I recently learnt about "augmented" filtrations, i.e. filtrations of sigma-algebras $(\mathcal{F}_i)_{i\in I}$ that are both complete and right-continuous. I didn't pay much attention to it as it seemed like assuming to have a standard filtration was a good way to make things working, at no cost: in particular, of course we can enlarge the sigma algebras in the filtrations such that all null sets of $\mathcal{F}$ are contained everywhere, in particular in $\mathcal{F}_0$ (this is "completeness"). Adding null sets cannot possibly add useful information, random variables that were measurable before are still measurable in the larger sigma-algebras, and results such the existence of cadlag versions of martingales follow in a smoother way.
Then I thought back of it and got confused. Pick for example a Poisson process, with its natural filtration. If we make the filtration complete, i.e. we add to $\mathcal{F}_0$ all null-events concerning the process, we have in $\mathcal{F}_0$ all the events of type {the nth jump of the process occurs exactly at time t}, for all natural n and all positive real t. These are events with measure 0, but they completely define the whole trajectory of the process! This disturbs me: $\mathcal{F}_0$ is only supposed to contain information about the process at time 0, plus potentially measure-zero events that should not carry too much information, but now the whole family of null events we added to $\mathcal{F}_0$ is threatening to spoil the whole trajectory of the process!
What am I missing?