My questions will be given at the end, let me just give some definitions first.
The counting process $\{ N(t), t \geq 0 \} $ is said to be a non homogenous Poisson process with intensity function $\lambda(t), t \geq 0$ if:
$N(0)= 0$
$\{ N(t),t \geq 0 \}$ has independent increments
$P\{ N(t+h) - N(t) \geq 2 \} = o(h)$
$P\{ N(t+h) - N(t) =1 \} =\lambda(t)h + o(h)$
Time sampling and ordinary poisson process generates a nonhomgenous poission process. Let $\{N(t),t\geq 0 \} $ be a poisson process with rate $\lambda$ and suppose that an event occurring at time $t$ is , independently of what has occurred prior to $t$, counted with probability $p(t)$,then the counting process $\{N_c(t), t \geq 0 \} $ is a nonhomogenous poisson process with intensity function $\lambda(t) = \lambda\cdot p(t)$
The book now shows that a time sampling satisfies the axioms above.verifying the last axiom we calculate
$P\{ N_c(t+h) - N_c(t) =1 \}=P\{ N_c(t,t+h)=1\} = \cdots =P\{N_c(t,t+h) = 1 \mid N(t,t+h) = 1 \}\lambda h + o(h) = p(t)\lambda h + o(h) $
Now, what i don't understand is how $P\{N_c(t,t+h) = 1 \mid N(t,t+h) = 1 \} = p(t)$, the way the book defined $p(t)$ (see above). First of all we are conditioning on that the regular poisson process is one, and also the above says that there is 1 event happening in the intervall $(t,t+h)$ , but $p(t)$ is the probability that an event occur at time $t$.
Could someone clarify this?
Events are occurring under the homogeneous process $N$ at rate $\lambda$. Each such event, occurring at time $t$ say, is then either counted in the $N_c$ process, with probability $p(t)$, or ignored with probability $1-p(t)$. So it's not correct to say that $p(t)$ is the probability that an event occurs at time $t$ in process $N_c$.
Mathematically, we can say:
$$p(t) = P(\text{Event at time $t$ is counted}\mid\text{Event occurs under process $N$ at time $t$}).$$
This is close to the line in the proof you refer to but there is a time interval of size $h$, so it is only an approximation
$$P\{N_c(t,t+h) = 1 \mid N(t,t+h) = 1 \} = p(t).$$
If the event occured at time $t_0 \in [t,t+h]$ then the true value for the RHS would be $p(t_0)$. But for small $h$, $p(t)$ is a reasonable approximation. Also, the "error" in this approximation is accounted for in the term $o(h)$ allowing the use of the equality sign.