Proof of The Mean $L^1$ Distance Between the Mollified Empirical Measure and the Mollified Measure

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms and the rules for adding inequalities, for multiplying them by nonnegative or positive real numbers, and for handling absolute values, from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field, are used without further mention; so are the facts that a square of a real number is nonnegative (claim 2 of Nonnegativity of Squares in an Ordered Field), that $|t|^{2}=t^{2}$, and that for nonnegative reals $s,t$ one has $s<t$ exactly when $s^{2}<t^{2}$ (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and that $0<c^{n}$ for real $c>0$ and natural $n$ (claims 5 and 4 of Properties of Natural Number Powers in a Field). For the Euclidean norm we use $d_{E}(y,y')=\lVert y-y'\rVert$ (claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and the triangle inequality (claim 6 there). Integrals of nonnegative measurable functions obey claim 1 of Linearity and Monotonicity of the Lebesgue Integral and those of integrable functions obey claim 2 there. For a nonnegative integrable function its real integral equals its integral as a nonnegative measurable function, the positive part being the function and the negative part $0$ (Integrable Function and the Lebesgue Integral); conversely a nonnegative measurable real function with finite integral is integrable (Measure Spaces and the Lebesgue Integral: Standing Notation §integral). These facts are used without further mention.

Step 0 (Notation). Write $Q=\mu^{\otimes N}$, $a=\eta_{\varepsilon}*\mu$ and, for $x\in\mathbb{R}^{dN}$, $a_{x}=\eta_{\varepsilon}*\mu^{N}_{x}$, which is defined since $\mu^{N}_{x}\in\mathcal{P}(\mathbb{R}^{d})$ by The Empirical Measure of a Configuration of N Particles §empirical. By Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density, $a$ and every $a_{x}$ are Borel, nonnegative and integrable with respect to $\lambda_{d}$ with integral $1$. Put $c_{\varepsilon}=(\varepsilon^{-1})^{d}$, $v=S\,c_{\varepsilon}N^{-1}$ and $c=\kappa_{d}R^{d}$; these are positive, since $0<\varepsilon^{-1}$, $0<S$, $0<N^{-1}$ (claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), $0<\kappa_{d}$ (The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §constant) and $0<R$. Let $B=\bar{B}(0,R)$; it is Borel with $\lambda_{d}(B)=c$ by The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §borel and The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §value, and $B^{c}=\mathbb{R}^{d}\setminus B$ is Borel; a point $y$ lies in $B^{c}$ exactly when $R<d_{E}(0,y)=\lVert y\rVert$. The measures $Q$ and $\mu$ are probability measures and $\lambda_{d}$ is $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, so the Tonelli statement of Tonelli and Fubini Theorems applies to each of the pairs $(Q,\lambda_{d})$ and $(\mu,\lambda_{d})$, with the Borel $\sigma$-algebras. For every $\lambda\in\mathbb{R}$, $\sum_{k=1}^{N}\lambda=\lambda N$ by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field.

Step 1 (Measurability). Let $x\in\mathbb{R}^{dN}$ and $y\in\mathbb{R}^{d}$. The function $x'\mapsto\eta_{\varepsilon}(y-x')$ is Borel by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density, so Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral (with $q=d$) gives

with the block maps $\mathfrak{p}_{k}$ of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks. Let $P_{1}=\mathrm{pr}^{dN,d}_{1}$, $P_{2}=\mathrm{pr}^{dN,d}_{2}$ and $\iota=\iota^{dN,d}$ be the coordinate projections of $\mathbb{R}^{dN+d}$ and the concatenation (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs). The maps $\mathfrak{p}_{k}\circ P_{1}$ and $P_{2}$ are Borel, by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and composition (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); so by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel (with $m=dN+d$) and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the function $\hat{\Delta}(w)=\sum_{k=1}^{N}N^{-1}\eta_{\varepsilon}(P_{2}(w)-\mathfrak{p}_{k}(P_{1}(w)))-a(P_{2}(w))$ is Borel on $\mathbb{R}^{dN+d}$, and so are $|\hat{\Delta}|$, $\hat{\Delta}^{2}$ (claims 4 and 3 there), $w\mapsto\mathbf{1}_{B^{c}}(P_{2}(w))$, which is the indicator of the Borel set $P_{2}^{-1}(B^{c})$ (claim 1 there), and the products of the latter with $|\hat\Delta|$, with $\sum_{k}N^{-1}\eta_{\varepsilon}(P_{2}-\mathfrak{p}_{k}\circ P_{1})$ and with $a\circ P_{2}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, $\iota$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R}^{dN+d})$, so by claim 4 of Borel Measurability and Bounded Integration on a Metric Space the compositions with $\iota$ of all these functions are measurable for the product $\sigma$-algebra. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and (1), $\hat{\Delta}(\iota(x,y))=\Delta(x,y)$, where $\Delta(x,y)=a_{x}(y)-a(y)$. Hence the functions

are nonnegative and measurable with respect to $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$; this proves the first assertion. By the Tonelli statement applied to $F$, the function $x\mapsto\int F(x,y)\,\lambda_{d}(dy)$ is measurable, the function $E(y)=\int F(x,y)\,Q(dx)$ is measurable, and

Since $a_{x}$ and $a$ are nonnegative, $F(x,y)\le a_{x}(y)+a(y)$, so $\int F(x,y)\,\lambda_{d}(dy)\le1+1=2$; thus the function $x\mapsto\int F(x,y)\,\lambda_{d}(dy)$ takes values in $[0,2]$ and is a Borel real function (Measure Spaces and the Lebesgue Integral: Standing Notation §measurable), proving the second assertion.

Step 2 (Pointwise variance). Fix $y\in\mathbb{R}^{d}$ and let $\varphi_{y}(x')=\eta_{\varepsilon}(y-x')$. By Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density and Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, $\varphi_{y}$ is Borel with $0\le\varphi_{y}\le c_{\varepsilon}S$, so it is bounded, and $\int\varphi_{y}\,d\mu=a(y)$; by (1), $\int\varphi_{y}\,d\mu^{N}_{x}=a_{x}(y)$ for every $x$ (Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral). Apply Mean-Square Deviation of the Empirical Average of a Bounded Borel Function under a Tensor Power with $q=d$, $\rho=\mu$ and $\varphi=\varphi_{y}$: its number written $a$ there is $a(y)$, its function $g$ is $x\mapsto a_{x}(y)$ by Mean-Square Deviation of the Empirical Average of a Bounded Borel Function under a Tensor Power §average, and Mean-Square Deviation of the Empirical Average of a Bounded Borel Function under a Tensor Power §variance gives

Multiplying $0\le\varphi_{y}\le c_{\varepsilon}S$ by $\varphi_{y}\ge0$ gives $\varphi_{y}^{2}\le c_{\varepsilon}S\,\varphi_{y}$ pointwise, so $\int\varphi_{y}^{2}\,d\mu\le c_{\varepsilon}S\,a(y)$ and

Step 3 (Inside the ball). Let $\theta=\sqrt{v\,c^{-1}}$; then $0\le\theta$ and $\theta^{2}=v\,c^{-1}>0$, so $0<\theta$. For all $x,y$, $0\le(F(x,y)-\theta)^{2}=\Delta(x,y)^{2}-2\theta F(x,y)+\theta^{2}$, hence

Integrating in $x$ against the probability measure $Q$, the constant $\frac{\theta}{2}$ having integral $\frac{\theta}{2}Q(\mathbb{R}^{dN})=\frac{\theta}{2}$ by The Integral of an Indicator Function is the Measure of the Set, and using (3),

Multiplying by $\mathbf{1}_{B}(y)$ and integrating in $y$, with $\int\mathbf{1}_{B}\,d\lambda_{d}=\lambda_{d}(B)=c$ (The Integral of an Indicator Function is the Measure of the Set) and $\int\mathbf{1}_{B}a\,d\lambda_{d}\le\int a\,d\lambda_{d}=1$,

Both $\theta c$ and $v\,\theta^{-1}$ are nonnegative with square $\theta^{2}c^{2}=v\,c=v^{2}\theta^{-2}$, so both equal $\sqrt{v\,c}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. Since $v\,c=\kappa_{d}R^{d}S\,c_{\varepsilon}N^{-1}$,

Step 4 (Outside the ball). Let $\hat{H}(w)=\mathbf{1}_{B^{c}}(\mathrm{pr}_{2}(w))\,\eta_{\varepsilon}(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$ for $w\in\mathbb{R}^{d+d}$; as in Step 1, $\hat H$ is Borel, and $H=\hat{H}\circ\iota^{d,d}$, $H(x',y)=\mathbf{1}_{B^{c}}(y)\eta_{\varepsilon}(y-x')$, is nonnegative and measurable for $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{d})$. Let $\tau(x')=\int H(x',y)\,\lambda_{d}(dy)$. The Tonelli statement for $(\mu,\lambda_{d})$ and $H$ shows that $\tau$ is Borel and, since $\int H(x',y)\,\mu(dx')=\mathbf{1}_{B^{c}}(y)a(y)$ by the multiple rule,

Since $H(x',y)\le\eta_{\varepsilon}(y-x')$, Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel gives $\tau(x')\le1$. Let $A=\{x'\in\mathbb{R}^{d}:(R-1)^{2}\le\lVert x'\rVert^{2}\}$. If $x'\notin A$, then $\lVert x'\rVert^{2}<(R-1)^{2}$ with $0<R-1$, so $\lVert x'\rVert<R-1$; for every $y\in B^{c}$ the triangle inequality gives $\lVert y-x'\rVert\ge\lVert y\rVert-\lVert x'\rVert>R-(R-1)=1\ge\varepsilon$, so $\eta_{\varepsilon}(y-x')=0$, $\eta_{\varepsilon}$ being a mollifier kernel of radius $\varepsilon$ (Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel and condition 3 of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$); thus $H(x',\cdot)=0\cdot\mathbf{1}_{\mathbb{R}^{d}}$ and $\tau(x')=0$. Hence $0\le\tau\le\mathbf{1}_{A}$ pointwise. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov, applied on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ to the nonnegative Borel function $x'\mapsto\lVert x'\rVert^{2}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and the positive number $t=(R-1)^{2}$, the set $A$ is Borel and $(R-1)^{2}\mu(A)\le\int\lVert x'\rVert^{2}\,\mu(dx')=M_{2}(\mu)$, a real number as $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). With The Integral of an Indicator Function is the Measure of the Set,

Now fix $x\in\mathbb{R}^{dN}$. By (1), $G_{1}(x,\cdot)=\sum_{k=1}^{N}N^{-1}H(\mathfrak{p}_{k}(x),\cdot)$, a linear combination of nonnegative Borel functions with finite integrals $\tau(\mathfrak{p}_{k}(x))$, hence of integrable functions; by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, $\int G_{1}(x,y)\,\lambda_{d}(dy)=\sum_{k=1}^{N}N^{-1}\tau(\mathfrak{p}_{k}(x))$. The functions $\tau\circ\mathfrak{p}_{k}$ are Borel with values in $[0,1]$; by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §particle-laws, $(\mathfrak{p}_{k})_{\#}Q=\mu$, so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives $\int\tau\circ\mathfrak{p}_{k}\,dQ=\int\tau\,d\mu$, and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear with Step 0 gives

Also $\int G_{2}(x,y)\,\lambda_{d}(dy)=\int\mathbf{1}_{B^{c}}a\,d\lambda_{d}$ does not depend on $x$, so by (5) its integral against the probability measure $Q$ is $\int\tau\,d\mu$ as well. Since $G_{0}\le G_{1}+G_{2}$, the multiple rule ($\mathbf{1}_{B^{c}}(y)E(y)=\int G_{0}(x,y)\,Q(dx)$), the Tonelli statement for $G_{0}$, monotonicity and additivity give, with (6),

Step 5 (Conclusion). Pointwise $E=\mathbf{1}_{B}E+\mathbf{1}_{B^{c}}E$, so by (2), additivity, (4) and (7),

which is the asserted bound.