Definition 3.1.3 (Augmentation).label Let $(\Omega, \bracs{\cf_t}, \bp)$ be a filtered probability space and $\mathcal{N}$ be the collection of all $\bp$-null sets in $\cf = \sigma(\bracs{\cf_t|t \ge 0})$. For each $t \ge 0$, let $\ol{\cf}_{t} = \sigma(\cf_{t} \cup \mathcal{N})$, then the filtration $\bracsn{\ol{\cf}_t|t \ge 0}$ is the $\bp$-augmentation of $\bracs{\mathcal{F}_t}$.