$p(t,w)$ is a poisson point process and $(Z,\ \mathscr{B}(Z))$ is an abstract separable space, and for any $U\in\mathscr{B}(Z)$, $N_{p}\large(\left(0,\ t\right]\large\times U,\ w)$ is a poisson counting measure w.r.t. the poisson point process $p(t,\ w)$. Also, $\widehat{N_p}(t,\ w)={N_p}(t,\ w)-\widetilde{N_p}(t,\ w)$ is the compensated measure. Then I have a question:
A $\mathscr{F}_t$-predictable process $f(t,\ z,\ w)$ satisfying that $$\mathbb{E}\Big[\int_{0}^{t+}\int_{Z}|f(s,\ ,z\ ,w)|\widehat{N_p}(t,\ w)\Big]<\infty$$ then the process $$x(t)=\int_{0}^{t+}\int_{Z}f(s,\ ,z\ ,w)\widetilde{N_p}(t,\ w)$$ is a $\mathscr{F}_t$-martingale, I do not know what is the reason, could you give the details?
(ps: I have this question when I read the page 33-35 of the book "Theory of stochastic differential equations with jumps and applications -Rong Situ")