Finite-window spatial observations and configurations

Fix an integer $d\ge 1$ and a bounded Borel set $W\subset {\mathbb{R}}^d$. Under Countable Choice every Borel subset of ${\mathbb{R}}^d$ is Lebesgue measurable (Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable), so Lebesgue measure ${\mathrm{\ell }}_d$ is defined on each such set. Write $\mathrm{\mid }A\mathrm{\mid }:={\mathrm{\ell }}_d(A)$ for Borel $A$, and ${\mathcal{B}}_W:=\{W\cap A:A\in \mathcal{B}({\mathbb{R}}^d)\}$ for the trace Borel sigma-algebra on $W$. Every member of ${\mathcal{B}}_W$ is Borel in ${\mathbb{R}}^d$ because $W$ is Borel. Define ${\mathrm{\ell }}_W:{\mathcal{B}}_W\to [0,+\mathrm{\infty }]$ by ${\mathrm{\ell }}_W(A):={\mathrm{\ell }}_d(A)$. If $(A_k)$ is pairwise disjoint in ${\mathcal{B}}_W$, then each $A_k$ and their union are Borel in ${\mathbb{R}}^d$, so countable additivity of ${\mathrm{\ell }}_d$ gives ${\mathrm{\ell }}_W\text{\hspace{0.17em}⁣}\left({\bigcup }_k{A}_k\right)={\mathrm{\ell }}_d\text{\hspace{0.17em}⁣}\left({\bigcup }_k{A}_k\right)={\sum }_k{\mathrm{\ell }}_d(A_k)={\sum }_k{\mathrm{\ell }}_W(A_k),$ and ${\mathrm{\ell }}_W(\mathrm{\varnothing })=0$. Thus ${\mathrm{\ell }}_W$ is a measure on the trace sigma- algebra. Since $W$ is bounded, ${\mathrm{\ell }}_W(W)<\mathrm{\infty }$ (Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure). The only choice axiom used to regard ${\mathrm{\ell }}_d$ as a measure in this library is Countable Choice, as recorded in Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume.

The finite configuration space on $W$, denoted $\mathcal{C}(W)$, is the measurable subspace of locally finite counting measures on ${\mathbb{R}}^d$ consisting of those $\eta$ with $\eta (W^c)=0$ and $\eta ({\mathbb{R}}^d)<\mathrm{\infty }$. It is measurable in the ambient evaluation sigma-algebra because both $\eta \mapsto \eta (W^c)$ and $\eta \mapsto \eta ({\mathbb{R}}^d)$ are evaluation maps for Borel sets. Thus each $\eta \in \mathcal{C}(W)$ has a representation $\eta ={\sum }_{i=1}^n{\delta }_{x_i}$ for a finite $n\ge 0$ and $x_i\in W$; repeated locations are allowed and represent multiplicity. Its sigma-algebra is the trace of the evaluation sigma-algebra: it is generated by $\eta \mapsto \eta (B)$ for Borel $B\subseteq W$.

For a finite ambient configuration $\tilde{\eta }$, its observation in $W$ is $\eta =\tilde{\eta }{\mathrm{\mid }}_W$, where $\eta (B)=\tilde{\eta }(B\cap W)$. The observation includes precisely the locations belonging to $W$: a boundary location is included when and only when it belongs to $W$. This convention does not replace $W$ by its interior or its closure.

This definition allows multiplicities and makes no simplicity claim. The separate finite-intensity Poisson simplicity argument is in A finite Poisson process with atomless intensity is simple.