Proof of The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. Throughout, $\Pi(\cdot,\cdot)$ and $I$ are the couplings and their quadratic cost; by The Quadratic Wasserstein Distance on Euclidean Space (i), for $\alpha,\beta\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the number $W_{2}(\alpha,\beta)$ is nonnegative, $W_{2}(\alpha,\beta)^{2}$ is the greatest lower bound of $J(\alpha,\beta)=\{I(\pi):\pi\in\Pi(\alpha,\beta)\}$, a set of nonnegative real numbers by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii), and $W_{2}(\alpha,\beta)^{2}\le I(\pi)$ for every $\pi\in\Pi(\alpha,\beta)$, so that $W_{2}(\alpha,\beta)\le\sqrt{I(\pi)}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. We use the following approximation: for $\alpha,\beta\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and real $\varepsilon>0$ there is $\pi\in\Pi(\alpha,\beta)$ with $\sqrt{I(\pi)}<W_{2}(\alpha,\beta)+\varepsilon$. Indeed, with $W=W_{2}(\alpha,\beta)$ and $\eta=2W\varepsilon+\varepsilon^{2}$, which is positive because $0\le2W\varepsilon$ by claim 5 of Elementary Arithmetic in an Ordered Field ($W$ being nonnegative and $0<2\varepsilon$ by claims 8 and 5 of Elementary Order Arithmetic in an Ordered Field), $0<\varepsilon^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, and claim 3 there, claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ gives $\pi\in\Pi(\alpha,\beta)$ with $I(\pi)<W^{2}+\eta=(W+\varepsilon)^{2}$ (claim 5 of Zero Products and Elementary Identities in a Field), whence $\sqrt{I(\pi)}<W+\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 1. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (ii), $\pi\mapsto\sigma_{\#}\pi$ is a bijection of $\Pi(\mu,\nu)$ onto $\Pi(\nu,\mu)$ with $I(\sigma_{\#}\pi)=I(\pi)$; hence $J(\mu,\nu)=J(\nu,\mu)$, so the two sets have the same greatest lower bound, and $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$ by The Quadratic Wasserstein Distance on Euclidean Space (i), the nonnegative square root being unique by Existence and Uniqueness of the Nonnegative Square Root.

Claim 2. If $\mu=\nu$, then $(\mathrm{id},\mathrm{id})_{\#}\mu\in\Pi(\mu,\mu)$ has quadratic cost $0$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iv), so $W_{2}(\mu,\mu)^{2}\le0$; being nonnegative, $W_{2}(\mu,\mu)^{2}=0$, and $W_{2}(\mu,\mu)=0$ by claim 3 of Zero Products and Elementary Identities in a Field. Conversely suppose $W_{2}(\mu,\nu)=0$, and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and Lipschitz with some constant $L\ge0$ from $(\mathbb{R}^{d},d_{E})$ to $\mathbb{R}$ with the absolute-value metric, the notion of Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound. Put $\Delta=|\int f\,d\mu-\int f\,d\nu|$, the integrals existing by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii) since $f$ is bounded and Borel (continuous by A Lipschitz Map is Uniformly Continuous and Continuous, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)). For every real $\varepsilon>0$ the approximation gives $\pi\in\Pi(\mu,\nu)$ with $\sqrt{I(\pi)}<\varepsilon$, and then $\Delta\le L\sqrt{I(\pi)}\le L\varepsilon$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (viii) and claim 5 of Elementary Arithmetic in an Ordered Field. If $L=0$ this gives $\Delta\le0$; if $0<L$ and $0<\Delta$ held, the choice $\varepsilon=\Delta L^{-1}\cdot2^{-1}$, positive by claims 7, 5 and 8 of Elementary Order Arithmetic in an Ordered Field, would give $\Delta\le\Delta\cdot2^{-1}<\Delta$ by claim 8 there, which is impossible. Hence $\Delta=0$, that is, $\int f\,d\mu=\int f\,d\nu$ for every bounded Lipschitz $f$. Now $(\mathbb{R}^{d},d_{E})$ is a nonempty metric space whose Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i), and $\mu,\nu$ are finite measures on it; so $\mu=\nu$ by claim 1 of Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits.

Claim 3. Let $\varepsilon\in\mathbb{R}$ satisfy $0<\varepsilon$. By the approximation there are $\pi_{12}\in\Pi(\mu,\nu)$ and $\pi_{23}\in\Pi(\nu,\lambda)$ with

$$\sqrt{I(\pi_{12})}<W_{2}(\mu,\nu)+\varepsilon,\qquad\sqrt{I(\pi_{23})}<W_{2}(\nu,\lambda)+\varepsilon ,$$

both costs being finite. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (vi) there is a Borel $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with finite image $F=T(\mathbb{R}^{d})$ and $\int\lVert T(y)-y\rVert^{2}\,\nu(dy)\le\varepsilon^{2}$, so that $\sqrt{\int\lVert T(y)-y\rVert^{2}\,\nu(dy)}\le\varepsilon$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; put $\rho=T_{\#}\nu\in\mathcal{P}(\mathbb{R}^{d})$, which satisfies $\rho(\mathbb{R}^{d}\setminus F)=0$ by the same clause. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (v), $\pi'_{12}=(\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})_{\#}\pi_{12}\in\Pi(\mu,\rho)$ and $\pi'_{23}=(T\circ\mathrm{pr}_{1},\mathrm{pr}_{2})_{\#}\pi_{23}\in\Pi(\rho,\lambda)$ have finite costs with

$$\sqrt{I(\pi'_{12})}\le\sqrt{I(\pi_{12})}+\varepsilon,\qquad\sqrt{I(\pi'_{23})}\le\sqrt{I(\pi_{23})}+\varepsilon ,$$

the second by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to the bound on $I(\pi'_{23})$ recorded there. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (vii) applied to $\rho$, $F$, $\pi'_{12}$ and $\pi'_{23}$, there is $\pi_{13}\in\Pi(\mu,\lambda)$ with $\sqrt{I(\pi_{13})}\le\sqrt{I(\pi'_{12})}+\sqrt{I(\pi'_{23})}$. Combining these inequalities by claims 2 and 3 of Elementary Order Arithmetic in an Ordered Field,

$$W_{2}(\mu,\lambda)\le\sqrt{I(\pi_{13})}<W_{2}(\mu,\nu)+W_{2}(\nu,\lambda)+\varepsilon+\varepsilon+\varepsilon+\varepsilon .$$

Write $S=W_{2}(\mu,\nu)+W_{2}(\nu,\lambda)$. If $S<W_{2}(\mu,\lambda)$ held, let $\gamma=W_{2}(\mu,\lambda)-S$, positive by claim 3 of Elementary Arithmetic in an Ordered Field and claim 1 of Elementary Order Arithmetic in an Ordered Field, and choose $\varepsilon=\gamma\cdot2^{-1}\cdot2^{-1}\cdot2^{-1}$, positive by claim 8 of Elementary Order Arithmetic in an Ordered Field; three applications of that claim give $\varepsilon+\varepsilon+\varepsilon+\varepsilon=\gamma\cdot2^{-1}<\gamma$, so the display yields $W_{2}(\mu,\lambda)<S+\gamma=W_{2}(\mu,\lambda)$, which is impossible. Hence $W_{2}(\mu,\lambda)\le S$.

Claim 4. By Metric Space, a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is a real-valued function of two arguments that is nonnegative, vanishes exactly on pairs of equal points, is symmetric, and satisfies the triangle inequality; $W_{2}$ has these properties by The Quadratic Wasserstein Distance on Euclidean Space (i) and claims 2, 1 and 3. So $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ is a metric space.