A question on the "usual" conditions of a probability space?

Question

I am reading a paper where it states the following:

"Let $(\Omega, \mathcal{F},\{{\mathcal{F}_t}\}_{t≥0}, \mathbb{P})$ be a complete probability space with a filtration $\{{\mathcal{F}_t}\}_{t≥0}$ satisfying the usual conditions (i.e., it is increasing and right continuous while $\mathcal{F}_0$ contains all $\mathbb{P}$-null sets)"

Can some explain these conditions further? What are $\mathbb{P}$-null sets?

Answer 1

The usual conditions for a filtration $\{\mathcal{F}\}_{t\geq 0}$ are that:

1. It is right-continuous: $\mathcal{F}_t = \bigcap_{s>t} \mathcal{F}_s$, and
2. $\mathcal{F}_0$ contains all the null sets of $\mathcal{F}$ (i.e. it is complete)

Condition (2) means that if $A \in \mathcal{F}$ and $\mathbb{P}(A)=0$, then for all $B \subseteq A$ we have $\mathbb{P}(B) = 0$. Condition 1 is nice as it allows us to deal with processes with cadlag paths. Condition 2 oftentimes makes analysis a bit easier without it being too strict a condition, as any filtration can be enlarged to be complete simply by augmenting it with the subsets of sets with probability zero.