Proof of Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named where it is used.

The infimum. By The Quadratic Wasserstein Distance on Euclidean Space (i) the set $\{I(\sigma):\sigma\in\Pi(\mu,\nu)\}$ is a nonempty set of real numbers bounded below by $0$, its greatest lower bound $c_{*}=\inf\{I(\sigma):\sigma\in\Pi(\mu,\nu)\}$ is a nonnegative real number, and $W_{2}(\mu,\nu)$ is its nonnegative square root; hence $W_{2}(\mu,\nu)^{2}=c_{*}$ by Existence and Uniqueness of the Nonnegative Square Root. Since $c_{*}$ is a lower bound of that set, every $\sigma\in\Pi(\mu,\nu)$ satisfies $c_{*}\le I(\sigma)$; so the final assertion of the statement follows once a coupling of cost $c_{*}$ is produced.

A minimising sequence. Let $\iota_{\mathbb{N}}$ be the embedding of The Canonical Map from the Natural Numbers to a Field of the natural numbers in $\mathbb{R}$; for $k\in\mathbb{N}$ the number $t_{k}=\iota_{\mathbb{N}}(k)$ is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so $t_{k}^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and $c_{*}<c_{*}+t_{k}^{-1}$ by claim 1 of that lemma. By Lower Bound and Greatest Lower Bound the number $c_{*}+t_{k}^{-1}$, being larger than the greatest lower bound, is not a lower bound of $\{I(\sigma):\sigma\in\Pi(\mu,\nu)\}$; choose, for each $k\in\mathbb{N}$, a coupling $\pi_{k}\in\Pi(\mu,\nu)$ with

$$I(\pi_{k})<c_{*}+t_{k}^{-1}.$$

Extraction. By Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment (ii) the family $\Pi(\mu,\nu)$ is tight in $(\mathbb{R}^{m+m},d_{E})$. The set $\{\pi_{k}:k\in\mathbb{N}\}$ of terms of the sequence $(\pi_{k})_{k\in\mathbb{N}}$ is a subset of $\Pi(\mu,\nu)$, so a compact set witnessing the tightness of $\Pi(\mu,\nu)$ for a given tolerance witnesses it for that subset as well; hence the sequence $(\pi_{k})_{k\in\mathbb{N}}$ is tight in the sense of Tight Family of Borel Measures on a Metric Space (ii). Its terms belong to $\mathcal{P}(\mathbb{R}^{m+m})$, so by Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence there are a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and a measure $\pi\in\mathcal{P}(\mathbb{R}^{m+m})$ such that $(\pi_{k_{j}})_{j\in\mathbb{N}}$ converges weakly to $\pi$ on $(\mathbb{R}^{m+m},d_{E})$.

By The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence, applied to the sequence $(\pi_{k_{j}})_{j\in\mathbb{N}}$, whose terms lie in $\Pi(\mu,\nu)$, we get $\pi\in\Pi(\mu,\nu)$. In particular $I(\pi)=J(\pi)$ in the notation of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit.

The cost of the limit. Let $\varepsilon\in\mathbb{R}$ be positive. By The Archimedean Property of the Real Numbers there is $N\in\mathbb{N}$ with $\varepsilon^{-1}<t_{N}$, the inverse $\varepsilon^{-1}$ existing and being positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Let $j\in\mathbb{N}$ satisfy $N\le j$. By Strictly Increasing Sequences of Natural Numbers Dominate Their Index we have $j\le k_{j}$, so $N\le k_{j}$ and therefore $t_{N}\le t_{k_{j}}$: by the trichotomy of claim 3 of Properties of the Order on the Natural Numbers either $N=k_{j}$, and the two are equal, or $N<k_{j}$, and claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field gives $t_{N}<t_{k_{j}}$; hence $\varepsilon^{-1}<t_{k_{j}}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Multiplying by the positive number $\varepsilon$ and then by the positive number $t_{k_{j}}^{-1}$, using claim 10 of Elementary Order Arithmetic in an Ordered Field twice, gives $1<\varepsilon\,t_{k_{j}}$ and then $t_{k_{j}}^{-1}<\varepsilon$. Consequently

$$J(\pi_{k_{j}})=I(\pi_{k_{j}})<c_{*}+t_{k_{j}}^{-1}<c_{*}+\varepsilon ,$$

so $J(\pi_{k_{j}})\le c_{*}+\varepsilon$ for every $j$ with $N\le j$.

Since $\varepsilon$ was an arbitrary positive real number, Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit (iii), applied to the sequence $(\pi_{k_{j}})_{j\in\mathbb{N}}$, its weak limit $\pi$ and the real number $c_{*}$, gives $J(\pi)\le c_{*}$, that is, $I(\pi)\le c_{*}$. Together with $c_{*}\le I(\pi)$ from the first paragraph this gives

$$I(\pi)=c_{*}=W_{2}(\mu,\nu)^{2},$$

so $\pi$ is an optimal coupling of $\mu$ and $\nu$ in the sense of Optimal Coupling of Two Probability Measures with Finite Second Moment (i). Finally, any $\pi\in\Pi(\mu,\nu)$ with $I(\pi)=W_{2}(\mu,\nu)^{2}=c_{*}$ satisfies $I(\pi)\le I(\sigma)$ for every $\sigma\in\Pi(\mu,\nu)$, because $c_{*}$ is a lower bound of $\{I(\sigma):\sigma\in\Pi(\mu,\nu)\}$.