Second factorial and reduced second factorial moment measures

Assume $\eta$ is a stationary point process as in Stationarity, finite intensity, and spatial ergodicity. It is locally square integrable if $\mathbb{E}[\eta (K{)}^2]<\mathrm{\infty }$ for every compact $K\subseteq {\mathbb{R}}^d$. For a realized counting measure $\mu ={\sum }_{i\in J}{\delta }_{X_i}$, with occurrences indexed separately even when locations coincide, as allowed by Random measures and locally finite point processes, define its second factorial measure on Borel $C\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ by

${\mu }^{(2)}(C):={\sum }_{i,j\in J,i\mathrm{\ne }j}{\mathbf{1}}_C(X_i,X_j)\mathrm{.}$

The second factorial moment measure is ${\alpha }^{(2)}(C):=\mathbb{E}[{\eta }^{(2)}(C)]$. With $Q=[0,1{)}^d$, the reduced second factorial moment measure is

${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(D):=\mathbb{E}{\sum }_{i,j\in J,i\mathrm{\ne }j}{\mathbf{1}}_Q(X_i){\mathbf{1}}_D(X_j-X_i),D\subseteq {\mathbb{R}}^d\text{ Borel}\mathrm{.}$

The local theorem Factorial and reduced factorial moment measures are well-defined and locally finite ↗ proves measurability and local finiteness under local square integrability. Occurrence indices, rather than distinct locations, are essential when multiplicities are present.

Axiom convention. AC and Countable Choice are carried by the transitive point-process supplier dependencies; the occurrence-indexed formulas themselves make no selections.