Factorial and reduced factorial moment measures are well-defined and locally finite

Fix a realization $\mu ={\sum }_{i\in J}{\delta }_{X_i}$. Define ${\mu }^{(2)}:={\sum }_{i,j\in J,i\mathrm{\ne }j}{\delta }_{(X_i,X_j)}$ directly as a countable sum of Dirac measures; [F4] makes it a measure, and it is finite on compact subsets of ${\mathbb{R}}^{2d}$ because $\mu (K{)}^2<\mathrm{\infty }$ for a compact $K$ containing both coordinate projections. On every bounded Borel $C$, this agrees with $(\mu \otimes \mu )(C)-({\mathrm{\Delta }}_{\mathrm{\#}}\mu )(C)$, where $\mathrm{\Delta }(x)=(x,x)$: both terms are finite there and the second removes exactly the equal-index pairs. Thus no subtraction of infinite values is used.

Let $K_n=[-n,n{]}^d$. For each Borel $C\subseteq {\mathbb{R}}^{2d}$, the evaluations $(\mu \otimes \mu )(C\cap (K_n\times K_n))$ and ${\mathrm{\Delta }}_{\mathrm{\#}}\mu (C\cap (K_n\times K_n))$ are measurable functions of $\mu$: the first is obtained by two successive integrations against the evaluation kernel [F3], and the second is evaluation on the Borel preimage under the continuous diagonal map [F1]. Both are finite; their difference is nonnegative and, by step 1.1, equals ${\mu }^{(2)}(C\cap (K_n\times K_n))$. These values increase to ${\mu }^{(2)}(C)$ because $K_n\times K_n$ increases to all of ${\mathbb{R}}^{2d}$. Hence [F5] makes $\mu \mapsto {\mu }^{(2)}(C)$ measurable.

For pairwise disjoint Borel $C_j$, countable additivity of each ${\eta }^{(2)}$ gives ${\eta }^{(2)}({\bigcup }_j{C}_j)={\sum }_j{\eta }^{(2)}(C_j)$ pointwise; [F5] passes the increasing partial sums through expectation. Thus ${\alpha }^{(2)}$ is a measure. If $C$ is bounded, choose compact $K$ with $C\subseteq K\times K$; then ${\eta }^{(2)}(C)\le \eta (K{)}^2$, so [F6] gives ${\alpha }^{(2)}(C)<\mathrm{\infty }$.

For Borel $D\subseteq {\mathbb{R}}^d$, $G_D=\{(x,y):x\in Q,y-x\in D\}$ is Borel because $(x,y)\mapsto y-x$ is continuous and $Q$ is Borel. The map $D\mapsto {\eta }^{(2)}(G_D)$ is the pushforward under $(x,y)\mapsto y-x$ of the restriction of ${\eta }^{(2)}$ to $Q\times {\mathbb{R}}^d$, so it is a measure; it is measurable in $\eta$ by step 2.1. Taking expectations and using [F5] shows ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}$ is a measure. If $D$ is bounded, $G_D$ is contained in $K\times K$ for some compact $K$ containing $\bar{Q}$ and $\bar{Q}+\bar{D}$, so ${\eta }^{(2)}(G_D)\le \eta (K{)}^2$ and [F6] makes its expectation finite.

Steps 1.1–4.1 establish the occurrence-indexed factorial measures, their measurability, and local finiteness. Multiplicity is retained because only equal occurrence indices are removed in step 1.1. The axioms [A1]–[A2] are inherited supplier assumptions and are not used by this construction.