I am facing the following statement:
Let $L$ be a piecewise constant stochastic process and let $(\mathfrak{F}_t)_{t \in \mathbb{R}}$ be the completed filtration generetad by $L$. Then the usual conditions are satisfied.
The usual conditions here mean that the filtration is right-continuous and that $\mathfrak{F}_0$ contains all nullsets.
That $\mathfrak{F}_0$ contains all nullsets follows from the choice of the completed filtration. But how do I show that the filtration is right-continuous?
Right continuity means that $$ \mathfrak{F}_t = \bigcap_{s > t} \mathfrak{F}_s $$ is satisfied.