Consider the following definition.
Definition.
Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid s\leq t\right\}\right)$ be the natural filtration associated to $W$. The augmented filtration $\mathcal{G}^{W}$ is defined as $$ \mathcal{G}^{W}_t=\sigma\left(\mathcal{F}^W_t \bigcup \mathcal{N}\right), $$ where $\mathcal{N}$ is the family of $\mathbb{P}$-negligible events $$ \mathcal{N}=\left\{F\in\mathcal{F}\mid \mathbb{P}\left[F\right]=0\right\}. $$Just few questions:
Which is the rational behind this definition?
I cannot see why the set $\mathcal{N}$ is to be added to the natural filtration of the Brownian motion, how is it possible to prove that it is not already included in the original filtration $\mathcal{F}^W$?
According to the corresponding wikipedia page, the augmented filtration is both left-continuous and right-continuous, while the natural filtration is (obviously) only left-continuous. Is it possible to find a reference for this proof?
