Proof of Existence of an Optimal Coupling of Two Borel Probability Measures with Finite Second Moment on a Hilbert Space

Each result cited below is universally quantified over the data in its own statement and is applied with the data named at the point of citation.

Step 1 (The infimum). Put $J=\{I(\sigma):\sigma\in\Pi(\mu,\nu)\}$. By The Quadratic Wasserstein Distance on a Hilbert Space §distance, whose preamble uses 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 §product and 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 §cost-finite, $J$ is a nonempty set of nonnegative real numbers, its greatest lower bound $c=\inf J$ is a nonnegative real number, and $W_{2}(\mu,\nu)$ is the nonnegative square root of $c$; hence $W_{2}(\mu,\nu)^{2}=c$ by Existence and Uniqueness of the Nonnegative Square Root. Since $c$ is a lower bound of $J$, every $\sigma\in\Pi(\mu,\nu)$ satisfies $c\le I(\sigma)$. It therefore suffices to produce $\pi\in\Pi(\mu,\nu)$ with $I(\pi)\le c$.

Step 2 (A minimising sequence). For $j\in\mathbb{N}$ let $t_{j}=\iota(j)$, where $\iota:\mathbb{N}\to\mathbb{R}$ is the canonical map. By claims 2 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $1\le t_{j}$ and $t_{j}^{-1}$ exists with $0<t_{j}^{-1}\le1$. For each $j\in\mathbb{N}$, claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, applied to the set $J$, which is nonempty and bounded below by $0$, and to the positive number $t_{j}^{-1}$, provides a coupling; choose one, $\pi_{j}\in\Pi(\mu,\nu)$, with

$$I(\pi_{j})<c+t_{j}^{-1}.$$

Thus every $I(\pi_{j})$ is a real number with $0\le I(\pi_{j})<c+1$.

Step 3 (Extraction). The space $X\times X$ is a real Hilbert space with distance $d$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, so $(X\times X,d)$ is a metric space by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight the set $\Pi(\mu,\nu)$ is tight in $(X\times X,d)$. The set $\{\pi_{j}:j\in\mathbb{N}\}$ of terms of the sequence $(\pi_{j})_{j\in\mathbb{N}}$ is a subset of $\Pi(\mu,\nu)$, so for each tolerance a compact set witnessing the tightness of $\Pi(\mu,\nu)$ witnesses it for that subset as well; hence the sequence $(\pi_{j})_{j\in\mathbb{N}}$ is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence. Its terms are Borel measures on $(X\times X,d)$ of total mass $1$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures. Hence Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence, applied to the metric space $(X\times X,d)$ and the sequence $(\pi_{j})_{j\in\mathbb{N}}$, gives a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and a Borel measure $\pi$ on $(X\times X,d)$ with $\pi(X\times X)=1$, that is $\pi\in\mathcal{P}(X\times X)$, such that $\pi_{k_{j}}\Rightarrow\pi$.

Step 4 (The limit is a coupling). Put $\mu_{j}=\mu$ and $\nu_{j}=\nu$ for every $j\in\mathbb{N}$. For every bounded continuous $f:X\to\mathbb{R}$ the sequence $\bigl(\int_{X}f\,d\mu_{j}\bigr)_{j\in\mathbb{N}}$ is constant with value $\int_{X}f\,d\mu$, and therefore converges to $\int_{X}f\,d\mu$, the distance from each term to that value being $0$; so $\mu_{j}\Rightarrow\mu$ in the sense of Weak Convergence of Finite Borel Measures on a Metric Space, and likewise $\nu_{j}\Rightarrow\nu$. Since $\pi_{k_{j}}\in\Pi(\mu_{j},\nu_{j})$ for every $j$ and $\pi_{k_{j}}\Rightarrow\pi$, Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §closed, applied to the sequences $(\mu_{j})$, $(\nu_{j})$, $(\pi_{k_{j}})$ and the measures $\mu,\nu,\pi$, gives $\pi\in\Pi(\mu,\nu)$.

Step 5 (The cost of the limit). Put $a_{j}=I(\pi_{k_{j}})$ for $j\in\mathbb{N}$. By Step 2 each $a_{j}$ is real with $|a_{j}|\le c+1$, and $0<c+1$, so $(a_{j})_{j\in\mathbb{N}}$ is bounded. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §lsc, applied in the situation of Step 4, $I(\pi)<\infty$ and $I(\pi)\le\ell$, where $\ell=\liminf_{j}a_{j}$ is the limit inferior.

We show $\ell\le c$. Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ be given; the following choices are made in this order. First, by claim 3 of The Archimedean Property of the Real Numbers choose $N\in\mathbb{N}$ with $0<t_{N}^{-1}<\varepsilon$. Second, by claim 3 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence choose $M\in\mathbb{N}$ with $\ell-\varepsilon<a_{m}$ for every $m\ge M$. Third, let $m$ be the larger of $N$ and $M$. By Strictly Increasing Sequences of Natural Numbers Dominate Their Index, $m\le k_{m}$, so $N\le k_{m}$; hence $t_{N}\le t_{k_{m}}$, with equality if $N=k_{m}$ and by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field if $N<k_{m}$, and so $t_{k_{m}}^{-1}\le t_{N}^{-1}$, both numbers being positive. By Step 2,

$$\ell-\varepsilon<a_{m}=I(\pi_{k_{m}})<c+t_{k_{m}}^{-1}\le c+t_{N}^{-1}<c+\varepsilon ,$$

so $\ell<c+2\varepsilon$. As $\varepsilon>0$ was arbitrary, $\ell\le c$: otherwise $c<\ell$, and the choice $\varepsilon=(\ell-c)/2$ would give $\ell<\ell$.

Step 6 (Conclusion). Steps 4 and 5 give $\pi\in\Pi(\mu,\nu)$ with $I(\pi)\le\ell\le c$, and Step 1 gives $c\le I(\pi)$. Hence $I(\pi)=c=W_{2}(\mu,\nu)^{2}$, which is the assertion.