Poisson factorial moment measure (${\alpha }^{(k)}={\mu }^{\otimes k}$)

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 $g$,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.