A pair-correlation density integrates to Ripley K

Apply [F2] with $D=B[0,r]$ and divide by ${\rho }^2>0$. By the K definition [F1], $K(r)={\rho }^{-2}{\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(B[0,r])\mathrm{.}$

Suppose $\eta$ is simple. The event $A_2=\{\mu :\mu (\{0\})\ge 2\}$ is measurable by the evaluation sigma-algebra. In the refined Campbell identity [F3], take $h(x,\mu )={\mathbf{1}}_Q(x){\mathbf{1}}_{A_2}(\mu )$ with $Q=[0,1{)}^d$. On the probability-one event of simple realizations, every counted occurrence $x$ has $({\theta }_x\eta )(\{0\})=\eta (\{x\})=1$, so the left side is zero. The right side is $\rho \mathrm{\mid }Q\mathrm{\mid }{P}^0(A_2)=\rho P^0(A_2)$, hence $P^0(A_2)=0$. Combining this with the at-least-one origin property in [F3] gives $\mu (\{0\})=1$ $P^0$-almost surely. Therefore $(\mu -{\delta }_0)(B[0,r])=\mu (B[0,r]\setminus \{0\})$ $P^0$-almost surely, and the definition [F1] gives the stated simple-process formula.

The density clause in [F1] applied to the same closed ball gives ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(B[0,r])={\rho }^2{\int }_{B[0,r]}g(u)du$. Substitution in step 1.1 proves the asserted integral formula.

At the endpoint $r=0$, the closed-ball convention gives $B[0,0]=\{0\}$, so $K(0)={\rho }^{-1}{\mathbb{E}}^0[\mu (\{0\})-1]$; this is $0$ for a simple process, while any other co-located occurrences contribute in the multiplicity-bearing case. The proof uses no density-existence claim beyond the stated hypothesis. AC and Countable Choice [A1]–[A2] are inherited supplier assumptions, not additional derivation steps.