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For a Poisson point process of intensity measure $\mu$, its $k$th Poisson factorial moment measure is ${\alpha }^{(k)}={\mu }^{\otimes k}$. Writing its counting measure as $N={\sum }_i{\delta }_{Z_i}$, the identity means that for nonnegative measurable function $g$,The indices select distinct point occurrences. Their locations need not be different when $\mu$ has an atom of a measure. For a simple point process, the locations are different as well.^[ref]

For example, if all intensity is concentrated at one point $a$, with $\mu (\{a\})=\lambda$, then $N(\{a\})$ has Poisson distribution. There are $N(\{a\})(N(\{a\})-1)$ ordered pairs of distinct occurrences there, with expected value ${\lambda }^2$, even though both locations equal $a$. Counting only pairs of different locations would give zero and fail the identity.

For $k=2$, separating equal-index terms from distinct-index terms gives the second-moment calculation in Campbell's theorem. Dividing the distinct-pair sum by two instead counts unordered pairs, as in the intersection point process of a Poisson line process.