Proof of The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms and the rules for adding inequalities, for multiplying them by nonnegative or positive real numbers, and for handling absolute values, from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field, are used without further mention; so is $0<N$, by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Integrals of integrable functions obey claim 2 of Linearity and Monotonicity of the Lebesgue Integral (linearity and monotonicity), finite sums of integrable functions obey Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, a bounded Borel real function on a Euclidean space is integrable with respect to every probability measure on it (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), and the integral of a constant $t$ against a probability measure is $t$ (claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space); these facts are used without further mention. Write $\mathcal{G}=\mathcal{G}_{\Phi,\varepsilon}$, $c=c_{N,\varepsilon}$ and $g_{0}(\nu)=\int_{\mathbb{R}^{d}}f\,d\nu$ for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so that $g_{\varepsilon}=g_{0}+\mathcal{G}$ by The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling §mean-field and $c(x)=N\,g_{\varepsilon}(\mu^{N}_{x})$ by The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling §particle; $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for every $x\in\mathbb{R}^{dN}$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment.

Step 1 (Regularity). By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=d$ and applied to $f$ and its bound $b$, $f$ is Borel, $g_{0}$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $|g_{0}(\nu)|\le b$ for every $\nu$. By The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost §bound, $0\le\mathcal{G}(\nu)\le L$; hence $|g_{\varepsilon}(\nu)|\le|g_{0}(\nu)|+|\mathcal{G}(\nu)|\le b+L$. Let $D$ be a nonnegative real number with $|\partial_{i}\eta(z)|\le D$ for every $z\in\mathbb{R}^{d}$ and $i\in[d]$, which exists by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel; by The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost §lipschitz there is a nonnegative real $K$ (the constant named there) with $|\mathcal{G}(\mu)-\mathcal{G}(\nu)|\le K\,W_{2}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $\zeta\in\mathbb{R}$ be positive. Choose first $\delta_{1}>0$ such that $|g_{0}(\mu)-g_{0}(\nu)|<\zeta/2$ whenever $W_{2}(\mu,\nu)<\delta_{1}$ (Uniformly Continuous Map Between Metric Spaces, the metric of The Absolute Value Metric on the Real Line being $|s-t|$), then put $\delta_{2}=\zeta\,\bigl(2(K+1)\bigr)^{-1}>0$ and $\delta=\min\{\delta_{1},\delta_{2}\}$. If $W_{2}(\mu,\nu)<\delta$, then $|\mathcal{G}(\mu)-\mathcal{G}(\nu)|\le K\delta_{2}=\tfrac{\zeta}{2}\,K(K+1)^{-1}<\tfrac{\zeta}{2}$ and therefore $|g_{\varepsilon}(\mu)-g_{\varepsilon}(\nu)|<\zeta$. So $g_{\varepsilon}$ is uniformly continuous. By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §empirical, applied to $g_{\varepsilon}$ with the bound $b+L$, the function $e_{\varepsilon}:\mathbb{R}^{dN}\to\mathbb{R}$, $e_{\varepsilon}(x)=g_{\varepsilon}(\mu^{N}_{x})$, is uniformly continuous with $|e_{\varepsilon}(x)|\le b+L$. Hence $|c(x)|=N|e_{\varepsilon}(x)|\le N(b+L)$; and given a positive $\zeta$, a $\delta>0$ with $|e_{\varepsilon}(x)-e_{\varepsilon}(x')|<\zeta N^{-1}$ whenever $\lVert x-x'\rVert<\delta$ gives $|c(x)-c(x')|<\zeta$ for such $x,x'$. This proves claim 1.

Step 2 (Measurability and the linear part). By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §empirical applied to $g_{0}$ with the bound $b$, the function $e_{0}(x)=g_{0}(\mu^{N}_{x})=\int_{\mathbb{R}^{d}}f\,d\mu^{N}_{x}$ is uniformly continuous on $\mathbb{R}^{dN}$ with $|e_{0}(x)|\le b$. By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=dN$ and applied to $e_{0}$ and to $e_{\varepsilon}$, both are Borel; so $x\mapsto\mathcal{G}(\mu^{N}_{x})=e_{\varepsilon}(x)-e_{0}(x)$ is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with values in $[0,L]$ by The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost §bound. This is the first assertion of claim 2. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral, read with $q=d$,

Let $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. Each $f\circ\mathfrak{p}_{k}$ is Borel, $\mathfrak{p}_{k}$ being Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear and a composition of Borel maps being Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and $|f\circ\mathfrak{p}_{k}|\le b$, so it is integrable with respect to $P$. By (1) and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, $\int e_{0}\,dP=\frac{1}{N}\sum_{k=1}^{N}\int f\circ\mathfrak{p}_{k}\,dP$. By Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, read with $q=d$, whose probability measure $A_{P}$ is $P^{[1]}$ by The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal, the Borel function $f$ is integrable with respect to $P^{[1]}$ and $\frac{1}{N}\sum_{k=1}^{N}\int f\circ\mathfrak{p}_{k}\,dP=\int f\,dP^{[1]}$. Hence

Step 3 (The tensor-averaged cost). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$; then $\mu^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments. By The Tensor-Averaged Running Cost of a Configuration-Space Cost §cost and $c=N\,e_{\varepsilon}=N\,e_{0}+N\,\mathcal{G}(\mu^{N}_{\cdot})$, where all three functions are bounded and Borel by Step 2,

By (2) with $P=\mu^{\otimes N}$ and $(\mu^{\otimes N})^{[1]}=\mu$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor), the first integral on the right is $\int_{\mathbb{R}^{d}}f\,d\mu$. This proves claim 2.

Step 4 (The defect). Let $\varepsilon\le1$, $S$, $\mu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and $R>1$ be as in claim 3. Write $Q=\mu^{\otimes N}$, and let $a=\eta_{\varepsilon}*\mu$ and $a_{x}=\eta_{\varepsilon}*\mu^{N}_{x}$, $x\in\mathbb{R}^{dN}$, be the mollified densities, 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; by that clause they are Borel and nonnegative. Let $\Delta(x,y)=|a_{x}(y)-a(y)|$, the function written $F$ in The Mean L^1 Distance Between the Mollified Empirical Measure and the Mollified Measure §fluctuation, and $\Psi(x)=\int_{\mathbb{R}^{d}}\Delta(x,y)\,\lambda_{d}(dy)$. That clause, applied with the present $d$, $N$, $\eta$, $S$, $\varepsilon$, $\mu$ and $R$ (its setting Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation underlies the present one, with the same dimension $d$), shows that $\Psi$ is Borel with values in $[0,2]$ and

First, $\Phi(s)-\Phi(t)\le L|s-t|$ for all $s,t\in[0,\infty)$: if $t\le s$ this is the Lipschitz bound of Convex Lipschitz Integrands §integrand with $a=t$, $b=s$; if $s\le t$, the same bound with $a=s$, $b=t$ gives $0\le\Phi(t)-\Phi(s)$, so $\Phi(s)-\Phi(t)\le0\le L|s-t|$. Now fix $x\in\mathbb{R}^{dN}$. By Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §cost, $\Phi\circ a_{x}$ and $\Phi\circ a$ are Borel and integrable with respect to $\lambda_{d}$, and their integrals are $\mathcal{G}(\mu^{N}_{x})$ and $\mathcal{G}(\mu)$ by The Mollified Density Cost of a Probability Measure with Finite Second Moment §cost. The function $y\mapsto\Delta(x,y)$ is Borel by claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, nonnegative, and has finite integral $\Psi(x)\le2$, so it is integrable (Measure Spaces and the Lebesgue Integral: Standing Notation §integral) with real integral $\Psi(x)$ (Integrable Function and the Lebesgue Integral). Since $\Phi(a_{x}(y))-\Phi(a(y))\le L\,\Delta(x,y)$ for every $y$, linearity and monotonicity give $\mathcal{G}(\mu^{N}_{x})-\mathcal{G}(\mu)\le L\,\Psi(x)$. By The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost §jensen, $\mathcal{G}(\mu)\le\mathcal{G}_{\Phi}(\mu)$, hence

Both $x\mapsto\mathcal{G}(\mu^{N}_{x})$ (Step 2) and $\Psi$ are bounded and Borel, so integrating against the probability measure $Q$ and using (3) and $0\le L$ gives $\int\mathcal{G}(\mu^{N}_{x})\,Q(dx)-\mathcal{G}_{\Phi}(\mu)\le L\int\Psi\,dQ\le L\,B$. By claim 2, $\int\mathcal{G}(\mu^{N}_{x})\,Q(dx)=\tilde{c}_{N,\varepsilon}(\mu)-\int_{\mathbb{R}^{d}}f\,d\mu$, which gives claim 3.