The Cell Integral of a Translated Periodic Function

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; Euclidean space $\mathbb{R}^{n}$ with the difference $y-x$ of its points, Lebesgue measure $\lambda_{n}$ on $\mathcal{B}(\mathbb{R}^{n})$, the integer lattice $\mathbb{Z}^{n}$, the cell $Q$ and $\mathbb{Z}^{n}$-periodicity of a map on $\mathbb{R}^{n}$ are the ones fixed there. Measurability of a map on $\mathbb{R}^{n}$ means measurability with respect to $\mathcal{B}(\mathbb{R}^{n})$, integrals over $\mathbb{R}^{n}$ are integrals with respect to $\lambda_{n}$ in the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$, and $\mathbf{1}_{A}$ is the indicator of $A\subseteq\mathbb{R}^{n}$, with $\mathbf{1}_{A}v$ the pointwise product of the indicator and a map $v$.

Let $w:\mathbb{R}^{n}\to\mathbb{R}$ be measurable and $\mathbb{Z}^{n}$-periodic, and suppose that $\mathbf{1}_{Q}w$ is integrable. Let $x\in\mathbb{R}^{n}$, and let $w_{x}:\mathbb{R}^{n}\to\mathbb{R}$ be the map $w_{x}(y)=w(y-x)$.

(i) Then $w_{x}$ is measurable and $\mathbb{Z}^{n}$-periodic, the map $\mathbf{1}_{Q}w_{x}$ is integrable, and

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,w_{x}\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,w\,d\lambda_{n} .$$