Proof of The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous

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

Throughout, $N$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers; by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field it is positive and its multiplicative inverse $N^{-1}$ exists and is positive, and $\tfrac{b}{N}=b\,N^{-1}$. The distance on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ is $W_{2}$, a metric by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions, so its values are nonnegative by Metric Space; on $\mathbb{R}$ the distance is $d_{\mathbb{R}}(s,t)=|s-t|$ of The Absolute Value Metric on the Real Line.

Step 1 (The integral functional at the configuration level). By N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level, $dN$ is a natural number with $1\le dN$. Let $0_{\mathbb{R}^{dN}}$ be the origin; then $0\le|c(0_{\mathbb{R}^{dN}})|$ by claim 1 of Properties of the Absolute Value in an Ordered Field and $|c(0_{\mathbb{R}^{dN}})|\le b$ by hypothesis, so $0\le b$ by transitivity of the order, and $b$ is a bound for $c$ in the sense of Bounded Real-Valued Function on a Set; thus $c$ is bounded and uniformly continuous. Applying Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral with $m=dN$, $c$ is Borel and integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and the function

is uniformly continuous and satisfies $|I(\rho)|\le b$ for every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the tensor power $\mu^{\otimes N}$ lies 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, and by the formula of the tensor-averaged cost

Step 2 (Bound). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By claim 4 of Properties of the Absolute Value in an Ordered Field, $|\tilde{c}(\mu)|=|N^{-1}|\,|I(\mu^{\otimes N})|$, and $|N^{-1}|=N^{-1}$ by Absolute Value in an Ordered Field, since $0\le N^{-1}$. Since $|I(\mu^{\otimes N})|\le b$ by Step 1 and $0\le N^{-1}$, claim 5 of Elementary Arithmetic in an Ordered Field gives

which is The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous §bound.

Step 3 (A square root of $N$). Since $0\le N$, Existence and Uniqueness of the Nonnegative Square Root gives $s\in\mathbb{R}$ with $0\le s$ and $s^{2}=N$. If $s=0$ then $s^{2}=0\cdot0=0\ne N$; hence $s\ne0$, so $0<s$, and by claim 7 of Elementary Order Arithmetic in an Ordered Field the inverse $s^{-1}$ exists and $0<s^{-1}$. Moreover $N\,s^{-1}s^{-1}=(s\,s^{-1})(s\,s^{-1})=1$ by the field axioms.

Step 4 (Uniform continuity). Let $\varepsilon\in\mathbb{R}$ be positive. By claim 5 of Elementary Order Arithmetic in an Ordered Field, $0<N\varepsilon$. By Step 1 and Uniformly Continuous Map Between Metric Spaces, applied to $I$ on the metric space $(\mathcal{P}_{2}(\mathbb{R}^{dN}),W_{2})$ with values in $(\mathbb{R},d_{\mathbb{R}})$ and with $N\varepsilon$ in place of $\varepsilon$, there is a positive $\eta\in\mathbb{R}$ such that all $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ with $W_{2}(\rho,\rho')<\eta$ satisfy $|I(\rho)-I(\rho')|<N\varepsilon$. Put $\delta=\eta\,s^{-1}$, which is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field and Step 3.

Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfy $W_{2}(\mu,\nu)<\delta$. Since $0\le W_{2}(\mu,\nu)$ and $0\le\delta$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\mu,\nu)^{2}<\delta^{2}$, and then claim 10 of Elementary Order Arithmetic in an Ordered Field (with the positive factor $N$) gives $N\,W_{2}(\mu,\nu)^{2}<N\delta^{2}$. By Step 3, $N\delta^{2}=\eta^{2}\,N\,s^{-1}s^{-1}=\eta^{2}$. By Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §tensor, read with $q=d$ as N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles provides,

Since $0\le W_{2}(\mu^{\otimes N},\nu^{\otimes N})$ and $0\le\eta$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\mu^{\otimes N},\nu^{\otimes N})<\eta$, and therefore $|I(\mu^{\otimes N})-I(\nu^{\otimes N})|<N\varepsilon$ by the choice of $\eta$ (both tensor powers lie in $\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Step 1). By Step 1, distributivity, claim 4 of Properties of the Absolute Value in an Ordered Field and $|N^{-1}|=N^{-1}$ (Step 2),

and claim 10 of Elementary Order Arithmetic in an Ordered Field (with the positive factor $N^{-1}$) gives that this is less than $N^{-1}(N\varepsilon)=\varepsilon$. Thus $d_{\mathbb{R}}(\tilde{c}(\mu),\tilde{c}(\nu))<\varepsilon$ whenever $W_{2}(\mu,\nu)<\delta$. As $\varepsilon$ was arbitrary, $\tilde{c}$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Uniformly Continuous Map Between Metric Spaces, which is The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous §uniform.