Marked periodic continuation and local restriction

Let $\mathrm{\Omega }$ be the positional configuration space of Poisson Configuration Reference. The marked configuration space ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$ consists of marked point sets $\xi \subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ with spatial projection $q=\{x\in {\mathbb{R}}^d:\mathrm{\exists }p\in {\mathbb{R}}^d,(x,p)\in \xi \}$ that is simple and locally finite, and with exactly one mark $p_x$ paired to each $x\in q$. Its sigma-algebra is generated by the marked counts $N_{A\times C}(\xi )=\mathrm{\#}\{(x,p)\in \xi :x\in A,p\in C\}$ for bounded Borel $A\subseteq {\mathbb{R}}^d$ and Borel $C\subseteq {\mathbb{R}}^d$. The positional projection is $\pi (\xi )=q\in \mathrm{\Omega }$. For a bounded Borel set $\mathrm{\Delta }$, the marked local restriction is ${\xi }_{\mathrm{\Delta }}=\{(x,p)\in \xi :x\in \mathrm{\Delta }\}$ with the induced count sigma-algebra; write $N_{\mathrm{\Delta }}(\xi )=\mathrm{\#}{\xi }_{\mathrm{\Delta }}$. A marked-local observable on $\mathrm{\Delta }$ is a real measurable function of ${\xi }_{\mathrm{\Delta }}$; it is bounded when its supremum norm is finite.

Let $d\ge 1$, $L>0$, and ${\mathrm{\Lambda }}_L=[0,L{)}^d$. A finite marked configuration consists of a finite simple set of positions $q\subseteq {\mathrm{\Lambda }}_L$ and a mark map $p:q\to {\mathbb{R}}^d$, as in Interacting marked phase-space and reference measures. Its marked periodic continuation is ${\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}:=\{(x+Lk,p_x):x\in q,k\in {\mathbb{Z}}^d\}\mathrm{.}$ The spatial positions in this set are distinct: if $x+Lk=y+Lj$ for $x,y\in [0,L{)}^d$, then coordinatewise $x-y=L(j-k)$ forces $j=k$ and $x=y$. Every bounded spatial region meets finitely many lattice translates of the finite set $q$, so the continuation has a locally finite spatial projection.

For a bounded Borel set $\mathrm{\Delta }\subseteq {\mathbb{R}}^d$, define its marked local restriction by ${\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}{\mathrm{\mid }}_{\mathrm{\Delta }}:=\{(y,p_x)\in {\omega }_{L,q,p}^{\mathrm{p}\mathrm{e}\mathrm{r}}:y\in \mathrm{\Delta }\}\mathrm{.}$ This is an element of ${\mathrm{\Omega }}^{\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}}$, with the induced sigma-algebra of marked counts above. If $\mathrm{\Delta }\subseteq {\mathrm{\Lambda }}_L$, the restriction is exactly the original marked points with positions in $\mathrm{\Delta }$.