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

Each result cited is universally quantified over the data in its own statement. In real expressions a natural number $N$ stands for its image $\iota_{\mathbb{R}}(N)$ under the canonical map of The Canonical Map from the Natural Numbers to a Field; by claims 2 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $1\le N$, $N\ne0$ and $\frac{1}{N}=N^{-1}$ exists. By The Empirical Measure of a Configuration of N Particles §empirical, $\mu^{N}_{x}=(\delta_{x})^{[1]}$, where $\delta_{x}\in\mathcal{P}(\mathbb{R}^{qN})$ by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure (in dimension $qN$). Each block map $\mathfrak{p}_{k}$ is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, so $\mathfrak{p}_{k}^{-1}(B)\in\mathcal{B}(\mathbb{R}^{qN})$ for $B\in\mathcal{B}(\mathbb{R}^{q})$.

Step 1 (Clause 1). Let $B\in\mathcal{B}(\mathbb{R}^{q})$. By The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal with $P=\delta_{x}$,

For each $k$, $x\in\mathfrak{p}_{k}^{-1}(B)$ exactly when $\mathfrak{p}_{k}(x)\in B$, so $\delta_{x}(\mathfrak{p}_{k}^{-1}(B))$ and $\mathbf{1}_{B}(\mathfrak{p}_{k}(x))$ are both $1$ if $\mathfrak{p}_{k}(x)\in B$ and both $0$ otherwise, by The Dirac Measure at a Point of Euclidean Space §dirac and Simple Function and Its Integral. Replacing the summands gives clause 1.

Step 2 (Clause 2). Comparing The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal with the function $A_{P}$ of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average for $P=\delta_{x}$ shows $\mu^{N}_{x}=A_{\delta_{x}}$. Let $f:\mathbb{R}^{q}\to[0,\infty]$ be Borel. For each $k$ and real $a$, $\{f\circ\mathfrak{p}_{k}>a\}=\mathfrak{p}_{k}^{-1}(\{f>a\})\in\mathcal{B}(\mathbb{R}^{qN})$, so $f\circ\mathfrak{p}_{k}$ is Borel in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable, and Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §integral (in dimension $qN$) gives $\int f\circ\mathfrak{p}_{k}\,d\delta_{x}=f(\mathfrak{p}_{k}(x))$. Substituting these values into the integration identity of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average gives $\int f\,d\mu^{N}_{x}=\frac{1}{N}\sum_{k=1}^{N}f(\mathfrak{p}_{k}(x))$. Now let $f:\mathbb{R}^{q}\to\mathbb{R}$ be Borel. Each $f\circ\mathfrak{p}_{k}$ is Borel as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), hence integrable with respect to $\delta_{x}$ with integral $f(\mathfrak{p}_{k}(x))$ by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §integral. By the last assertion of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, $f$ is integrable with respect to $A_{\delta_{x}}=\mu^{N}_{x}$ and the same identity holds, which after the same substitution is the stated one.

Step 3 (Clause 3). Let $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$ be Borel. Then $h^{\oplus}:\mathbb{R}^{qN}\to\mathbb{R}^{pN}$ is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map, so Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §pushforward (in dimension $qN$, with target dimension $pN$) gives $(h^{\oplus})_{\#}\delta_{x}=\delta_{h^{\oplus}(x)}$. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §pushforward, applied with $P=\delta_{x}$ (and $\rho=\mu^{N}_{x}$, a member of $\mathcal{P}(\mathbb{R}^{q})$ that the second identity there does not involve),

the last equality being The Empirical Measure of a Configuration of N Particles §empirical in dimension $p$.

Step 4 (Clause 4). By Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure, $\delta_{x}\in\mathcal{P}_{2}(\mathbb{R}^{qN})$ with $M_{2}(\delta_{x})=\lVert x\rVert^{2}$. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments with $P=\delta_{x}$ (and $\rho=\mu^{N}_{x}$ as in Step 3), $M_{2}(\mu^{N}_{x})=\frac{1}{N}M_{2}(\delta_{x})=\frac{1}{N}\lVert x\rVert^{2}$, a real number; hence $M_{2}(\mu^{N}_{x})<\infty$, $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, and $N\,M_{2}(\mu^{N}_{x})=N\cdot N^{-1}\lVert x\rVert^{2}=\lVert x\rVert^{2}$.

Step 5 (Clause 5). By Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure, $\delta_{x},\delta_{x'}\in\mathcal{P}_{2}(\mathbb{R}^{qN})$, so Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §marginal with $P=\delta_{x}$ and $P'=\delta_{x'}$ gives $N\,W_{2}(\mu^{N}_{x},\mu^{N}_{x'})^{2}\le W_{2}(\delta_{x},\delta_{x'})^{2}$. By Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §coupling, $W_{2}(\delta_{x},\delta_{x'})\le\lVert x-x'\rVert$, both sides being nonnegative, so $W_{2}(\delta_{x},\delta_{x'})^{2}\le\lVert x-x'\rVert^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Combining the two inequalities gives clause 5.

Step 6 (Clause 6). Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$. By Step 4, $\mu^{N}_{y}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ for every $y\in\mathbb{R}^{qN}$, so $g(y)=W_{2}(\mu^{N}_{y},\nu)$ is a nonnegative real number. Write $a=g(x)$, $b=g(x')$ and $D=W_{2}(\mu^{N}_{x},\mu^{N}_{x'})$. By The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, $a\le D+b$, and, using also The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry, $b\le W_{2}(\mu^{N}_{x'},\mu^{N}_{x})+a=D+a$. By claim 3 of Elementary Arithmetic in an Ordered Field (applied twice to each), the first inequality gives $a-b\le D$ and the second gives $-D\le a-b$; hence $|a-b|\le D$ by claim 6 of Properties of the Absolute Value in an Ordered Field. This is the stated inequality.

For the continuity, let $y,y'\in\mathbb{R}^{qN}$ and $w=W_{2}(\mu^{N}_{y},\mu^{N}_{y'})^{2}\ge0$. Since $1\le N$, claim 5 of Elementary Arithmetic in an Ordered Field gives $w=1\cdot w\le N\,w$, and clause 5 (Step 5, applied to $y,y'$) gives $N\,w\le\lVert y-y'\rVert^{2}$; so $w\le\lVert y-y'\rVert^{2}$, and $W_{2}(\mu^{N}_{y},\mu^{N}_{y'})\le\lVert y-y'\rVert$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. With the inequality just proved (applied to $y,y'$) and claim 2 of Properties of the Absolute Value in an Ordered Field,

Now fix $y\in\mathbb{R}^{qN}$, let $0<\varepsilon$ be given, and choose $\delta=\varepsilon$. If $y'\in\mathbb{R}^{qN}$ satisfies $d_{E}(y,y')<\delta$, where $d_{E}(y,y')=\lVert y-y'\rVert$ by Euclidean Space and Lebesgue Measure: Standing Notation §space, then the display and claim 2 of Elementary Order Arithmetic in an Ordered Field give $|g(y')-g(y)|<\varepsilon$, which is the distance of $g(y')$ and $g(y)$ in the metric of The Absolute Value Metric on the Real Line. Hence $g$ is continuous at every point of $\mathbb{R}^{qN}$, and it is Borel by claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, as recorded in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.