I shall show that for a point process (counting process) $\Phi((0,t])=\sum_{n \geq 1} \mathbf{1}_{\lbrace T_n \leq t \rbrace}$, \begin{align*} M_t = \Phi((0,t]) - \int_{0}^{t} \mathbf{1}_{\lbrace s < T_{\infty}\rbrace} \lambda (X_{s}) ds, \end{align*} is for all $t \geq 0$ a $\lbrace \mathcal{F}_t \rbrace$-martingale. The $T_1,T_2, \dots$ are jump times, $\int_{0}^{t} \mathbf{1}_{\lbrace s < T_{\infty}\rbrace} \lambda (X_{s}) ds $ the compensator of a point process and $X_s$ the value of a stochastic process $(X_t)$ at time $s$.
I've already shown $\mathbb{E}[|M_t|]< \infty$. Now I have to show the martingale property $E [M_s \mid \mathcal{F}_t] = M_t $ for all $s < t$ with $s,t \geq 0$ but I have problems to show it. I would be very thankful for your help.