A positive finite-window Gibbs density has its density-ratio Papangelou intensity

Extend $g$ and $h$ by zero outside $\mathcal{C}(W)$ and $W\times \mathcal{C}(W)$, respectively; [F5] makes these measurable extensions. For a test function $h$ as in the Statement, define on the domain where $(x,\zeta )$ contains an occurrence at $x$ $k(x,\zeta ):=g(\zeta )h(x,\zeta -{\delta }_x),$ and set $k=0$ off that domain. The domain and deletion map are measurable by [F2]; the extensions, addition operation, and $g,h$ are measurable by [F4, F5], so $k$ is a nonnegative measurable test function for [F1]. Also the addition map is measurable by [F4], and since $g$ is finite and strictly positive, its ratio ${\lambda }_g(u;\eta )=g(\eta +{\delta }_u)\mathrm{/}g(\eta )$ is a measurable function on $W\times \mathcal{C}(W)$.

The left side of [F1] with this $k$ is ${\mathbb{E}}_Q[g(\xi ){\sum }_{x\in \xi }h(x,\xi -{\delta }_x)]={\mathbb{E}}_P{\sum }_{x\in \eta }h(x,\eta -{\delta }_x)\mathrm{.}$ On the right side, $(u,\xi +{\delta }_u)$ is in the deletion domain and $(\xi +{\delta }_u)-{\delta }_u=\xi$, so the integrand is $g(\xi +{\delta }_u)h(u,\xi )=g(\xi ){\lambda }_g(u;\xi )h(u,\xi )$. Consequently [F1] gives, after Tonelli's interchange [F6], ${\mathbb{E}}_P{\sum }_{x\in \eta }h(x,\eta -{\delta }_x)={\mathbb{E}}_P{\int }_{W}h(u,\eta ){\lambda }_g(u;\eta ){\mathrm{\ell }}_W(du)\mathrm{.}$ which is the defining identity [F3].

Step 2.1 proves the claimed Papangelou identity for every admissible nonnegative $h$, hence proves the Statement.