $\Sigma$.

Subcategories

This category has only the following subcategory.

Pages in category "Filtrations of Sigma-Algebras"

The following 3 pages are in this category, out of 3 total.