The reduced second factorial moment measure is the Palm expectation after one-occurrence deletion

Write each realization with occurrence indices, $\eta ={\sum }_i{\delta }_{X_i}$. For each selected occurrence $i$, recentering and subtracting one unit atom gives $({\theta }_{X_i}\eta -{\delta }_0)(D)={\sum }_{j\mathrm{\ne }i}{\mathbf{1}}_D(X_j-X_i)\mathrm{.}$ Indeed, $X_i$ moves to $0$, and only the selected index is removed; if $X_j=X_i$ for another index $j$, that remaining occurrence is still counted at $0$.

Define on the product of location and configuration spaces $h(x,\mu )=\{\begin{array}{ll}{\mathbf{1}}_Q(x)(\mu (D)-{\mathbf{1}}_D(0)),& \mu (\{0\})\ge 1,\\ 0,& \mu (\{0\})=0.\end{array}$ This is nonnegative and product measurable: the two cases are measurable by the configuration evaluation sigma-algebra, and on the first case $\mu -{\delta }_0$ is a counting measure. For every counted occurrence $x=X_i$, [F2] puts ${\theta }_x\eta$ in the first case, and $h(x,{\theta }_x\eta )={\mathbf{1}}_Q(x)({\theta }_x\eta -{\delta }_0)(D)$.

Summing the identity in step 1.1 over the selected indices with $X_i\in Q$ and taking expectations, [F1] gives ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(D)=\mathbb{E}\int {\mathbf{1}}_Q(x)({\theta }_x\eta -{\delta }_0)(D)\eta (dx)\mathrm{.}$ This remains valid for extended nonnegative values by monotone summation.

Apply [F3] to the integrand $h$ from step 1.2. The result is ${\alpha }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{(2)}(D)=\rho \int \int h(x,\mu )P^0(d\mu )dx=\rho \mathrm{\mid }Q\mathrm{\mid }\int (\mu -{\delta }_0)(D)P^0(d\mu )\mathrm{.}$ Since $\mathrm{\mid }Q\mathrm{\mid }=1$, this is the asserted identity. AC and Countable Choice [A1]–[A2] are inherited supplier assumptions, not additional steps in the occurrence calculation.