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

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, and let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$. Couplings, the sets $\Pi(\mu,\nu)$ and the quadratic cost $I(\pi)$ are those of that definition, the second moment $M_{2}$ and the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment are those of that definition, $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ are the coordinate projections for the splitting $d+d$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), and $\mathrm{id}$ denotes the identity map of $\mathbb{R}^{d}$, which is Borel. For Borel $S,T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ the function $x\mapsto\lVert S(x)-T(x)\rVert^{2}$ on $\mathbb{R}^{d}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), so its integral against any member of $\mathcal{P}(\mathbb{R}^{d})$ is defined in $[0,\infty]$. Finite subsets of $\mathbb{R}^{d}$ and their complements are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v). A function $f:\mathbb{R}^{d}\to\mathbb{R}$ is Lipschitz with constant $L$, for real $L\ge0$, if $|f(x)-f(y)|\le L\lVert x-y\rVert$ for all $x,y\in\mathbb{R}^{d}$, which is the notion of Lipschitz Map Between Metric Spaces for the Euclidean distance and the absolute-value metric, and $f$ is bounded in the sense of that definition. Then the following hold.

1. (Product coupling)(i) $\mu\boxtimes\nu\in\Pi(\mu,\nu)$; in particular $\Pi(\mu,\nu)$ is nonempty.

2. (Swap)(ii) Let $\sigma:\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$ be the swap, $\sigma((x,y))=(y,x)$. For every $\pi\in\Pi(\mu,\nu)$ one has $\sigma_{\#}\pi\in\Pi(\nu,\mu)$, $\sigma_{\#}(\sigma_{\#}\pi)=\pi$ and $I(\sigma_{\#}\pi)=I(\pi)$; thus $\pi\mapsto\sigma_{\#}\pi$ is a bijection of $\Pi(\mu,\nu)$ onto $\Pi(\nu,\mu)$ preserving the quadratic cost.

3. (Finiteness of the cost)(iii) If $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, then every $\pi\in\Pi(\mu,\nu)$ satisfies $I(\pi)\le2M_{2}(\mu)+2M_{2}(\nu)<\infty$. Conversely, if $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and some $\pi\in\Pi(\mu,\nu)$ has $I(\pi)<\infty$, then $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $M_{2}(\nu)\le2M_{2}(\mu)+2I(\pi)$.

4. (Push-forward couplings)(iv) Let $S,T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel. Then $(S,T)_{\#}\mu\in\Pi(S_{\#}\mu,T_{\#}\mu)$ and

$$I\bigl((S,T)_{\#}\mu\bigr)=\int_{\mathbb{R}^{d}}\lVert S(x)-T(x)\rVert^{2}\,\mu(dx).$$

In particular $(\mathrm{id},\mathrm{id})_{\#}\mu\in\Pi(\mu,\mu)$ has quadratic cost $0$.

5. (Modifying one marginal)(v) Let $\pi\in\Pi(\mu,\nu)$ and let $S,T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel. Then $(\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})_{\#}\pi\in\Pi(\mu,T_{\#}\nu)$ and $(S\circ\mathrm{pr}_{1},\mathrm{pr}_{2})_{\#}\pi\in\Pi(S_{\#}\mu,\nu)$. If moreover $I(\pi)<\infty$ and $\int_{\mathbb{R}^{d}}\lVert T(y)-y\rVert^{2}\,\nu(dy)<\infty$, then $\pi'=(\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})_{\#}\pi$ has finite quadratic cost and

$$\sqrt{I(\pi')}\le\sqrt{I(\pi)}+\sqrt{\int_{\mathbb{R}^{d}}\lVert T(y)-y\rVert^{2}\,\nu(dy)}\ ;$$

likewise, if $I(\pi)<\infty$ and $\int_{\mathbb{R}^{d}}\lVert S(x)-x\rVert^{2}\,\mu(dx)<\infty$, then $(S\circ\mathrm{pr}_{1},\mathrm{pr}_{2})_{\#}\pi$ has finite quadratic cost, at most the square of $\sqrt{I(\pi)}+\sqrt{\int_{\mathbb{R}^{d}}\lVert S(x)-x\rVert^{2}\,\mu(dx)}$.

6. (Quantisation)(vi) For every Borel $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose image $T(\mathbb{R}^{d})$ is a finite set, $T_{\#}\nu(\mathbb{R}^{d}\setminus T(\mathbb{R}^{d}))=0$. If $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\varepsilon\in\mathbb{R}$ satisfies $0<\varepsilon$, then there is a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose image is a finite set, such that $\int_{\mathbb{R}^{d}}\lVert T(y)-y\rVert^{2}\,\nu(dy)\le\varepsilon^{2}$.

7. (Gluing over a finitely supported measure)(vii) Let $\rho\in\mathcal{P}(\mathbb{R}^{d})$ satisfy $\rho(\mathbb{R}^{d}\setminus F)=0$ for some finite set $F\subseteq\mathbb{R}^{d}$, and let $\pi_{12}\in\Pi(\mu,\rho)$ and $\pi_{23}\in\Pi(\rho,\nu)$ satisfy $I(\pi_{12})<\infty$ and $I(\pi_{23})<\infty$. Then there is $\pi_{13}\in\Pi(\mu,\nu)$ with $I(\pi_{13})<\infty$ and

$$\sqrt{I(\pi_{13})}\le\sqrt{I(\pi_{12})}+\sqrt{I(\pi_{23})} .$$

8. (Lipschitz bound)(viii) Let $\pi\in\Pi(\mu,\nu)$ satisfy $I(\pi)<\infty$, and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$. Then $f$ is Borel, integrable with respect to $\mu$ and to $\nu$, and

$$\Bigl|\int_{\mathbb{R}^{d}}f\,d\mu-\int_{\mathbb{R}^{d}}f\,d\nu\Bigr|\le L\sqrt{I(\pi)} .$$