A filtration on $(\Omega,\mathcal{F},P)$ induced by $\mathbb{N}$ is a family $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ of sub σ-algebras of $\mathcal{F}$ (i.e. $\mathcal{F}_{n}$ is a σ-algebra and $\mathcal{F}_{n}\subset \mathcal{F}, \ \forall n\in\mathbb{N}$) s.t. $$\mathcal{F}_{n}\subset \mathcal{F}_{n+1}, \ \forall n\in\mathbb{N}$$

Let $(X_{n})_{n\in\mathbb{N}}$ be a Stochastic Process on $(\Omega,\mathcal{F},P)$, $\forall n\in\mathbb{N}$, let the natural Filtration be defined as $$\mathcal{F_{n}}^{X}=\sigma(X_{0},\dots,X_{n})$$

A filtration $(\mathcal{F}_{t})_{t\ge 0}$ satisfies the usual conditions if 1. $\mathcal{F}_{t}=\mathcal{F}_{t^{+}}$, $\forall t\ge 0$ where $$\mathcal{F_{t^{+}}}=\bigcap_{s>t}\mathcal{F}_{s}$$i.e. $(\mathcal{F}_{t})_{t\ge 0}$ is right-continuous and; 2. $\mathcal{F}_{t}$ is complete i.e. $$\forall A\in\mathcal{F},\forall B\in\mathcal{P}(\Omega),B\subset A \ \ \& \ \ P(A)=0\implies B\in\mathcal{F} \ (\text{hence }P(B)=0)$$and; 3. $\mathcal{F}_{0}$ contains all elements in $\mathcal{F}$ of $P$-measure $0$ i.e. $$\forall A\in\mathcal{F}\text{ s.t. }P(A)=0\text{ then }A\in\mathcal{F}_{0}$$