In Protter's book Stochastic Integration and Differential Equations, it defines usual hypotheses as follows:
A filtered complete probability space $(\Omega,\mathcal{F},\{\mathcal{F_t}\},P)$ is said to satisfy the usual hypotheses if
(i) $\mathcal{F}_0$ contains all the P-null sets of $\mathcal{F}$;
(ii) $\{\mathcal{F_t}\}$ is right-continuous.
I want to know what is 'P-null sets of $\mathcal{F}$'. Do we need that a P-null sets of $\mathcal{F}$ must be in $\mathcal{F}$?
In Le Gall's book GTM274, it defines as follows:
Let $\{\mathcal{F_t}\}$ be a filtration and let $\mathcal{N}$ be the class of all $(\mathcal{F_\infty},P)$ negligible sets (i.e. $A\in\mathcal{N}$ if there exists an $A^{\prime}\in\mathcal{F_\infty}$ such $A\subset A^{\prime}$ and $P(A^{\prime})=0.$)
The filtration is said to be complete if $\mathcal{N}\subset\mathcal{F_0}$ (and thus $\mathcal{N}\subset\mathcal{F_t}$ for every t).
In this definition, we don't need set in $\mathcal{N}$ to be in $\mathcal{F}$. So I am really confusiong now.