Filtration

Let $(\Omega ,\mathcal{A},\mathbb{P})$ be a probability space, and $I$ be a totally ordered index. $\mathbb{F}:=({\mathcal{F}}_i{)}_{i\in I}$ is a filtration if:

1. ${\mathcal{F}}_i\subseteq \mathcal{A}$ is a sigma-algebra $\forall i\in I$.
2. $i\le k\Rightarrow {\mathcal{F}}_i\subseteq {\mathcal{F}}_k$.