Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used, let $q,N\in\mathbb{N}$. The block maps $\mathfrak{p}_{k}$ and product maps $h^{\oplus}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map respectively, $\mu^{N}_{x}$ is the empirical measure of $x\in\mathbb{R}^{qN}$, $\mathbf{1}_{B}$ is the indicator function of $B\subseteq\mathbb{R}^{q}$, Borel maps are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, the second moment $M_{2}$ and $\mathcal{P}_{2}$ are those of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, $W_{2}$ is the Wasserstein distance, and continuity of a real function on $\mathbb{R}^{qN}$ refers to the Euclidean distance and the metric of The Absolute Value Metric on the Real Line. Let $x,x'\in\mathbb{R}^{qN}$. Then the following hold.

1. (Values)¶ For every $B\in\mathcal{B}(\mathbb{R}^{q})$, $\mu^{N}_{x}(B)=\frac{1}{N}\sum_{k=1}^{N}\mathbf{1}_{B}(\mathfrak{p}_{k}(x))$.

2. (Integrals)¶ For every Borel $f:\mathbb{R}^{q}\to[0,\infty]$, $\int_{\mathbb{R}^{q}}f\,d\mu^{N}_{x}=\frac{1}{N}\sum_{k=1}^{N}f(\mathfrak{p}_{k}(x))$. Every Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable with respect to $\mu^{N}_{x}$, with the same identity.

3. (Push-forwards)¶ For $p\in\mathbb{N}$ and Borel $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$, $h_{\#}\mu^{N}_{x}=\mu^{N}_{h^{\oplus}(x)}$, the right side being the empirical measure of $h^{\oplus}(x)\in\mathbb{R}^{pN}$.

4. (Second moment)¶ $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and $N\,M_{2}(\mu^{N}_{x})=\lVert x\rVert^{2}$.

5. (Lipschitz dependence)¶ $N\,W_{2}(\mu^{N}_{x},\mu^{N}_{x'})^{2}\le\lVert x-x'\rVert^{2}$.

6. (Distance to a fixed measure)¶ For every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$, $|W_{2}(\mu^{N}_{x},\nu)-W_{2}(\mu^{N}_{x'},\nu)|\le W_{2}(\mu^{N}_{x},\mu^{N}_{x'})$, and the function $y\mapsto W_{2}(\mu^{N}_{y},\nu)$ on $\mathbb{R}^{qN}$ is continuous and Borel.