Papangelou conditional intensity on a finite window

Let $W$, ${\mathrm{\ell }}_W$, and $\mathcal{C}(W)$ be as in Finite-window spatial observations and configurations. Let $P_{\theta ,b}$ be a specified probability law on $\mathcal{C}(W)$; $b$ denotes a fixed exterior configuration when the model uses a boundary condition, and is omitted when it does not. For this fixed boundary convention, a jointly measurable function ${\lambda }_{\theta }:W\times \mathcal{C}(W)\to [0,+\mathrm{\infty }]$ is a Papangelou conditional intensity relative to ${\mathrm{\ell }}_W$ for this law and boundary convention; the notation ${\lambda }_{\theta }(u;\eta ,b)$ displays the fixed boundary label but $b$ is not an additional variable in its measurable domain. Omit $b$ when there is no exterior condition or when the fixed convention is clear from context. The defining identity holds when, for every nonnegative product-measurable $h:W\times \mathcal{C}(W)\to [0,\mathrm{\infty }]$, ${\mathbb{E}}_{\theta ,b}[{\sum }_{x\in \eta }h(x,\eta -{\delta }_x)]={\mathbb{E}}_{\theta ,b}[{\int }_{W}h(u,\eta ){\lambda }_{\theta }(u;\eta ,b){\mathrm{\ell }}_W(du)]\mathrm{.}$ The sum runs over atom occurrences, so $\eta -{\delta }_x$ deletes one occurrence; if $b$ is used, it is fixed on both sides. The identity is the definition of the Papangelou intensity used on this page. It does not assert that every finite-window law has such a function.

The deletion operation and the resulting measurable random sum use Deleting one occurrence from a locally finite counting measure is measurable. On the occurrence domain, the deletion-composition is measurable; extend it by zero off that measurable domain and integrate against the evaluation kernel $K(\eta ,B)=\eta (B)$. This integral is the displayed sum, and its measurability follows from the deletion map and Measurability of integration against a kernel, as used in the deletion lemma.