Proof of The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost

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. For $\mu\in\mathcal{P}(\mathbb{R}^{d})$ we write $\eta_{\varepsilon}*\mu$ for the mollified density of Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost. For a nonnegative integrable function on a measure space, its real integral equals its integral as a nonnegative measurable function, its positive part being the function itself and its negative part $0$ (Integrable Function and the Lebesgue Integral); conversely a nonnegative measurable real function whose integral in $[0,\infty]$ is finite is integrable, by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral. These two facts are used without further mention. The 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.

Step 1 (Claim 1). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By The Mollified Density Cost of a Probability Measure with Finite Second Moment §cost, $\mathcal{G}_{\Phi,\varepsilon}(\mu)=\int_{\mathbb{R}^{d}}\Phi\circ(\eta_{\varepsilon}*\mu)\,d\lambda_{d}$, and Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §cost gives $0\le\mathcal{G}_{\Phi,\varepsilon}(\mu)\le L$.

Step 2 (Claim 2). Let $D$ be as in claim 2, put $K_{0}=\varepsilon^{-1}(2+2^{d}d\,\kappa_{d}D)$, so that $K=L\,K_{0}$, and let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Write $a=\eta_{\varepsilon}*\mu$ and $b=\eta_{\varepsilon}*\nu$.

(2a) Reduction to the densities. 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 §cost, the functions $a$, $b$, $\Phi\circ a$ and $\Phi\circ b$ are Borel, nonnegative and integrable with respect to $\lambda_{d}$. For $y\in\mathbb{R}^{d}$, let $\alpha$ and $\beta$ be the smaller and the larger of $a(y),b(y)$; the Lipschitz bound of Convex Lipschitz Integrands §integrand gives $|\Phi(a(y))-\Phi(b(y))|=\Phi(\beta)-\Phi(\alpha)\le L(\beta-\alpha)=L\,|a(y)-b(y)|$. The functions $\Phi\circ a-\Phi\circ b$ and $a-b$ are integrable, their absolute values are measurable by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and have finite integrals, so they are integrable too; hence claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

(2b) An optimal coupling. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment (with $m=d$) there is a coupling $\pi\in\Pi(\mu,\nu)$ whose quadratic cost satisfies $I(\pi)=W_{2}(\mu,\nu)^{2}$. Its push-forwards by the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs are $\mu$ and $\nu$. Fix $y\in\mathbb{R}^{d}$. The function $x\mapsto\eta_{\varepsilon}(y-x)$ is Borel and integrable with respect to $\mu$ and to $\nu$ by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density, so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives that $w\mapsto\eta_{\varepsilon}(y-\mathrm{pr}_{1}(w))$ and $w\mapsto\eta_{\varepsilon}(y-\mathrm{pr}_{2}(w))$ are integrable with respect to $\pi$, with

Put $F(w,y)=|\eta_{\varepsilon}(y-\mathrm{pr}_{1}(w))-\eta_{\varepsilon}(y-\mathrm{pr}_{2}(w))|$. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

(2c) Measurability. Write $P_{1}=\mathrm{pr}^{d+d,d}_{1}$ and $P_{2}=\mathrm{pr}^{d+d,d}_{2}$ for the coordinate projections of $\mathbb{R}^{(d+d)+d}$ and $\iota'=\iota^{d+d,d}$ for the concatenation (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs). The maps $\mathrm{pr}_{1}\circ P_{1}$, $\mathrm{pr}_{2}\circ P_{1}$ and $P_{2}$ are Borel, as compositions of Borel maps (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, 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 $m=(d+d)+d$) and claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the function $\hat F(v)=|\eta_{\varepsilon}(P_{2}(v)-\mathrm{pr}_{1}(P_{1}(v)))-\eta_{\varepsilon}(P_{2}(v)-\mathrm{pr}_{2}(P_{1}(v)))|$ is Borel on $\mathbb{R}^{(d+d)+d}$. 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 $\mathcal{B}(\mathbb{R}^{d+d})\otimes\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R}^{(d+d)+d})$, so $F=\hat F\circ\iota'$, which holds by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, is measurable with respect to the product $\sigma$-algebra by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; it is nonnegative.

(2d) Tonelli. The measure $\pi$ is a probability measure and $\lambda_{d}$ is $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite. The Tonelli statement of Tonelli and Fubini Theorems, applied to $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$, $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and $F$, shows that $y\mapsto\int F(w,y)\,\pi(dw)$ is Borel and that

For each $w$, Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §translation with $x=\mathrm{pr}_{1}(w)$ and $x'=\mathrm{pr}_{2}(w)$ gives $\int F(w,y)\,\lambda_{d}(dy)\le K_{0}\,X(w)$, where $X(w)=\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions. Combining with (2), by monotonicity,

(2e) Cauchy-Schwarz. Since $\pi(\mathbb{R}^{d+d})=1$, $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$ is a probability space, on which $X$ and the constant $Y=1$ are random variables (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions for $Y$). Pointwise $X^{2}=\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert^{2}$, so $\mathbb{E}[X^{2}]=I(\pi)=W_{2}(\mu,\nu)^{2}<\infty$, and $\mathbb{E}[Y^{2}]=\pi(\mathbb{R}^{d+d})=1$ by The Integral of an Indicator Function is the Measure of the Set, with expectations as in Expectation, Variance, and Moments. Hence $X$ and $Y$ are square-integrable, with $\lVert X\rVert_{2}=W_{2}(\mu,\nu)$ and $\lVert Y\rVert_{2}=1$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root, since $W_{2}(\mu,\nu)\ge0$ and $1\ge0$. As $XY=X$ is nonnegative and integrable (Square-Integrable Random Variables and the Mean-Square Inner Product), claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm gives

With (1) and (3), $|\mathcal{G}_{\Phi,\varepsilon}(\mu)-\mathcal{G}_{\Phi,\varepsilon}(\nu)|\le L\,K_{0}\,W_{2}(\mu,\nu)=K\,W_{2}(\mu,\nu)$. In the terms of Lipschitz Map Between Metric Spaces, with the metric $W_{2}$ of The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric and the absolute-value metric, this is the Lipschitz property with constant $K$. This proves claim 2.

Step 3 (Claim 3). Let $\mu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and put $a=\eta_{\varepsilon}*\mu$. As recorded in The Density Cost of a Convex Lipschitz Integrand §cost, $\mu$ has a density $\rho$ with respect to $\lambda_{d}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities: $\rho:\mathbb{R}^{d}\to\mathbb{R}$ is Borel, $\rho\ge0$, and $\mu(A)=\int\mathbf{1}_{A}\rho\,d\lambda_{d}$ for every $A\in\mathcal{B}(\mathbb{R}^{d})$; moreover $\mathcal{G}_{\Phi}(\mu)=\int_{\mathbb{R}^{d}}\Phi\circ\rho\,d\lambda_{d}$. Thus $\mu$ is the measure with density $\rho$ with respect to $\lambda_{d}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables, and that claim gives

By Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand §composition, applied to $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and $h=\rho$, the function $\Phi\circ\rho$ is Borel and $0\le\Phi(\rho(x))\le L\,\rho(x)$ for every $x$.

(3a) A pointwise inequality. Fix $y\in\mathbb{R}^{d}$ and write $e_{y}(x)=\eta_{\varepsilon}(y-x)$; by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel (with $m=d$, $v$ constant equal to $y$ and $u$ the identity) $e_{y}$ is Borel, and it is nonnegative. By Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §density, $t_{0}=a(y)$ lies in $[0,\infty)$, and by (4) with $f=e_{y}$ the nonnegative Borel function $e_{y}\rho$ (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) has integral $t_{0}$ with respect to $\lambda_{d}$, so it is integrable with real integral $t_{0}$. By Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, $\eta_{\varepsilon}$ is a nonnegative Borel mollifier kernel, so by condition 4 of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ it is integrable with integral $1$; claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$, applied with $n=d$, $a=y$ and $f=\eta_{\varepsilon}$, gives $\int e_{y}\,d\lambda_{d}=\int\eta_{\varepsilon}(y-x)\,\lambda_{d}(dx)=\int\eta_{\varepsilon}\,d\lambda_{d}=1$, so $e_{y}$ is integrable with integral $1$. By Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand §support choose a real $s$ with $\Phi(t)\ge\Phi(t_{0})+s(t-t_{0})$ for all $t\in[0,\infty)$. Taking $t=\rho(x)$ and multiplying by $e_{y}(x)\ge0$,

The right side is integrable with integral $(\Phi(t_{0})-s\,t_{0})\cdot1+s\,t_{0}=\Phi(t_{0})$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral. The left side $q_{y}=e_{y}\cdot(\Phi\circ\rho)$ is Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) with $0\le q_{y}\le L\,e_{y}\rho$, so its integral is at most $L\,t_{0}<\infty$ and it is integrable. The monotonicity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

(3b) Integration in $y$. Let $\hat{Q}(w)=\eta_{\varepsilon}(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))\,\Phi(\rho(\mathrm{pr}_{1}(w)))$ for $w\in\mathbb{R}^{d+d}$; it is Borel by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel (with $m=d+d$, $u=\mathrm{pr}_{1}$, $v=\mathrm{pr}_{2}$), by composition (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps) and by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and nonnegative. As in (2c), $Q=\hat{Q}\circ\iota$ with the concatenation $\iota=\iota^{d,d}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{d})$, and $Q(x,y)=\eta_{\varepsilon}(y-x)\Phi(\rho(x))$. The Tonelli statement of Tonelli and Fubini Theorems for $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ twice and $Q$ gives

For each $x$, by the multiple rule in claim 1 of Linearity and Monotonicity of the Lebesgue Integral with the constant $\Phi(\rho(x))\ge0$ and by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, the inner integral on the right is $\Phi(\rho(x))\int\eta_{\varepsilon}(y-x)\,\lambda_{d}(dy)=\Phi(\rho(x))$; so the right side is $\int\Phi\circ\rho\,d\lambda_{d}=\mathcal{G}_{\Phi}(\mu)$. By (5) and monotonicity, the left side is at least $\int_{\mathbb{R}^{d}}\Phi\circ a\,d\lambda_{d}=\mathcal{G}_{\Phi,\varepsilon}(\mu)$, the function $\Phi\circ a$ being nonnegative, Borel and integrable by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §cost. Hence $\mathcal{G}_{\Phi,\varepsilon}(\mu)\le\mathcal{G}_{\Phi}(\mu)$, which is claim 3.