Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$

Let $n\ge1$ be a natural number and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$. Sums and negatives are those of the real vector space $\mathbb{R}^n$, and for $a\in\mathbb{R}^n$ and $B\subseteq\mathbb{R}^n$ we write

$$B+a=\{x+a:x\in B\},\qquad a-B=\{a-x:x\in B\}.$$

Measurability and integrals of $[0,\infty]$-valued functions are those of Lebesgue Integral of a Nonnegative Measurable Function, and integrable means integrable. Fix $a\in\mathbb{R}^n$.

1. (Sets) For every $B\in\mathcal{B}(\mathbb{R}^n)$ the sets $B+a$ and $a-B$ are Borel, and

$$\lambda_n(B+a)=\lambda_n(a-B)=\lambda_n(B).$$

2. (Nonnegative integrals) For every measurable $f:\mathbb{R}^n\to[0,\infty]$ the maps $x\mapsto f(x+a)$ and $x\mapsto f(a-x)$ are measurable and

$$\int_{\mathbb{R}^n}f(x+a)\,d\lambda_n(x)=\int_{\mathbb{R}^n}f(a-x)\,d\lambda_n(x)=\int_{\mathbb{R}^n}f\,d\lambda_n\qquad\text{in }[0,\infty].$$

3. (Integrable functions) Let $f:\mathbb{R}^n\to\mathbb{R}$ be measurable with respect to $\mathcal{B}(\mathbb{R}^n)$ and the Borel $\sigma$-algebra of the real line. Then $f$ is integrable with respect to $\lambda_n$ if and only if $x\mapsto f(x+a)$ is, if and only if $x\mapsto f(a-x)$ is, and in that case the two displayed identities of claim 2 hold in $\mathbb{R}$.