Mollification Converges Uniformly on Compact Subsets

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure, write $s<t$ to mean that $s\le t$ and $s\ne t$, write $t^{-1}$ for the multiplicative inverse of $t\ne 0$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Numbers. Let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^{n}$ and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{n}$; by The Open Subsets of a Metric Space Form a Topology the subsets open in $(\mathbb{R}^{n},d)$ form a topology, to which the topological notions below refer. Write $\bar B(x,r)$ for the closed ball in $(\mathbb{R}^{n},d)$ and let $\lambda_{n}$ be Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}^{n}$. Powers with a natural exponent are those of Natural Number Power of an Element of a Field, and $t\,y$ denotes the scalar multiple of $y\in\mathbb{R}^{n}$ in the real vector space $\mathbb{R}^{n}$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open in $(\mathbb{R}^{n},d)$ and let $f:\Omega\to\mathbb{R}$ be continuous on $\Omega$ as a map from $(\mathbb{R}^{n},d)$ into $(\mathbb{R},d_{\mathbb{R}})$. Let $\delta\in\mathbb{R}$ with $0<\delta$ and let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$. For $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ let $\rho_{\varepsilon}:\mathbb{R}^{n}\to\mathbb{R}$ be given by $\rho_{\varepsilon}(y)=(\varepsilon^{-1})^{n}\rho(\varepsilon^{-1}y)$, a mollifier kernel of radius $\varepsilon\delta$ by Rescaling a Mollifier Kernel; being smooth it is continuous on $\mathbb{R}^{n}$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and it vanishes at every $y$ with $\varepsilon\delta<\lVert y\rVert$, so the convolution $f*\rho_{\varepsilon}$ is defined, with kernel radius $\varepsilon\delta$, on the set $\Omega^{\varepsilon\delta}=\{x\in\mathbb{R}^{n}:\bar B(x,\varepsilon\delta)\subseteq\Omega\}$.

Let $K\subseteq\Omega$ be compact in $\mathbb{R}^{n}$.

Then the following hold.

1. (The convolution is eventually defined on $K$) There is $\varepsilon_{1}\in\mathbb{R}$ with $0<\varepsilon_{1}$ such that $K\subseteq\Omega^{\varepsilon\delta}$ for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon\le\varepsilon_{1}$.

2. (Uniform convergence on $K$) For every $\eta\in\mathbb{R}$ with $0<\eta$ there is $\varepsilon_{0}\in\mathbb{R}$ with $0<\varepsilon_{0}$ such that, for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon<\varepsilon_{0}$, one has $K\subseteq\Omega^{\varepsilon\delta}$ and

$$\bigl|(f*\rho_{\varepsilon})(x)-f(x)\bigr|<\eta\qquad\text{for every }x\in K .$$