Proof of The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance

Each result cited is universally quantified over the data in its own statement, and is used here with the dimension $m$ and with the measures named in the statement.

Claim 1. Let $\pi\in\Pi(\mu,\nu)$. The maps $\mathrm{pr}_{1}^{m}$ and $\mathrm{pr}_{2}^{m}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), and the map $y\mapsto\lVert y\rVert^{2}$ on $\mathbb{R}^{m}$ is a nonnegative Borel function by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i) the push-forwards of $\pi$ by $\mathrm{pr}_{1}^{m}$ and $\mathrm{pr}_{2}^{m}$ are $\mu$ and $\nu$, so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) gives

$$\int_{\mathbb{R}^{m+m}}\lVert\mathrm{pr}_{1}^{m}(z)\rVert^{2}\,\pi(dz)=\int_{\mathbb{R}^{m}}\lVert y\rVert^{2}\,\mu(dy)=M_{2}(\mu),\qquad\int_{\mathbb{R}^{m+m}}\lVert\mathrm{pr}_{2}^{m}(z)\rVert^{2}\,\pi(dz)=M_{2}(\nu),$$

the second moments being those of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i). Both are finite because $\mu$ and $\nu$ lie in $\mathcal{P}_{2}(\mathbb{R}^{m})$. Hence, by the description of $L^{2}(\pi;\mathbb{R}^{m})$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), the classes of $\mathrm{pr}_{1}^{m}$ and $\mathrm{pr}_{2}^{m}$ belong to $L^{2}(\pi;\mathbb{R}^{m})$, with $\lVert\mathrm{pr}_{1}^{m}\rVert_{\pi}^{2}=M_{2}(\mu)$ and $\lVert\mathrm{pr}_{2}^{m}\rVert_{\pi}^{2}=M_{2}(\nu)$, the squares of the norms being the integrals just computed because the nonnegative square root of a nonnegative real number squares back to it, by Existence and Uniqueness of the Nonnegative Square Root.

The difference of the two classes is the class of the pointwise difference $\mathrm{pr}_{1}^{m}-\mathrm{pr}_{2}^{m}$, by the vector space operations of The Space of Square-Integrable Random Vectors (ii) applied on the probability space $(\mathbb{R}^{m+m},\mathcal{B}(\mathbb{R}^{m+m}),\pi)$ as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), so

$$\lVert\mathrm{pr}_{1}^{m}-\mathrm{pr}_{2}^{m}\rVert_{\pi}^{2}=\int_{\mathbb{R}^{m+m}}\lVert\mathrm{pr}_{1}^{m}(z)-\mathrm{pr}_{2}^{m}(z)\rVert^{2}\,\pi(dz)=I(\pi),$$

the last equality being Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii). This proves claim 1; in particular $I(\pi)$ is finite, as 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) also gives.

Claim 2. Write $c=\bigl|\sqrt{M_{2}(\mu)}-\sqrt{M_{2}(\nu)}\bigr|$, a nonnegative real number by claim 1 of Properties of the Absolute Value in an Ordered Field. Let $\pi\in\Pi(\mu,\nu)$, a nonempty set 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 (i). The space $L^{2}(\pi;\mathbb{R}^{m})$ is a real inner product space by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), so the reverse triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (ii) applied to the two elements of claim 1 gives

$$\bigl|\lVert\mathrm{pr}_{1}^{m}\rVert_{\pi}-\lVert\mathrm{pr}_{2}^{m}\rVert_{\pi}\bigr|\le\lVert\mathrm{pr}_{1}^{m}-\mathrm{pr}_{2}^{m}\rVert_{\pi}.$$

By claim 1 the three norms occurring here are the nonnegative real numbers whose squares are $M_{2}(\mu)$, $M_{2}(\nu)$ and $I(\pi)$, so by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root they are $\sqrt{M_{2}(\mu)}$, $\sqrt{M_{2}(\nu)}$ and $\sqrt{I(\pi)}$; the displayed inequality therefore reads $c\le\sqrt{I(\pi)}$. Both sides being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $c^{2}\le I(\pi)$.

Since $\pi$ was an arbitrary member of $\Pi(\mu,\nu)$, the real number $c^{2}$ is a lower bound of the set $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$, hence is at most its greatest lower bound, which is a lower bound of that set no smaller than any other by the definition of a greatest lower bound. By The Quadratic Wasserstein Distance on Euclidean Space (i) the number $W_{2}(\mu,\nu)$ is the nonnegative square root of that greatest lower bound, so the bound equals $W_{2}(\mu,\nu)^{2}$ by Existence and Uniqueness of the Nonnegative Square Root. Hence $c^{2}\le W_{2}(\mu,\nu)^{2}$, and since $c$ and $W_{2}(\mu,\nu)$ are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $c\le W_{2}(\mu,\nu)$, which is claim 2.

Claim 3. Assume $W_{2}(\mu,\nu)\le R$ with $R$ nonnegative. By claim 3 of Properties of the Absolute Value in an Ordered Field,

$$\sqrt{M_{2}(\nu)}-\sqrt{M_{2}(\mu)}\le\Bigl|\sqrt{M_{2}(\mu)}-\sqrt{M_{2}(\nu)}\Bigr|,$$

using claim 2 of that lemma for the symmetry of the absolute value; the right-hand side is at most $W_{2}(\mu,\nu)$ by claim 2 above, hence at most $R$ by the transitivity of the order of the ordered field $\mathbb{R}$. Adding $\sqrt{M_{2}(\mu)}$ to both sides, by the compatibility of the order with addition (an axiom of Ordered Field), and cancelling the two occurrences of $\sqrt{M_{2}(\mu)}$ on the left by the associativity, inverse and identity axioms of a field, gives

$$\sqrt{M_{2}(\nu)}\le\sqrt{M_{2}(\mu)}+R .$$

Both sides are nonnegative, the left-hand side by Existence and Uniqueness of the Nonnegative Square Root and the right-hand side as a sum of nonnegative numbers by claim 2 of Elementary Arithmetic in an Ordered Field, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

$$M_{2}(\nu)=\Bigl(\sqrt{M_{2}(\nu)}\Bigr)^{2}\le\Bigl(\sqrt{M_{2}(\mu)}+R\Bigr)^{2},$$

the first equality again by Existence and Uniqueness of the Nonnegative Square Root. This is claim 3.