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

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Boundedness of a real function is that of Bounded Real-Valued Function on a Set. Uniform continuity of a real function on $\mathbb{R}^{m}$ refers to the Euclidean distance, and on $\mathcal{P}_{2}(\mathbb{R}^{m})$ to $W_{2}$, with the metric of The Absolute Value Metric on the Real Line on $\mathbb{R}$.

1. (Integrals)¶ Let $m\in\mathbb{N}$, let $c:\mathbb{R}^{m}\to\mathbb{R}$ be bounded and uniformly continuous, and let $b\in\mathbb{R}$ satisfy $|c(x)|\le b$ for every $x\in\mathbb{R}^{m}$. Then $c$ is Borel and integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$, and the function $\rho\mapsto\int_{\mathbb{R}^{m}}c\,d\rho$ on $\mathcal{P}_{2}(\mathbb{R}^{m})$ is uniformly continuous and satisfies $\bigl|\int_{\mathbb{R}^{m}}c\,d\rho\bigr|\le b$ for every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$.

2. (Costs of the empirical measure)¶ Let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be bounded and uniformly continuous, and let $b\in\mathbb{R}$ satisfy $|g(\nu)|\le b$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then the function $x\mapsto g(\mu^{N}_{x})$ on $\mathbb{R}^{dN}$ is uniformly continuous and satisfies $|g(\mu^{N}_{x})|\le b$ for every $x\in\mathbb{R}^{dN}$; here $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment.