The Mean $L^1$ Distance Between the Mollified Empirical Measure and the Mollified Measure

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used, let $d$ be the dimension of that setting and let $N\in\mathbb{N}$. 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. The letter $\eta$, which denotes vector fields in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, denotes here a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$. Let $S$ be a positive real number with $\eta(z)\le S$ for every $z\in\mathbb{R}^{d}$ (such an $S$ exists by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel), let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon\le1$, and for $\nu\in\mathcal{P}(\mathbb{R}^{d})$ let $\eta_{\varepsilon}*\nu$ be the mollified density of $\nu$ formed with the rescaled kernel $\eta_{\varepsilon}$ of Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with second moment $M_{2}(\mu)$, let $\mu^{\otimes N}\in\mathcal{P}(\mathbb{R}^{dN})$ be its tensor power, and for $x\in\mathbb{R}^{dN}$ let $\mu^{N}_{x}\in\mathcal{P}(\mathbb{R}^{d})$ be the empirical measure of $x$, tensor powers and empirical measures being read with their dimension parameter equal to $d$. Let $R$ be a real number with $1<R$. The natural numbers $N$ and $d$ occurring as real factors are read through the canonical map, 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 $\sqrt{t}$ is the nonnegative square root of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces.

(Fluctuation bound)¶ The function $F:\mathbb{R}^{dN}\times\mathbb{R}^{d}\to\mathbb{R}$,

is measurable with respect to the product $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$; the function $x\mapsto\int_{\mathbb{R}^{d}}F(x,y)\,\lambda_{d}(dy)$ is Borel on $\mathbb{R}^{dN}$ with values in $[0,2]$; and