Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$

Let $n$ be a natural number with $1\le n$, let $\mathbb{N}$ be the natural numbers, and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $x-y$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ with $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; compactness refers to the topology of the open sets of $(\mathbb{R}^{n},d_{E})$, a topology by Metric Open Sets Form a Topology. Let $\mathcal{B}(\mathbb{R}^{n})$ be the Borel $\sigma$-algebra and $\lambda_{n}$ Lebesgue measure on it. For $m\in\mathbb{N}$ let $[m]$ be the initial segment determined by $m$ and $[m]^{n}$ the set of $n$-tuples in $[m]$; $|X|$ denotes the number of elements of a finite set $X$. Powers with natural exponent are those of Natural Number Power of an Element of a Field. Finally let $e=(1,\dots,1)\in\mathbb{R}^{n}$ and put $\sigma_{n}=\lVert e\rVert$, so that $0<\sigma_{n}$ and $\sigma_{n}^{2}=n$ by claims 1 and 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

Fix $c\in\mathbb{R}^{n}$ and $s\in\mathbb{R}$ with $0<s$, and put

For $m\in\mathbb{N}$ and $j\in[m]^{n}$ put

natural numbers being read as real numbers as in claim 3 of The Real Numbers and Standard Notation. Then the following hold.

1. (The grid) ¶ Let $m\in\mathbb{N}$. The set $[m]^{n}$ is nonempty and finite; the sets $Q_{m,j}$ with $j\in[m]^{n}$ are pairwise disjoint members of $\mathcal{B}(\mathbb{R}^{n})$ whose union is $B$; and for every $j\in[m]^{n}$,

2. (Refinement) ¶ Let $m\in\mathbb{N}$ and $j'\in[2m]^{n}$. Then there is exactly one $j\in[m]^{n}$ with $Q_{2m,j'}\subseteq Q_{m,j}$.

3. (Grid hull) ¶ Let $m\in\mathbb{N}$ and let $K\subseteq B$ be nonempty. Put

Then $J_{m}$ is nonempty and finite, $E_{m}\in\mathcal{B}(\mathbb{R}^{n})$, $K\subseteq E_{m}\subseteq B$, and

4. (Exhaustion of a compact set) ¶ Let $K\subseteq B$ be nonempty and compact in $\mathbb{R}^{n}$, and let $(m_{k})_{k\in\mathbb{N}}$ be the sequence in $\mathbb{N}$ determined by $m_{1}=1$ and $m_{k+1}=2m_{k}$. Write $E_{k}^{\ast}=E_{m_{k}}$ for the grid hull of claim 3 at mesh index $m_{k}$. Then

and for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $k\in\mathbb{N}$ with