The quadratic Wasserstein space is complete

For each integer $k\ge 1$, the Cauchy property gives a threshold N_k such that $W_2({\mu }_n,{\mu }_m)<2^{-k}$ whenever $n,m\ge N_k$. Set $n_1=N_1$ and recursively $n_{k+1}=\max \{N_{k+1},n_k+1\}$. Then $n_k<n_{k+1}$ and $W_2({\mu }_{n_k},{\mu }_{n_{k+1}})\le 2^{-k}$. By AC choose an optimal coupling ${\pi }_k$ of these consecutive laws for each k using [F1].

Put $S=({\mathbb{R}}^d{)}^{\{1,2,\dots \}}$, the set of sequences of Euclidean vectors indexed by positive integers, and equip $S$ with $D(x,y)={\sum }_{j\ge 1}{2}^{-j}\min (1,\mathrm{\mid }{x}_j-y_j\mathrm{\mid })$. This metric induces the product topology: each coordinate projection is continuous from the bound on its summand, and a product neighborhood restricting finitely many coordinates contains a sufficiently small D-ball. Conversely, given a D-ball radius, make the geometric tail less than half that radius and restrict the finitely many remaining coordinates so their summands total less than the other half; this gives a product neighborhood inside the D-ball. A D-Cauchy sequence is Cauchy in each coordinate, so coordinate completeness gives a limit in S; bounding the first finitely many coordinate terms and then the geometric tail proves convergence in D. Sequences of rational vectors which are zero after finitely many coordinates form a countable dense subset, since the tail is arbitrarily small and each of finitely many coordinates can be approximated by rationals. Thus S is complete and separable, hence Polish by definition.

Set $Q_1={\pi }_1$, the probability on the first two coordinates. For $N\ge 2$, assume $Q_{N-1}$ is a probability on coordinates $1,\dots ,N$ with adjacent pair marginals ${\pi }_1,\dots ,{\pi }_{N-1}$ and last marginal ${\mu }_{n_N}$. Regard it as a law on ${\mathbb{R}}^{d(N-1)}\times {\mathbb{R}}^d$, and regard ${\pi }_N$ as a law on ${\mathbb{R}}^d\times {\mathbb{R}}^d$; they have the common second/first marginal ${\mu }_{n_N}$. The gluing statement [F1] gives $Q_N$ on coordinates $1,\dots ,N+1$ preserving all earlier adjacent marginals and adding ${\pi }_N$. Let $I_N:({\mathbb{R}}^d{)}^{N+1}\to S$ send $(x_1,\dots ,x_{N+1})$ to $(x_1,\dots ,x_{N+1},0,0,\dots )$; this map is continuous for the product topology. Define ${\eta }_N=(I_N{)}_{\mathrm{\#}}{Q}_N$, that is, ${\eta }_N(B)=Q_N(I_N^{-1}(B))$ for Borel $B\subset S$. By [F7] it is a Borel probability, and its first $N+1$ coordinates have law $Q_N$ while every later coordinate equals zero.

The $j$th marginal of any ${\eta }_N$ is either ${\mu }_{n_j}$ (when $N+1\ge j$) or ${\delta }_0$. Each fixed probability ${\mu }_{n_j}$ is tight by [F2]; AC chooses a compact $K_j\subset {\mathbb{R}}^d$ containing zero with ${\mu }_{n_j}(K_j^c)<\epsilon 2^{-j-1}$. Countable subadditivity gives ${\eta }_N(({\prod }_j{K}_j{)}^c)\le {\sum }_{j\ge 1}\epsilon 2^{-j-1}<\epsilon$ for every $N$. The product $K={\prod }_j{K}_j$ is compact in $S$: it is closed and complete, and it is totally bounded because for any radius one first makes the metric tail small and then takes finite nets in the finitely many compact coordinates; [F5] applies. Hence $({\eta }_N)$ is tight. By [F2] and Prokhorov, a subsequence converges weakly to a probability $\eta$ on $S$.

For fixed $k$, all sufficiently late ${\eta }_N$ have their $(k,k+1)$ coordinate marginal equal to ${\pi }_k$. The coordinate projection $S\to {\mathbb{R}}^{2d}$ is continuous, so the projected laws of ${\eta }_N$ converge weakly to the corresponding marginal of $\eta$; because they are eventually exactly ${\pi }_k$, uniqueness of weak limits [F6] shows that this marginal is ${\pi }_k$. Let $X_k(x)=x_k$ be the coordinate maps on $(S,\eta )$. By [F4] and the pair marginal identity, ${\int }_S\mathrm{\mid }{X}_{k+1}-X_k{\mathrm{\mid }}^{2}d\eta =\int \mathrm{\mid }x-y{\mathrm{\mid }}^{2}d{\pi }_k=W_2({\mu }_{n_k},{\mu }_{n_{k+1}}{)}^2\le 4^{-k}\mathrm{.}$ Therefore $\int \mathrm{\mid }{X}_{k+1}-X_k\mathrm{\mid }d\eta \le 2^{-k}$ by Cauchy–Schwarz. Monotone convergence [F4] shows $\int {\sum }_k\mathrm{\mid }{X}_{k+1}-X_k\mathrm{\mid }d\eta \le {\sum }_k{2}^{-k}<\mathrm{\infty }$, so this sum is finite almost everywhere. The convergence set is Borel; its pointwise limit there, extended by zero on the complement, is a Borel map $X_{\mathrm{\infty }}:S\to {\mathbb{R}}^d$.

Minkowski [F3] gives, for $m>n$, $\mathrm{\parallel }{X}_m-X_n{\mathrm{\parallel }}_{L^2(\eta )}\le {\sum }_{k=n}^{m-1}\mathrm{\parallel }{X}_{k+1}-X_k{\mathrm{\parallel }}_2\le {\sum }_{k=n}^{m-1}{2}^{-k}$. Fatou's lemma [F3] as $m\to \mathrm{\infty }$ yields $\mathrm{\parallel }{X}_{\mathrm{\infty }}-X_n{\mathrm{\parallel }}_2\le {\sum }_{k=n}^{\mathrm{\infty }}{2}^{-k}$. Since $X_1$ has law ${\mu }_{n_1}\in {\mathcal{P}}_2$, the same inequality at $n=1$ and Minkowski imply $\int \mathrm{\mid }{X}_{\mathrm{\infty }}{\mathrm{\mid }}^{2}d\eta <\mathrm{\infty }$. Let ${\mu }_{\mathrm{\infty }}=(X_{\mathrm{\infty }}{)}_{\mathrm{\#}}\eta$, a Borel probability by [F7]. The map $x\mapsto (X_n(x),X_{\mathrm{\infty }}(x))$ is Borel, so its pushforward is a Borel probability by [F7]; its marginals are the laws of $X_n$ and $X_{\mathrm{\infty }}$, namely ${\mu }_{n_n}$ and ${\mu }_{\mathrm{\infty }}$. Thus ${\mu }_{\mathrm{\infty }}\in {\mathcal{P}}_2$, and this joint law is a coupling, so $W_2({\mu }_{n_n},{\mu }_{\mathrm{\infty }})\le \mathrm{\parallel }{X}_n-X_{\mathrm{\infty }}{\mathrm{\parallel }}_2\le {\sum }_{k=n}^{\mathrm{\infty }}{2}^{-k}⟶0.$ The original Cauchy sequence converges to the same limit: for any $ϵ>0$, choose n large enough that the displayed tail bound is below $ϵ\mathrm{/}2$ and $n_n$ exceeds the Cauchy threshold for $ϵ\mathrm{/}2$; then $W_2({\mu }_m,{\mu }_{n_n})<ϵ\mathrm{/}2$ for all sufficiently large m and apply the triangle inequality [F1].