I want to show that there is a non-homogeneous Poisson process with a certain intensity function, but I have some problems while showing that this Poisson process satisfies the axioms(?). I am using the axioms as follows:

- $N(0) = 0$
- if $s\leq t$, then $N(s)\leq N(t)$
- etc

as usual. Now the problem is given by:

Let $X_1,X_2,\dots$ be i.i.d. continuous random variables with common probability density function $f_{X_1}$ and let $Z \xrightarrow{d}Po(\lambda)$ be independent of $X_1,X_2,\dots$. Now define the random point set $\mathcal{P} = \{X_1,\dots,X_Z\}$ ($\mathcal{P}$ = $\emptyset$ if $Z = 0$).

Show that $\mathcal{P}$ is an inhomogeneous Poisson point process with intensity function $\lambda\cdot f_{X_1}$.

Sorry if my question is not clear enough, and thank you!