The probability generating functional of a point process $N(\cdot)$ on the real line is defined as $$G[f]=\mathbb{E}\bigg[\exp \bigg\lbrace\int \log f(t)N(dt)\bigg\rbrace\bigg]$$ for a suitable class of real functions $0\le f\le 1.$
How are probability generating functionals useful?
For example, the probability generating functional for a Poisson process with a mean measure $\nu(\cdot)$ is $$\exp\bigg(\int (f(t)-1)\nu(dt)\bigg).$$
What information does this function give us about the Poisson point process?