= Poisson factorial moment measure
{c}
{title2=$\alpha^{(k)}=\mu^{\otimes k}$}
For a <Poisson point process> with <intensity measure> $\mu$, ordered tuples of distinct atoms have factorial moment <measure> $\mu^{\otimes k}$. Thus for any nonnegative <measurable> function $g$,
$$
\mathbb E\sum_{z_1,\ldots,z_k\text{ distinct}}g(z_1,\ldots,z_k)=\int g\,d\mu^{\otimes k}.
$$
For $k=2$, separating diagonal terms from distinct pairs gives the second moment of a <Poisson> sum. Dividing by two instead counts unordered pairs, which is essential for geometric intersection intensities.
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