Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d$ be a natural number with $1\le d$, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and let $\kappa_{d}=\lambda_{d}(\bar{B}(0,1))$, where $0$ is the zero vector of $\mathbb{R}^{d}$, be the normalising constant of closed balls. Natural numbers occurring as real factors are read through the canonical map into $\mathbb{R}$, powers with natural exponent are those of Natural Number Power of an Element of a Field, $t^{-1}$ is the multiplicative inverse of a real $t\ne0$, and $2=1+1$. The partial derivatives $\partial_{i}$, $i\in[d]$, are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, the set $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and continuity of a real function on $\mathbb{R}^{d}$ is that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §extrema, with the subset written $S$ there read as $\mathbb{R}^{d}$ (the letter $S$ is used below for a bound of the kernel).

Let $\eta:\mathbb{R}^{d}\to\mathbb{R}$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$, let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, and let $\eta_{\varepsilon}:\mathbb{R}^{d}\to\mathbb{R}$ be its rescaling as in Rescaling a Mollifier Kernel,

For $\mu\in\mathcal{P}(\mathbb{R}^{d})$ and $y\in\mathbb{R}^{d}$ we write

the integral of the function $x\mapsto\eta_{\varepsilon}(y-x)$, which by claim 3 is Borel and integrable with respect to $\mu$; the function $\eta_{\varepsilon}*\mu:\mathbb{R}^{d}\to\mathbb{R}$ is the mollified density of $\mu$. Then the following hold.

1. (The kernel)¶ $\eta_{\varepsilon}$ is a mollifier kernel of radius $\varepsilon$ on $\mathbb{R}^{d}$; $\eta$ and $\eta_{\varepsilon}$ are continuous and Borel; and $\eta$ is a test function on $\mathbb{R}^{d}$. There is a positive real number $S$ with $\eta(z)\le S$ for every $z\in\mathbb{R}^{d}$, and for every real $S$ with this property, $0\le\eta_{\varepsilon}(z)\le(\varepsilon^{-1})^{d}S$ for every $z\in\mathbb{R}^{d}$. There is a nonnegative real number $D$ with $|\partial_{i}\eta(z)|\le D$ for every $z\in\mathbb{R}^{d}$ and every $i\in[d]$, and for every real $D$ with this property and all $z,z'\in\mathbb{R}^{d}$,

For every natural number $m$ and all Borel maps $u,v:\mathbb{R}^{m}\to\mathbb{R}^{d}$, the function $w\mapsto\eta_{\varepsilon}(v(w)-u(w))$ on $\mathbb{R}^{m}$ is Borel. For every $x\in\mathbb{R}^{d}$, the function $y\mapsto\eta_{\varepsilon}(y-x)$ on $\mathbb{R}^{d}$ is Borel, nonnegative and integrable with respect to $\lambda_{d}$, and

2. (Translation estimate)¶ Let $D$ be a real number as in claim 1. For all $x,x'\in\mathbb{R}^{d}$ the function $y\mapsto|\eta_{\varepsilon}(y-x)-\eta_{\varepsilon}(y-x')|$ on $\mathbb{R}^{d}$ is Borel, and

3. (The mollified density)¶ Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$. For every $y\in\mathbb{R}^{d}$ the function $x\mapsto\eta_{\varepsilon}(y-x)$ is Borel, bounded and integrable with respect to $\mu$, so that $(\eta_{\varepsilon}*\mu)(y)$ is a real number. The function $\eta_{\varepsilon}*\mu$ is continuous and Borel; for every real $S$ as in claim 1,

and $\eta_{\varepsilon}*\mu$ is integrable with respect to $\lambda_{d}$, with

4. (The cost of the mollified density)¶ Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$, let $L\in\mathbb{R}$ be nonnegative, and let $\Phi$ be a convex Lipschitz integrand with constant $L$. Then $\Phi\circ(\eta_{\varepsilon}*\mu):\mathbb{R}^{d}\to\mathbb{R}$ is Borel, $0\le\Phi\bigl((\eta_{\varepsilon}*\mu)(y)\bigr)\le L\,(\eta_{\varepsilon}*\mu)(y)$ for every $y\in\mathbb{R}^{d}$, and $\Phi\circ(\eta_{\varepsilon}*\mu)$ is integrable with respect to $\lambda_{d}$, with