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

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Uniform continuity of a real function on $\mathbb{R}^{dN}$ refers to the Euclidean distance, and on $\mathcal{P}_{2}(\mathbb{R}^{d})$ to $W_{2}$, with the metric of The Absolute Value Metric on the Real Line on $\mathbb{R}$. Let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be uniformly continuous, let $b\in\mathbb{R}$ satisfy $|c(x)|\le b$ for every $x\in\mathbb{R}^{dN}$, and let $\tilde{c}$ be the tensor-averaged cost of $c$, defined because $c$ is Borel and bounded by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral; $\tfrac{b}{N}$ is the product of $b$ with the multiplicative inverse of $N$.

1. (Bound)¶ $|\tilde{c}(\mu)|\le\tfrac{b}{N}$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Uniform continuity)¶ $\tilde{c}$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$.