The Integral over the Unit Cell of a Product of One-Variable Functions

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the initial segments $[n]$, Euclidean space $\mathbb{R}^{n}$, the half-open unit cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ together with the notation $\int_{\mathbb{T}^{n}}v\,dx$, and the integral and the notion of an integrable map are the ones fixed there. Let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of the real line and let $\lambda$ be Lebesgue measure on it. Finite products of real numbers are the finite products of that definition, formed in the field of real numbers.

Put

$$J=\{t\in\mathbb{R}:0\le t<1\}.$$

Then $J$ is an interval, since $0\le y$ and $y<1$ whenever $x\le y\le z$ with $x,z\in J$; hence $J\in\mathcal{B}(\mathbb{R})$ by Borel Sigma-Algebra on the Real Line. The closed intervals $[0,1]$ and $[1,1]=\{1\}$ are intervals too, hence also lie in $\mathcal{B}(\mathbb{R})$, and claim 4 of Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness gives $\lambda([0,1])=1$ and $\lambda(\{1\})=0$. Since $\{1\}\subseteq[0,1]$, $\lambda([0,1])$ is finite and $J=[0,1]\setminus\{1\}$, claim 3 of Basic Properties of a Measure gives $\lambda(J)=1-0=1$. Let $(J,\mathcal{B}_{J},\lambda_{J})$ denote the restriction of $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ to $J$ furnished by claim 1 of that lemma, so that $\mathcal{B}_{J}=\{A\in\mathcal{B}(\mathbb{R}):A\subseteq J\}$ and $\lambda_{J}(A)=\lambda(A)$. By the description of the cell recorded in The Half-Open Unit Cell Tiles Euclidean Space (i), a point $x\in\mathbb{R}^{n}$ lies in $Q$ if and only if $x_{i}\in J$ for every $i\in[n]$.

Let $g_{i}:J\to\mathbb{R}$ be measurable with respect to $\mathcal{B}_{J}$, for every $i\in[n]$. Then the following hold.

1. (The product function)(i) Setting

$$G(x)=\prod_{i=1}^{n}g_{i}(x_{i})\qquad(x\in Q)$$

defines a map $G:Q\to\mathbb{R}$, and $G$ is measurable with respect to $\mathcal{B}_{Q}$.

2. (The integral of the product)(ii) Suppose in addition that for every $i\in[n]$ there is a real number $M_{i}$ with $|g_{i}(t)|\le M_{i}$ for every $t\in J$, where $|\,\cdot\,|$ is the absolute value. Then each $g_{i}$ is integrable with respect to $\lambda_{J}$, the map $G$ is integrable with respect to $\lambda_{Q}$, and

$$\int_{\mathbb{T}^{n}}G\,dx=\prod_{i=1}^{n}\int_{J}g_{i}\,d\lambda_{J}.$$