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 Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; compactness refers to the topology of the open sets of $(\mathbb{R}^{n},d_{E})$, a topology by The Open Subsets of a Metric Space 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 Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

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

$$B=\{x\in\mathbb{R}^{n}:c_{i}\le x_{i}<c_{i}+s\ \text{ for every }i\in[n]\}.$$

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

$$Q_{m,j}=\Bigl\{x\in\mathbb{R}^{n}:c_{i}+(j_{i}-1)\tfrac{s}{m}\le x_{i}<c_{i}+j_{i}\tfrac{s}{m}\ \text{ for every }i\in[n]\Bigr\},$$

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) (i) 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 $j\in[m]^{n}$,

$$\lambda_{n}(Q_{m,j})=\Bigl(\frac{s}{m}\Bigr)^{n},\qquad \lVert x-y\rVert\le\sigma_{n}\,\frac{s}{m}\ \text{ for all }x,y\in Q_{m,j}.$$

2. (Refinement) (ii) 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) (iii) Let $m\in\mathbb{N}$ and let $K\subseteq B$ be nonempty. Put

$$J_{m}=\{j\in[m]^{n}:Q_{m,j}\cap K\ne\varnothing\},\qquad E_{m}=\bigcup_{j\in J_{m}}Q_{m,j}.$$

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

$$\lambda_{n}(E_{m})=|J_{m}|\Bigl(\frac{s}{m}\Bigr)^{n}<\infty .$$

4. (Exhaustion of a compact set) (iv) 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

$$K\subseteq E_{k+1}^{\ast}\subseteq E_{k}^{\ast}\ \text{ for every }k\in\mathbb{N},\qquad \bigcap_{k\in\mathbb{N}}E_{k}^{\ast}=K,$$

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

$$\lambda_{n}(E_{k}^{\ast})\le\lambda_{n}(K)+\varepsilon .$$