Filtration

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a Probability Space, let $(I, \leq)$ be a Total Ordering. Let $\mathcal{B}{i} \subset \mathcal{A}$ be Sub-Sigma Algebras of $\mathcal{A}$. Then $(\mathcal{B}{i}){i \in I}$ is a Filtration if $\mathcal{B}{k} \subset \mathcal{B}_{l}$ when $k \leq l$.

That is, the Function $f: I \to {\mathcal{B}{i}: i \in I}$ defined by $f(i) = \mathcal{B}{i}$ Non-Decreasing Function.