Due to the general theory of stochastic processes by Dellacherie and Meyer we know that split any stopping time into an accesible and totally inaccessible part, that is
$T=T_I\wedge T_A$ on $\{T<\infty\}$ and with $I,A\in F_{T-}$.
If we consider a Poisson Process $N$ in its natural filtration $\mathcal{F}^N$, I was wondering whether we could proove that
$\Delta N_{T_I}=1$ and $\Delta N_{T_A}=0$ almost surely for all $\mathcal{F}^N$-stopping times $T$?
Or in other words (the jump times are totally inacessible by Meyer's theorem) don't we have that times at jumps are the only inaccessible ones?
Many thanks for ideas how to prove this or a counter example.