Rescaling a Mollifier Kernel

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers, and let $\delta,\varepsilon\in\mathbb{R}$ satisfy $0<\delta$ and $0<\varepsilon$. Write $t^{-1}$ for the multiplicative inverse of $t\ne 0$, and let powers with a natural exponent be those of Natural Number Power of an Element of a Field. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, so that $t\,y$ denotes the scalar multiple of $y\in\mathbb{R}^{n}$ by $t\in\mathbb{R}$.

Let $\rho:\mathbb{R}^{n}\to\mathbb{R}$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, and let $\rho_{\varepsilon}:\mathbb{R}^{n}\to\mathbb{R}$ be given by

$$\rho_{\varepsilon}(y)=\bigl(\varepsilon^{-1}\bigr)^{n}\,\rho\bigl(\varepsilon^{-1}y\bigr)\qquad(y\in\mathbb{R}^{n}).$$

Then $\rho_{\varepsilon}$ is a mollifier kernel of radius $\varepsilon\delta$ on $\mathbb{R}^{n}$.