Scaling of Lebesgue Measure and the Lebesgue Integral on $\mathbb{R}^n$

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

$$cB=\{cx:x\in B\}.$$

Powers with a natural exponent are those of Natural Number Power of an Element of a Field. Measurability and integrals of $[0,\infty]$-valued functions are those of Lebesgue Integral of a Nonnegative Measurable Function, with the conventions of Measure, Measure Space, and Probability Measure extended by $a\cdot\infty=\infty\cdot a=\infty$ for $0<a\le\infty$, and integrable means integrable. Fix a real number $c\ne0$ and write $|c|^{-n}$ for the multiplicative inverse of $|c|^{n}$.

1. (Sets) For every $B\in\mathcal{B}(\mathbb{R}^n)$ the set $cB$ is Borel, and

$$\lambda_n(cB)=|c|^{n}\,\lambda_n(B).$$

2. (Nonnegative integrals) For every measurable $f:\mathbb{R}^n\to[0,\infty]$ the map $x\mapsto f(cx)$ is measurable and

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

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 $x\mapsto f(cx)$ is, and in that case the identity of claim 2 holds in $\mathbb{R}$.