Elementary sets: the finite unions of half-open boxes in ${\mathbb{R}}^n$

Fix $n\ge 1$. A subset $E\subseteq {\mathbb{R}}^n$ is an elementary set when there are a natural number $m$ and a list $B_0,\dots ,B_{m-1}$ of half-open boxes (Half-open boxes in ${\mathbb{R}}^n$ and their volume), that is a function $j\mapsto B_j$ on $\{j\in \mathbb{N}:j<m\}$ whose values are half-open boxes, with

$E={\bigcup }_{j<m}{B}_j\mathrm{.}$

Write ${\mathcal{E}}_n$ for the family of all elementary subsets of ${\mathbb{R}}^n$.

The list is part of the data of the presentation and not of the set: one elementary set has many presentations, and nothing below reads a presentation off a set. At $m=0$ the union is empty, so $\mathrm{\varnothing }\in {\mathcal{E}}_n$; at $m=1$ every half-open box is elementary, ${\mathbb{R}}^n=(-\mathrm{\infty },+\mathrm{\infty }{]}^n$ included. The boxes of a presentation are not required to be disjoint or nonempty.

• Every elementary set does admit a disjoint presentation, by a common coordinate grid built from the parameters of the given list (Every elementary set is a finite disjoint union of half-open boxes, and any finitely many boxes admit a common grid refinement). That is a theorem about ${\mathcal{E}}_n$, not part of the definition, and keeping it out of the definition is what lets a presentation be produced by hand wherever one is convenient.

• ${\mathcal{E}}_n$ is an algebra and not merely a ring of sets (The elementary sets form an algebra of subsets of ${\mathbb{R}}^n$ containing every half-open box), and that is exactly what the infinite parameters of Half-open boxes in ${\mathbb{R}}^n$ and their volume buy: with real parameters only, every member of ${\mathcal{E}}_n$ would be bounded and the family would not contain ${\mathbb{R}}^n$.