Stationarity, finite intensity, and spatial ergodicity

Work with a point process $\eta$ on ${\mathbb{R}}^d$ and its law $P_{\eta }$ on the evaluation-measurable configuration space from Random measures and locally finite point processes. Recentering ${\theta }_x$ is as in Recentering a point configuration at a spatial location. The process is stationary if ${\theta }_x\eta$ has law $P_{\eta }$ for every $x\in {\mathbb{R}}^d$.

Put $Q=[0,1{)}^d$. Its intensity is $\rho :=\mathbb{E}[\eta (Q)]$. It has finite positive intensity when $0<\rho <\mathrm{\infty }$. The item A stationary point process of finite intensity has mean measure rho times Lebesgue measure proves that its mean measure is $\rho {\lambda }_d$ under the stated measure-theoretic assumptions.

The translation-invariant sigma-algebra is ${\mathcal{I}}_{\eta }:=\{A\text{ in the configuration sigma-algebra}:P_{\eta }({\theta }_x^{-1}A\mathrm{△}A)=0\text{ for every }x\in {\mathbb{R}}^d\}\mathrm{.}$ It is a sigma-algebra because symmetric-difference nullity is preserved by complements and countable unions. The stationary point process is ergodic under translations if every $A\in {\mathcal{I}}_{\eta }$ has $P_{\eta }(A)\in \{0,1\}$. This is the full ${\mathbb{R}}^d$ action convention, not ergodicity of one selected unit shift.

Axiom convention. This item carries AC and Countable Choice through the transitive well-definedness dependencies of the point-process supplier; stationarity and the scalar intensity definition themselves use no choices.