Let $E$ be a polish space and $N(E)$ be the space of finite integer value measures. It is known that for every $\mu \in N(E)$ exists $x_1, \dots, x_n \in E$ such that $$\mu = \sum_{i=1}^n \delta_{x_i}$$.
A point process is a random element on $N(E)$ therefore a point process is random collection of point on $E$.
We can construct a point process with a measure $p$ on $\mathbb{N}$ (this measure represents the random number of elements in the point process) and a family of measurable functions $(F^{(N)}_k \colon [0,1] \to E ; \quad N \in \mathbb{N}, \ k \leq N )$ such that the point process can be represented as $$ \sum_{k=1}^N \delta_{F^{N}_k(U)} $$ with $N \sim p$, $U \sim $Unif$([0,1])$ independent from each other.
Can every point process be represented this way?
thanks