The full Palm distribution defined by the refined Campbell identity

Assume AC and Countable Choice as carried by the point-process supplier. Let $\eta$ be a stationary process in the sense of Stationarity, finite intensity, and spatial ergodicity, with finite positive intensity $\rho$. Its full Palm distribution $P^0$ is the unique probability measure on the configuration space such that

$\mathbb{E}\int h(x,{\theta }_x\eta )\eta (dx)=\rho {\int }_{{\mathbb{R}}^d}\int h(x,\mu )P^0(d\mu )dx$

for every nonnegative function $h$ measurable for the product of the Borel and configuration evaluation sigma-algebras. This definition uses the full point-centered configuration, not conditioning on the probability-zero event that an atom occurs at the origin. Existence, uniqueness, and retention of the distinguished origin occurrence are proved in Existence and uniqueness of the full Palm law and the refined Campbell identity ↗.