Function of a Poisson Point Process

Function of a Poisson Point Process

Let $(P_s)$ be a Poisson point process on a state space $X$ with intensity
measure $\mu$. Let $f(s,x)$ be a non-negative integer valued function on
$[0,1]\times X$. Let $N = \sum_{s\leq 1} \mathbb 1_{P_s \not=\Delta}
f(s,P_s)$, where here $P_s = \Delta$ (the "graveyard point") if $P_s$ is
not an element of $X$, i.e. $s$ is not a jump time for $P$. For example,
we might have a family of sets $A_s \subset X$ and $f(s,x) = \mathbb 1_{x
\in A_s}$. What assumptions on $f$ are needed for $N$ to have a Poisson
distribution?
I tried to compute the characteristic functional of $N$. I also tried
looking at the process $N_t = \sum_{s\leq t} \mathbb 1_{P_s \not=\Delta}
f(s,P_s)$, which has independent increments, and using some sort of
general result about such processes. Neither of these approaches were
successful, however. Does anyone have any suggestions?