Van Hove sequences of spatial windows

Fix an integer $d\ge 1$ and assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). By Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable, every Borel subset of ${\mathbb{R}}^d$ is Lebesgue measurable under this assumption; by 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, $\mathrm{\mid }\cdot \mathrm{\mid }$ is the resulting complete Lebesgue measure. A Van Hove sequence of spatial windows is a sequence $(W_n{)}_{n\ge 0}$ of bounded Borel sets (The Borel sigma-algebra of a topological space) in ${\mathbb{R}}^d$ with $0<\mathrm{\mid }{W}_n\mathrm{\mid }<\mathrm{\infty }$ for every $n$, $\mathrm{\mid }{W}_n\mathrm{\mid }\to \mathrm{\infty }$, and $\frac{\mathrm{\mid }(W_n+K)\setminus W_n\mathrm{\mid }}{\mathrm{\mid }{W}_n\mathrm{\mid }}⟶0\text{for every compact }K\subseteq {\mathbb{R}}^d\mathrm{.}$ Here $\mathrm{\mid }E\mathrm{\mid }$ is Lebesgue measure when $E$ is Lebesgue measurable, and $W_n+K:=\{x+y:x\in W_n,y\in K\}$. As part of the condition, each boundary excess $(W_n+K)\setminus W_n$ in the display is required to be Lebesgue measurable. Thus the window volume diverges while every fixed compact spatial enlargement adds a negligible relative volume.