The Half-Open Unit Cell Tiles Euclidean Space

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, used here with a natural number $n$ satisfying $1\le n$, and in the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, whose measure space $(X,\mathcal{F},\mu)$ is instantiated throughout as $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$; this is a measure space because $\mathcal{B}(\mathbb{R}^{n})$ is a $\sigma$-algebra on $\mathbb{R}^{n}$ and $\lambda_{n}$ is a measure on it. The two settings fix the real numbers, the natural numbers, the integers and the initial segments $[n]$ by reference to the same definitions, so their readings agree. From the second we take the extended half-line $[0,\infty]$ with its arithmetic and order, measurability of maps into $\mathbb{R}$ and into $[0,\infty]$, the indicator $\mathbf{1}_{A}$ of a subset $A$, and the integral and the notion of an integrable map; from the first, Euclidean space with its norm, distance, balls, topology and the notions of open, closed, bounded and compact set.

Let $\mathbb{Z}^{n}$ be the integer lattice and let $\mathbb{Z}^{n}$-periodicity of a map on $\mathbb{R}^{n}$ be as defined there, for maps into $\mathbb{R}$ and for maps into $[0,\infty]$ alike. For $A\subseteq\mathbb{R}^{n}$ and $h\in\mathbb{R}^{n}$ put $A+h=\{x+h:x\in A\}$. Put

$$Q=\{x\in\mathbb{R}^{n}:0\le x_{i}<1\ \text{for every}\ i\in[n]\},\qquad \mathring{Q}=\{x\in\mathbb{R}^{n}:0<x_{i}<1\ \text{for every}\ i\in[n]\},$$ $$\overline{Q}=\{x\in\mathbb{R}^{n}:0\le x_{i}\le1\ \text{for every}\ i\in[n]\}.$$

Then the following hold.

1. (The cell)(i) $\mathring{Q}\subseteq Q\subseteq\overline{Q}$. The set $\mathring{Q}$ is open and is the interior of $Q$; the set $\overline{Q}$ is closed and compact and is the closure of $Q$; all three sets belong to $\mathcal{B}(\mathbb{R}^{n})$; and

$$\lambda_{n}(\mathring{Q})=\lambda_{n}(Q)=\lambda_{n}(\overline{Q})=1 .$$

2. (Tiling)(ii) For every $x\in\mathbb{R}^{n}$ there is exactly one $m\in\mathbb{Z}^{n}$ with $x-m\in Q$. Consequently the sets $Q+m$, for $m\in\mathbb{Z}^{n}$, are pairwise disjoint and their union is $\mathbb{R}^{n}$.

3. (The wrapping map)(iii) Let $\pi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map sending $x$ to $x-m$, where $m$ is the unique integer vector provided by claim 2; it is well defined by that claim. Then $\pi(x)\in Q$ for every $x\in\mathbb{R}^{n}$; $\pi(x)=x$ if and only if $x\in Q$; $\pi(x+k)=\pi(x)$ for every $x\in\mathbb{R}^{n}$ and every $k\in\mathbb{Z}^{n}$; the $i$th coordinate of $\pi(x)$ equals $x_{i}-\lfloor x_{i}\rfloor$ for every $i\in[n]$, where $\lfloor\,\cdot\,\rfloor$ is the integer part; and $\pi$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R}^{n})$.

4. (Cell integrals of a periodic function)(iv) Let $u:\mathbb{R}^{n}\to[0,\infty]$ be measurable and $\mathbb{Z}^{n}$-periodic and let $h\in\mathbb{R}^{n}$. Then $Q+h\in\mathcal{B}(\mathbb{R}^{n})$, the maps $\mathbf{1}_{Q}u$ and $\mathbf{1}_{Q+h}u$ are measurable, and

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,u\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,u\,d\lambda_{n}\qquad\text{in }[0,\infty].$$

5. (The integrable case)(v) Let $u:\mathbb{R}^{n}\to\mathbb{R}$ be measurable and $\mathbb{Z}^{n}$-periodic, let $h\in\mathbb{R}^{n}$, and suppose that $\mathbf{1}_{Q}u$ is integrable. Then $\mathbf{1}_{Q+h}u$ is integrable and

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,u\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,u\,d\lambda_{n}\qquad\text{in }\mathbb{R}.$$