Dyadic Cells in $\mathbb{R}^n$ and the Decomposition of an Open Set

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$ and notion of openness on $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$ with its conventions for $[0,\infty]$ and for sums of sequences there, and the constant $\sigma_{n}$ with $\sigma_{n}^{2}=n$, are all as fixed in that setting. As fixed in clause 1 of that setting, $\mathbb{Z}^{n}$ is the set of $n$-tuples of integers, countability is as defined there, and $2^{-k}$ denotes the multiplicative inverse of $2^{k}$ for $k\in\mathbb{N}$.

For $k\in\mathbb{N}$ and $j=(j_{1},\dots,j_{n})\in\mathbb{Z}^{n}$ put

$$Q_{k,j}=\{x\in\mathbb{R}^{n}: j_{i}2^{-k}\le x_{i}<(j_{i}+1)2^{-k}\ \text{ for every }i\in\{1,\dots,n\}\},$$

called the dyadic cell of generation $k$ and index $j$, and let $\mathcal{Q}=\{Q_{k,j}:k\in\mathbb{N},\ j\in\mathbb{Z}^{n}\}$ be the set of all dyadic cells. (The similar notation of Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$ refers to the cells of a uniform grid on a fixed half-open box, indexed by a finite set; the cells $Q_{k,j}$ introduced here are indexed by $\mathbb{Z}^{n}$ and cover all of $\mathbb{R}^{n}$.) Then the following hold.

1. (Each generation is a countable partition) (i) Let $k\in\mathbb{N}$. Every $x\in\mathbb{R}^{n}$ lies in exactly one set $Q_{k,j}$ with $j\in\mathbb{Z}^{n}$; the sets $Q_{k,j}$ with $j\in\mathbb{Z}^{n}$ are pairwise disjoint members of $\mathcal{B}(\mathbb{R}^{n})$ whose union is $\mathbb{R}^{n}$; the index set $\mathbb{Z}^{n}$ and the set $\mathbb{N}\times\mathbb{Z}^{n}$ are countable and infinite, and the map $(k,j)\mapsto Q_{k,j}$ from $\mathbb{N}\times\mathbb{Z}^{n}$ to $\mathcal{Q}$ is injective; and for every $j\in\mathbb{Z}^{n}$,

$$\lambda_{n}(Q_{k,j})=(2^{-k})^{n},\qquad \lVert x-y\rVert\le\sigma_{n}2^{-k}\ \text{ for all }x,y\in Q_{k,j}.$$

2. (Nesting) (ii) Let $k,k'\in\mathbb{N}$ with $k\le k'$ and let $j,j'\in\mathbb{Z}^{n}$. Then either $Q_{k',j'}\subseteq Q_{k,j}$ or $Q_{k',j'}\cap Q_{k,j}=\varnothing$; and there is exactly one $j''\in\mathbb{Z}^{n}$ with $Q_{k',j'}\subseteq Q_{k,j''}$. Consequently any two dyadic cells are either disjoint or one of them is contained in the other.

3. (Decomposition of an open set) (iii) Let $U\subseteq\mathbb{R}^{n}$ be open. Then there is a sequence $(P_{m})_{m\in\mathbb{N}}$ of pairwise disjoint members of $\mathcal{B}(\mathbb{R}^{n})$, each of which is either empty or a dyadic cell contained in $U$, whose union is $U$; and for every such sequence

$$\lambda_{n}(U)=\sum_{m\in\mathbb{N}}\lambda_{n}(P_{m}).$$

4. (Small cells around a point) (iv) Let $x\in\mathbb{R}^{n}$ and let $r\in\mathbb{R}$ with $0<r$. Then there is $k\in\mathbb{N}$ with $\sigma_{n}2^{-k}\le r$, and for every such $k$ the cell of generation $k$ containing $x$ satisfies

$$\lVert y-x\rVert\le r\qquad\text{for every }y\in Q_{k,j}\text{ with }x\in Q_{k,j}.$$