Proof of Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure

Each result cited is universally quantified over the data in its own statement.

Throughout, the metric on $\mathbb{R}$ is $d_{\mathbb{R}}(s,t)=|s-t|$ of The Absolute Value Metric on the Real Line, the Euclidean distance satisfies $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $\frac{t}{2}$ denotes $t\cdot2^{-1}$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Step 1 (Borel measurability, integrability and the bound in Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral). Let $m$, $c$ and $b$ be as there. First, $0\le|c(0_{\mathbb{R}^{m}})|\le b$ by claim 1 of Properties of the Absolute Value in an Ordered Field, so $0\le b$. Next, $c$ is continuous: given $x\in\mathbb{R}^{m}$ and a positive $\varepsilon$, the $\delta$ supplied for $\varepsilon$ by Uniformly Continuous Map Between Metric Spaces satisfies $|c(y)-c(x)|<\varepsilon$ for every $y$ with $d_{E}(x,y)<\delta$, which is the condition of Continuous Map Between Metric Spaces at $x$. Hence $c$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Being bounded (Bounded Real-Valued Function on a Set) and Borel, $c$ is integrable with respect to every member of $\mathcal{P}(\mathbb{R}^{m})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, in particular with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})\subseteq\mathcal{P}(\mathbb{R}^{m})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). For such $\rho$, the function $|c|$ is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and satisfies $|c|\le b\,\mathbf{1}_{\mathbb{R}^{m}}$ pointwise, so claim 2 and then claim 1 of Linearity and Monotonicity of the Lebesgue Integral, together with The Integral of an Indicator Function is the Measure of the Set and $\rho(\mathbb{R}^{m})=1$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), give

Step 2 (Choice of constants and a pointwise estimate). Let $\varepsilon\in\mathbb{R}$ be positive. The choices are made in this order. First, $0<\frac{\varepsilon}{2}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, and uniform continuity of $c$ (Uniformly Continuous Map Between Metric Spaces) supplies a positive $\delta$ with $|c(x)-c(y)|<\frac{\varepsilon}{2}$ for all $x,y\in\mathbb{R}^{m}$ with $\lVert x-y\rVert<\delta$. Second, put $\beta=b+b+1$; then $0\le b+b$ by claim 2 of Elementary Arithmetic in an Ordered Field, $b+b\le\beta$ by claims 1 and 3 there (as $\beta-(b+b)=1$), and $0<\beta$ by claims 6 and 3 of Elementary Order Arithmetic in an Ordered Field (adding $0<1$ and $0\le b+b$). Third, $0<\delta\delta$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, so $(\delta\delta)^{-1}$ and $\beta^{-1}$ exist and are positive by claim 7 there; put $\tau=\beta(\delta\delta)^{-1}$ and $\eta=\frac{\varepsilon}{2}\,\delta\delta\,\beta^{-1}$, both positive by claim 5 there, so that $\tau\eta=\frac{\varepsilon}{2}$. Fourth, put $r=\sqrt{\eta}$, the nonnegative square root of Existence and Uniqueness of the Nonnegative Square Root; $r\ne0$ since $r^{2}=\eta\ne0$, so $0<r$.

We claim that for all $x,y\in\mathbb{R}^{m}$,

Note $0\le\tau\lVert x-y\rVert^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 5 of Elementary Arithmetic in an Ordered Field. If $\lVert x-y\rVert<\delta$, then $|c(x)-c(y)|<\frac{\varepsilon}{2}\le\frac{\varepsilon}{2}+\tau\lVert x-y\rVert^{2}$. Otherwise $\delta\le\lVert x-y\rVert$, the order being total; then $\delta\delta\le\lVert x-y\rVert^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and multiplying by $0\le(\delta\delta)^{-1}$ and then by $0\le\beta$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\beta\le\tau\lVert x-y\rVert^{2}$. By claims 5 and 2 of Properties of the Absolute Value in an Ordered Field, $|c(x)-c(y)|\le|c(x)|+|c(y)|\le b+b\le\beta$, and since $0\le\frac{\varepsilon}{2}$ we again obtain (2.1).

Step 3 (Uniform continuity of $\rho\mapsto\int c\,d\rho$). With $\varepsilon$ and the constants of Step 2 fixed, let $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{m})$ satisfy $W_{2}(\rho,\rho')<r$. Since $W_{2}(\rho,\rho')$ and $r$ are nonnegative (The Quadratic Wasserstein Distance on Euclidean Space §distance), claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\rho,\rho')^{2}<\eta$. As $1\le m$ by claim 4 of Properties of the Order on the Natural Numbers, Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment supplies $\pi\in\Pi(\rho,\rho')$ with $I(\pi)=W_{2}(\rho,\rho')^{2}$, so $I(\pi)$ is a real number with $I(\pi)<\eta$. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, $(\mathrm{pr}_{1})_{\#}\pi=\rho$ and $(\mathrm{pr}_{2})_{\#}\pi=\rho'$, so by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and Step 1 the functions $c\circ\mathrm{pr}_{1}$ and $c\circ\mathrm{pr}_{2}$ are integrable with respect to $\pi$, with integrals $\int c\,d\rho$ and $\int c\,d\rho'$. Put $f=c\circ\mathrm{pr}_{1}-c\circ\mathrm{pr}_{2}$. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral, $f$ is integrable with $\int f\,d\pi=\int c\,d\rho-\int c\,d\rho'$ and $|\int f\,d\pi|\le\int|f|\,d\pi$. By (2.1) with $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$,

where $|f|$ is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and the last function is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. Claim 1 of Linearity and Monotonicity of the Lebesgue Integral, The Integral of an Indicator Function is the Measure of the Set, $\pi(\mathbb{R}^{m+m})=1$ and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost therefore give $\int|f|\,d\pi\le\frac{\varepsilon}{2}+\tau I(\pi)$. Since $0<\tau$ and $I(\pi)<\eta$, claim 10 of Elementary Order Arithmetic in an Ordered Field gives $\tau I(\pi)<\tau\eta=\frac{\varepsilon}{2}$, and then claims 1 and 2 there and claim 8 there give

As $\varepsilon$ was arbitrary and $r$ depends only on $\varepsilon$, $c$ and $b$, the function $\rho\mapsto\int c\,d\rho$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{m})$ for $W_{2}$ and $d_{\mathbb{R}}$ by Uniformly Continuous Map Between Metric Spaces. With Step 1 this proves Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral.

Step 4 (The empirical measure is $1$-Lipschitz in the configuration). Write $\bar{N}=\iota_{\mathbb{R}}(N)$ for the real number of The Canonical Map from the Natural Numbers to a Field, which is how the factor $N$ in Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §lipschitz is read. Since $1\in[N]$ by claim 4 of Properties of the Order on the Natural Numbers and each summand $1$ of $\bar{N}=\sum_{k=1}^{N}1$ is nonnegative by claim 1 of Elementary Arithmetic in an Ordered Field, claim 6 of Properties of Finite Sums gives $1\le\bar{N}$. Let $x,x'\in\mathbb{R}^{dN}$ and $w=W_{2}(\mu^{N}_{x},\mu^{N}_{x'})$, defined since $\mu^{N}_{x},\mu^{N}_{x'}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment (with $q=d$, by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles). As $0\le w^{2}$ (claim 5 of Elementary Arithmetic in an Ordered Field, $0\le w$), claim 5 of Elementary Arithmetic in an Ordered Field and Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §lipschitz give $w^{2}=1\cdot w^{2}\le\bar{N}w^{2}\le\lVert x-x'\rVert^{2}$, and since $0\le w$ and $0\le\lVert x-x'\rVert$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

Step 5 (Proof of Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §empirical). Let $g$ and $b$ be as there. The bound $|g(\mu^{N}_{x})|\le b$ is the hypothesis on $g$ applied to $\nu=\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For uniform continuity, let $\varepsilon\in\mathbb{R}$ be positive and choose, by Uniformly Continuous Map Between Metric Spaces for $g$, a positive $\delta$ with $|g(\nu)-g(\nu')|<\varepsilon$ for all $\nu,\nu'\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\nu')<\delta$. If $x,x'\in\mathbb{R}^{dN}$ satisfy $d_{E}(x,x')=\lVert x-x'\rVert<\delta$, then $W_{2}(\mu^{N}_{x},\mu^{N}_{x'})<\delta$ by (4.1) and claim 2 of Elementary Order Arithmetic in an Ordered Field, hence $|g(\mu^{N}_{x})-g(\mu^{N}_{x'})|<\varepsilon$. By Uniformly Continuous Map Between Metric Spaces the function $x\mapsto g(\mu^{N}_{x})$ is uniformly continuous on $\mathbb{R}^{dN}$ for the Euclidean distance and $d_{\mathbb{R}}$, which completes the proof.