Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $\mu,\nu\in\mathcal{P}(X)$. Couplings, the sets $\Pi(\mu,\nu)$ and the quadratic cost $I(\pi)$ are those of that definition, and the second moment $M_{2}$ and the set $\mathcal{P}_{2}(X)$ of probability measures with finite second moment are those of that definition. For Borel $S,T:X\to X$ the function $x\mapsto|S(x)-T(x)|^{2}$ is Borel, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing and claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space, so its integral against any member of $\mathcal{P}(X)$ is defined in $[0,\infty]$. A finite set $F\subseteq X$ is closed in $(X,d)$, since every $x\in X\setminus F$ has an open ball about it of radius $1$ (if $F$ is empty) or of radius the least of the distances $d(x,a)$, $a\in F$, which misses $F$; hence $F$ and $X\setminus F$ are Borel by claim 1 of Borel Measurability and Bounded Integration on a Metric Space. A function $f:X\to\mathbb{R}$ is Lipschitz with constant $L$, for real $L\ge0$, if $|f(x)-f(y)|\le L|x-y|$ for all $x,y\in X$, which is the notion of Lipschitz Map Between Metric Spaces for the distance $d$ and the absolute-value metric. Square roots are the nonnegative square roots of Existence and Uniqueness of the Nonnegative Square Root, and $\mathrm{id}_{X}$ is the identity map of $X$. Then the following hold.

1. (Product coupling)¶ The product measure $\mu\otimes\nu$ belongs to $\Pi(\mu,\nu)$; in particular $\Pi(\mu,\nu)$ is nonempty.

2. (Swap)¶ Let $\sigma:X\times X\to X\times X$ be the swap $\sigma(x,y)=(y,x)$, which is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, since $\sigma=(\pi_{2},\pi_{1})$. For every $\pi\in\Pi(\mu,\nu)$ one has $\sigma_{\#}\pi\in\Pi(\nu,\mu)$, $\sigma_{\#}(\sigma_{\#}\pi)=\pi$ and $I(\sigma_{\#}\pi)=I(\pi)$.

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

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

$$I\bigl((S,T)_{\#}\mu\bigr)=\int_{X}|S(x)-T(x)|^{2}\,\mu(dx).$$

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

5. (Modifying one marginal)¶ Let $\pi\in\Pi(\mu,\nu)$ and let $S,T:X\to X$ be Borel. Then $\pi'=(\pi_{1},T\circ\pi_{2})_{\#}\pi\in\Pi(\mu,T_{\#}\nu)$ and $\pi''=(S\circ\pi_{1},\pi_{2})_{\#}\pi\in\Pi(S_{\#}\mu,\nu)$. If moreover $I(\pi)<\infty$ and $\int_{X}|T(y)-y|^{2}\,\nu(dy)<\infty$, then $I(\pi')<\infty$ and

$$\sqrt{I(\pi')}\le\sqrt{I(\pi)}+\sqrt{\int_{X}|T(y)-y|^{2}\,\nu(dy)}\ ;$$

likewise, if $I(\pi)<\infty$ and $\int_{X}|S(x)-x|^{2}\,\mu(dx)<\infty$, then $I(\pi'')<\infty$ and $\sqrt{I(\pi'')}\le\sqrt{I(\pi)}+\sqrt{\int_{X}|S(x)-x|^{2}\,\mu(dx)}$.

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

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

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

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

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

9. (Moment bound)¶ If $\mu,\nu\in\mathcal{P}_{2}(X)$ and $\pi\in\Pi(\mu,\nu)$, then $\bigl|\sqrt{M_{2}(\mu)}-\sqrt{M_{2}(\nu)}\bigr|\le\sqrt{I(\pi)}$.