Consider a planar point process $X$ and call $N\_A = \text{Card}\big( X \cap A\big)$ the number of points inside the subset $A \subset \mathbb{R}^2$. If one knows the law of $(N\_{A\_1}, \ldots, N\_{A\_r})$ for any sets $A\_1, \ldots, A\_r$, then the process is completely characterized. I recently learned that it in fact suffices to know $f(A)=P(N\_A=0)$ (called the void-probability function) for any set $A$ in order to completely characterize the law of $X$.


Intuitively, I do not understand why such a result is true. Indeed, the knowledge of the function $f$ brings some information in the correlation structure of the process $X$: nevertheless, I still fail to understand how the function $f$ can encode the whole correlation structure of the process. Any thoughts on this ?



a point process is characterized by its void probabilities
This is only true for **simple** point processes (no duplicate points).


By the inclusion-exclusion principle, $f$ determines the joint distribution of several (disjoint) sets being empty or occupied. If the process is simple this allows recovering the law of $(N\_{A\_1},\dots,N\_{A\_r})$ as a limit over finer partitions.


