Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $\mu,\nu\in\mathcal{P}(X)$, and let $\pi_{1},\pi_{2}:X\times X\to X$ be the coordinate maps of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, which are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma. In this definition and its dependents the letter $I$ denotes the quadratic cost below, not the identity form of Real Hilbert Spaces: Standing Notation and Background §forms.

1. (Coupling)¶ A coupling of $\mu$ and $\nu$ is a $\pi\in\mathcal{P}(X\times X)$ with $(\pi_{1})_{\#}\pi=\mu$ and $(\pi_{2})_{\#}\pi=\nu$. The set of couplings of $\mu$ and $\nu$ is denoted $\Pi(\mu,\nu)$.

2. (Quadratic cost)¶ The quadratic cost of $\pi\in\Pi(\mu,\nu)$ is

$$I(\pi)=\int_{X\times X}|\pi_{1}(z)-\pi_{2}(z)|^{2}\,\pi(dz)\in[0,\infty],$$

the integral of a nonnegative function that is continuous on $(X\times X,d)$, hence Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space: the maps $\pi_{1},\pi_{2}$ are linear with $|\pi_{i}z|\le|z|$ by Properties of the Product of Two Real Inner Product Spaces §coordinates, so $z\mapsto\pi_{1}(z)-\pi_{2}(z)$ is Lipschitz with constant $2$, and $x\mapsto|x|^{2}$ is continuous on $X$ as recorded in The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment.