In discrete time, I've read that a convention for any filtration is to assume that $\mathcal{F}_{t}=\mathcal{F}_{s} \ \forall s \in [t,t+1[$.
Wouldn't this make $\mathcal{F}_{t+}=\cap_{s>t} \mathcal{F}_s=\mathcal{F}_t$, and so every filtration in discrete time be right-continuous?