Existence and uniqueness of the full Palm law and the refined Campbell identity

For each configuration event $A$ and Borel set $B\subseteq {\mathbb{R}}^d$, define $C_A(B):=\mathbb{E}\int {\mathbf{1}}_B(x){\mathbf{1}}_A({\theta }_x\eta )\eta (dx)\mathrm{.}$ The integrand is measurable by [F3], so this is defined. For disjoint $B_j$, the indicators of their finite unions increase to the indicator of their union; monotone convergence [F5] and countable additivity of each $\eta$ give $C_A({\bigcup }_j{B}_j)={\sum }_j{C}_A(B_j)$. Also $0\le C_A(B)\le \mathbb{E}\eta (B)=\rho {\lambda }_d(B)$ by [F2], so $C_A$ is locally finite.

For any $y\in {\mathbb{R}}^d$, recentering has the flow property ${\theta }_{x+y}={\theta }_x{\theta }_y$. Changing the location variable in the integral gives $C_A(B+y)=\mathbb{E}\int {\mathbf{1}}_B(u){\mathbf{1}}_A({\theta }_u({\theta }_y\eta ))({\theta }_y\eta )(du)=C_A(B),$ where the last equality uses stationarity of $\eta$. Thus $C_A$ is translation invariant.

By [F4] there is a constant $c_A=C_A(Q)$ such that $C_A(B)=c_A{\lambda }_d(B)$ for every Borel $B$. For disjoint configuration events $A_j$, monotone convergence in the indicator sum gives $c_{{\cup }_j{A}_j}={\sum }_j{c}_{A_j}$. Also $c_{\mathcal{N}}=\mathbb{E}\eta (Q)=\rho$ and $c_{\mathrm{\varnothing }}=0$. Hence $P^0(A):=c_A\mathrm{/}\rho$ is a probability measure on the configuration sigma-algebra.

For Borel $B$ and configuration event $A$, the definition gives $\mathbb{E}\int {\mathbf{1}}_B(x){\mathbf{1}}_A({\theta }_x\eta )\eta (dx)=C_A(B)=\rho {\lambda }_d(B)P^0(A)\mathrm{.}$ Define the Campbell measure $\mathrm{\Pi }(E)=\mathbb{E}\int {\mathbf{1}}_E(x,{\theta }_x\eta )\eta (dx)$ on the product space. Joint measurability [F3] and monotone convergence [F5] make $\mathrm{\Pi }$ a measure. It is sigma-finite since $\mathrm{\Pi }(Q_n\times \mathcal{N})=\mathbb{E}\eta (Q_n)=\rho \mathrm{\mid }{Q}_n\mathrm{\mid }<\mathrm{\infty }$ for the increasing bounded cubes $Q_n=[-n,n{)}^d$. The product measure $\rho ({\lambda }_d\otimes P^0)$ is sigma-finite by [F6] and agrees with $\mathrm{\Pi }$ on rectangles by the displayed identity; uniqueness on the generating rectangle pi-system therefore gives equality of these measures. Integrating any nonnegative product-measurable $h$ against them proves the refined Campbell identity.

If $P_1^0$ is another probability law satisfying the refined identity, use $h(x,\mu )={\mathbf{1}}_Q(x){\mathbf{1}}_A(\mu )$ to obtain $\rho P_1^0(A)=C_A(Q)=\rho P^0(A)$ for every configuration event $A$; since $\rho >0$, $P_1^0=P^0$.

Let $A_0=\{\mu :\mu (\{0\})=0\}$. For every counted occurrence $x$ of $\eta$, the recentered configuration ${\theta }_x\eta$ has an occurrence at $0$, so the left side of the refined identity with $h(x,\mu )={\mathbf{1}}_Q(x){\mathbf{1}}_{A_0}(\mu )$ is zero. Its right side is $\rho P^0(A_0)$ because $\mathrm{\mid }Q\mathrm{\mid }=1$. Hence $P^0(A_0)=0$: the full Palm law retains at least the distinguished occurrence at the origin, even when other occurrences coincide there.

The constructed law satisfies the defining identity by step 4.1, is unique by step 5.1, and has the origin-atom property by step 5.2. AC and Countable Choice [A1]–[A2] are inherited supplier assumptions; the construction itself uses no choice beyond the Lebesgue-measure supplier.