Mecke identity for locally finite Poisson point processes

First assume $\lambda :=\nu ({\mathbb{R}}^d)<\mathrm{\infty }$. If $\lambda =0$, then $\eta ({\mathbb{R}}^d)=0$ almost surely by [F1], so both identities are zero. If $\lambda >0$, put $Q_{\nu }=\nu \mathrm{/}\lambda$ and $p_n=e^{-\lambda }{\lambda }^n\mathrm{/}n!$. For each $n$, let $P_n$ be the law of ${\sum }_{j=1}^n{\delta }_{X_j}$ when $(X_1,\dots ,X_n)$ has product law $Q_{\nu }^{\otimes n}$, with $P_0$ the point mass at the zero measure. The countable weighted sum $P^{\ast }={\sum }_{n\ge 0}{p}_n{P}_n$ is a probability measure by the normalized Poisson masses [F3] and the countable weighted-sum result [F7]; evaluation maps of the finite Dirac sum are measurable by [F2].

Now let $\nu$ be sigma-finite and locally finite. Partition ${\mathbb{R}}^d$ into the explicit bounded Borel cells $A_1=[-1,1{)}^d$ and $A_n=[-n,n{)}^d\setminus [-(n-1),n-1{)}^d$ for $n\ge 2$. Let ${\eta }_i$ be the restriction of $\eta$ to $A_i$ and ${\eta }_{-i}$ its restriction to $A_i^c$. Each ${\eta }_i$ is a finite-intensity Poisson process. The two random configurations ${\eta }_i$ and ${\eta }_{-i}$ are independent: finite evaluation cylinders refine to disjoint Borel pieces in $A_i$ and $A_i^c$, whose counts are independent by [F1], and the generating rectangle pi-system extends this factorization to their evaluation sigma-algebras by [F4].

For disjoint Borel sets $B_1,\dots ,B_m$, add their complement as $B_0$ and put $q_j=Q_{\nu }(B_j)$. On the $n$th component the vector of counts has multinomial mass $n!{\prod }_{j=0}^m{q}_j^{k_j}\mathrm{/}{k}_j!$ when ${\sum }_j{k}_j=n$. Mixing by $p_n$ and summing $k_0$ gives $P^{\ast }(N(B_j)=k_j,1\le j\le m)={\prod }_{j=1}^m{e}^{-\nu (B_j)}\frac{\nu (B_j{)}^{k_j}}{k_j!},$ using the exponential series; hence $P^{\ast }$ has the defining independent Poisson counts [F1]. Any other Poisson process with finite $\nu$ has the same finite-dimensional evaluation laws: refine any finite family of Borel sets into its finitely many disjoint Boolean atoms, apply [F1] to their counts, and recover the original counts by summation. Evaluation cylinders form a generating pi-system for the configuration sigma-algebra [F2], so probability-measure uniqueness [F4] shows that $\eta$ has law $P^{\ast }$.

For any nonnegative product-measurable $f$, represent $\eta$ by the mixture law from steps 1.1–1.2. Pathwise the integral against its finite counting measure is a sum over occurrences, so Tonelli [F6] gives $\mathbb{E}\int f(x,\eta )\eta (dx)={\sum }_{n\ge 1}{p}_n{\int }_{({\mathbb{R}}^d{)}^n}{\sum }_{i=1}^{n}f(x_i,{\sum }_{j=1}^n{\delta }_{x_j})Q_{\nu }^{\otimes n}(d\mathbf{x})\mathrm{.}$ Coordinate-permutation invariance of $Q_{\nu }^{\otimes n}$ makes the $n$ summands equal. Reindexing with $m=n-1$ and $n{p}_n=\lambda p_{n-1}$ changes this to $\lambda {\sum }_{m\ge 0}{p}_m{\int }_{({\mathbb{R}}^d{)}^{m+1}}f(x_1,{\delta }_{x_1}+{\sum }_{j=2}^{m+1}{\delta }_{x_j})Q_{\nu }^{\otimes (m+1)}(d\mathbf{x})\mathrm{.}$ Integrating first in $x_1$ and using $\nu =\lambda Q_{\nu }$, the inner mixture over $m$ remaining points is $\mathbb{E}[f(x,\eta +{\delta }_x)]$ by steps 1.1–1.2. This proves the add-one identity for finite total intensity.

Fix $i$. Independence makes the joint law of $({\eta }_i,{\eta }_{-i})$ the product of their laws, as it agrees with that product on measurable rectangles [F4]. For each fixed background configuration $\mu$, the function $f_{\mu }(x,\zeta )=f(x,\zeta +\mu )$ is product-measurable by the addition evaluation formula in [F5]. Apply the finite-intensity add-one identity of step 3.1 to ${\eta }_i$ and $f_{\mu }$, then integrate over $\mu ={\eta }_{-i}$; this gives $\mathbb{E}\int f(x,\eta ){\eta }_i(dx)={\int }_{A_i}\mathbb{E}[f(x,\eta +{\delta }_x)]\nu (dx)\mathrm{.}$ Summing over $i$ and using [F6] proves the add-one identity for $\nu$.

Let $g$ be the nonnegative integrand in the deletion formula, and put $D=\{(x,\mu ):\mu (\{x\})\ge 1\}$. Define $H(x,\mu )=g(x,\mu -{\delta }_x)$ on $D$ and $H=0$ off $D$; [F5] makes $H$ measurable. Apply the add-one identity of step 4.1 to $H$. At every occurrence counted by $\eta (dx)$, $(x,\eta )\in D$ and $H(x,\eta )=g(x,\eta -{\delta }_x)$; for every $x$, $(x,\eta +{\delta }_x)\in D$ and $H(x,\eta +{\delta }_x)=g(x,\eta )$. This gives the deletion identity. Since $\eta -{\delta }_x$ removes one unit mass, all other co-located occurrences remain.

Steps 1.1–4.1 establish both forms, including the zero-intensity boundary case and multiplicity convention. AC and Countable Choice [A1]–[A2] are inherited supplier assumptions; the local Mecke derivation itself uses no selection, and no DC is used.