Proof of The Mollified N-Particle Cost Dominates N Times the Mollified Mean-Field Cost of the One-Particle Marginal

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 and for multiplying them by positive real numbers, from Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic 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 nonnegative measurable functions obey claim 1 of Linearity and Monotonicity of the Lebesgue Integral and those of integrable functions obey claim 2 there; 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 for a nonnegative integrable function its real integral equals its integral as a nonnegative measurable function (Integrable Function and the Lebesgue Integral). These facts are used without further mention.

Let $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and write $\mathcal{G}=\mathcal{G}_{\Phi,\varepsilon}$ for the mollified density cost with integrand $\Phi$, kernel $\eta$ and scale $\varepsilon$, so that $g_{\varepsilon}(\nu)=\int_{\mathbb{R}^{d}}f\,d\nu+\mathcal{G}(\nu)$ by The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling §mean-field; $\lambda_{d}$ is Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and $\eta_{\varepsilon}$ and the mollified densities $\eta_{\varepsilon}*\nu$ are those of Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost. Recall $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for $x\in\mathbb{R}^{dN}$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments, Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment).

Step 1 (Reduction to the density part). By The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling §particle, $c_{N,\varepsilon}(x)=N\int_{\mathbb{R}^{d}}f\,d\mu^{N}_{x}+N\,\mathcal{G}(\mu^{N}_{x})$, and by The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §identity both functions of $x$ on the right are bounded and Borel and

Since $N\,g_{\varepsilon}(P^{[1]})=N\int_{\mathbb{R}^{d}}f\,dP^{[1]}+N\,\mathcal{G}(P^{[1]})$ and $0<N$, the assertion is equivalent to

Step 2 (Joint measurability). Write $a=\eta_{\varepsilon}*P^{[1]}$ and $a_{x}=\eta_{\varepsilon}*\mu^{N}_{x}$ for $x\in\mathbb{R}^{dN}$; by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density they are Borel and nonnegative. Fix $y\in\mathbb{R}^{d}$ and let $\varphi_{y}(z)=\eta_{\varepsilon}(y-z)$; by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density and Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, $\varphi_{y}$ is Borel with $0\le\varphi_{y}\le(\varepsilon^{-1})^{d}S$ for a positive real $S$ as in that clause, so $\varphi_{y}$ is bounded. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral, read with $q=d$,

Let $\mathrm{pr}_{1}=\mathrm{pr}^{dN,d}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{dN,d}_{2}$ be the coordinate projections of $\mathbb{R}^{dN+d}$ and $\iota=\iota^{dN,d}$ the concatenation, as in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. The maps $\mathfrak{p}_{k}\circ\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are Borel, by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and composition (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); so by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, read with $m=dN+d$, and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the function $\hat{h}(w)=\sum_{k=1}^{N}N^{-1}\eta_{\varepsilon}\bigl(\mathrm{pr}_{2}(w)-\mathfrak{p}_{k}(\mathrm{pr}_{1}(w))\bigr)$ is Borel on $\mathbb{R}^{dN+d}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, $\iota$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R}^{dN+d})$, so by claim 4 of Borel Measurability and Bounded Integration on a Metric Space the function $h=\hat{h}\circ\iota$ on $\mathbb{R}^{dN}\times\mathbb{R}^{d}$ is measurable for the product $\sigma$-algebra; by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and (2), $h(x,y)=a_{x}(y)\ge0$. The measures $P$ and $\lambda_{d}$ are $\sigma$-finite, $P$ being a probability measure and $\lambda_{d}$ by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, so the product measure $P\otimes\lambda_{d}$ is defined; applying Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand §composition on the measure space $(\mathbb{R}^{dN}\times\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d}),P\otimes\lambda_{d})$ to $h$ shows that $\Phi\circ h$ is measurable for the product $\sigma$-algebra and nonnegative.

Step 3 (Jensen's inequality in the configuration variable). Fix $y\in\mathbb{R}^{d}$ and let $h_{y}(x)=h(x,y)=a_{x}(y)$. By (2), $h_{y}=\sum_{k=1}^{N}N^{-1}\varphi_{y}\circ\mathfrak{p}_{k}$, where each $\varphi_{y}\circ\mathfrak{p}_{k}$ is Borel and bounded, hence integrable with respect to $P$; so $h_{y}$ is Borel, nonnegative and integrable with respect to $P$, and

by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, then Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, read with $q=d$, whose measure $A_{P}$ is $P^{[1]}$ by The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal, and finally the definition of the mollified density in Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost. Applying Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand §jensen on the measure space $(\mathbb{R}^{dN},\mathcal{B}(\mathbb{R}^{dN}),P)$, which has $P(\mathbb{R}^{dN})=1$, to $h_{y}$ gives

Step 4 (Tonelli). By the Tonelli statement of Tonelli and Fubini Theorems, applied to the nonnegative $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable function $\Phi\circ h$ and the $\sigma$-finite measures $P$ and $\lambda_{d}$, the function $y\mapsto\int\Phi(a_{x}(y))\,P(dx)$ is Borel and

For each $x$, the inner integral on the right is $\mathcal{G}(\mu^{N}_{x})$, by The Mollified Density Cost of a Probability Measure with Finite Second Moment §cost and Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §cost, which make $\Phi\circ a_{x}$ nonnegative and integrable. On the left, (3) and monotonicity give the lower bound $\int_{\mathbb{R}^{d}}\Phi\circ a\,d\lambda_{d}$, which is $\mathcal{G}(P^{[1]})$ by the same two items. Hence $\mathcal{G}(P^{[1]})\le\int_{\mathbb{R}^{dN}}\mathcal{G}(\mu^{N}_{x})\,P(dx)$, the right side being the real integral of the bounded Borel function of The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §identity. This is (1), and the assertion follows by Step 1.