filtration

Definition

Let $(\Omega, \mathcal{A},P)$ be a probability space, $I$ an index set with total order $\leq$ (e.g. $\mathbb{N}$, $\mathbb{R}^+$, or subset of $\mathbb{R}^+$)
For every $i \in I$ let $\mathcal{F}_i$ be a sub-σ-algebra of $\mathcal{A}$. Then $\mathbb{F} := (\mathcal{F}_i)_{i \in I}$ is called a filtration, if $\mathcal{F}_k \subseteq \mathcal{F}_\ell$ for all $k \leq \ell$. i.e., filtrations are families of σ-algebras ordered non-decreasingly.

If $\mathbb{F}$ is a filtration, then $(\Omega, \mathcal{A}, \mathbb{F},P)$ is called a filtered probability space.

References

1. https://en.wikipedia.org/wiki/Filtration_(probability_theory)