Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers, and let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^{n}$, which is an open subset of itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $\lambda_{n}$ be Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}^{n}$, and let $\delta\in\mathbb{R}$ with $0<\delta$.

A map $\rho:\mathbb{R}^{n}\to\mathbb{R}$ is a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$ if the following four conditions hold.

1. (Smoothness) $\rho$ is smooth on $\mathbb{R}^{n}$.

2. (Nonnegativity) $0\le\rho(y)$ for every $y\in\mathbb{R}^{n}$.

3. (Support) $\rho(y)=0$ for every $y\in\mathbb{R}^{n}$ with $\delta<\lVert y\rVert$.

4. (Unit mass) $\rho$ is integrable with respect to $\lambda_{n}$, and

$$\int_{\mathbb{R}^{n}}\rho\,d\lambda_{n}=1.$$