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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?

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    The point is that the probability generating function uniquely determines the distribution of the entire point process. To see why this is true, consider any finite collection $\{ A_1,...,A_n \}$ of Borel subsets of $\Bbb R_+$ and any $r_1,...,r_n \in [0,1]$. Then define the function $f = r_1^{1_{A_1}} \cdots r_n^{1_{A_n}}$, and compute $G[f]$. As the $r_i$ vary, you should obtain the joint MGF of $(N(A_1),...,N(A_n))$ which determines its distribution. – 2017-02-13

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