Pair-correlation density and Ripley K-function

Assume AC and Countable Choice as carried by the point-process and Palm suppliers. Let $\eta$ be stationary and locally square integrable as in Stationarity, finite intensity, and spatial ergodicity and Second factorial and reduced second factorial moment measures, with finite positive intensity $\rho$ and reduced second factorial moment measure ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}$.

A pair-correlation density exists exactly when ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}$ is absolutely continuous with respect to Borel Lebesgue measure Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$. Local finiteness makes ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}$ sigma-finite, so A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density gives a density; positivity of the measure makes that density nonnegative almost everywhere, and replacing its values on the null exceptional set by zero gives a nonnegative Borel version. The density is unique up to Lebesgue-null sets. Normalize it by ${\rho }^2$ and call the resulting nonnegative Borel function $g$:

${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(D)={\rho }^2{\int }_{D}g(u)du$

for every Borel $D\subseteq {\mathbb{R}}^d$. The definition does not assume pair densities for all stationary processes.

For $r\ge 0$, let $B[0,r]=\{u\in {\mathbb{R}}^d:\mathrm{\parallel }u{\mathrm{\parallel }}_2\le r\}$ be the closed Euclidean ball. Ripley's K-function is

$K(r):={\rho }^{-1}\int (\mu -{\delta }_0)(B[0,r])P^0(d\mu ),$

where $P^0$ is the full Palm distribution from The full Palm distribution defined by the refined Campbell identity and ${\delta }_0$ removes one distinguished occurrence at the origin. The Palm law has an origin occurrence almost surely by Existence and uniqueness of the full Palm law and the refined Campbell identity, and the reduced-measure identity The reduced second factorial moment measure is the Palm expectation after one-occurrence deletion gives the equivalent expression $K(r)={\rho }^{-2}{\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(B[0,r])$. For a simple process as defined in Random measures and locally finite point processes, deleting ${\delta }_0$ is the same as excluding the origin location; for a multiplicity-bearing process other co-located occurrences remain.

Axiom convention. AC and Countable Choice are inherited through the point-process and Palm dependency closure; no further choice is used by these definitions.