Proof of The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist

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. Throughout, $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ are the coordinate projections and $\iota$ the concatenation map fixed there, $\Pi(\cdot,\cdot)$ denotes sets of couplings, $M_{2}$ and $\mathcal{P}_{2}(\mathbb{R}^{d})$ are the second moment and the measures with finite second moment, and $W_{2}$ is the quadratic Wasserstein distance. The results Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling and Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound are stated in the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation; by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions the sets $\mathcal{P}(\mathbb{R}^{m})$, $\mathcal{P}_{2}(\mathbb{R}^{m})$, $\Pi(\cdot,\cdot)$, $M_{2}$ and $W_{2}$ named there are exactly the objects just listed, read in dimension $m$, so these results apply to them. The dimension $d$ that The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data fixes, and in which Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling is stated, is an arbitrary natural number with $1\le d$, the results stated in that setting holding for each such choice; we take it to be the dimension $d$ of the torus.

Preliminaries.

(i) The cost function. Let $c_{\mathbb{T}}:\mathbb{R}^{d+d}\to\mathbb{R}$ be given by $c_{\mathbb{T}}(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))^{2}$. By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost it is Borel with values in $[0,d/4]$; it is therefore bounded, with bound $d/4$, and integrable with respect to every member of $\mathcal{P}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; and $I_{\mathbb{T}}(\gamma)=\int_{\mathbb{R}^{d+d}}c_{\mathbb{T}}\,d\gamma$ for all $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$, by the same clause.

(ii) A Lipschitz bound. Put $\Lambda=\sqrt{d}+\sqrt{d}$, a nonnegative real number by claim 2 of Elementary Arithmetic in an Ordered Field. For all $x,y,x',y'\in\mathbb{R}^{d}$,

$$\bigl|d_{\mathbb{T}}(x,y)^{2}-d_{\mathbb{T}}(x',y')^{2}\bigr|\le\sqrt{d}\,\lVert x-x'\rVert+\sqrt{d}\,\lVert y-y'\rVert. \qquad(\mathrm{E})$$

Indeed, by the triangle inequality for the absolute value (claim 5 of Properties of the Absolute Value in an Ordered Field) the left side is at most $|d_{\mathbb{T}}(x,y)^{2}-d_{\mathbb{T}}(x',y)^{2}|+|d_{\mathbb{T}}(x',y)^{2}-d_{\mathbb{T}}(x',y')^{2}|$. The first term is at most $\sqrt{d}\,\lVert x-x'\rVert$ by the second inequality of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry, $d_{\mathbb{T}}(x',y)=d_{\mathbb{T}}(y,x')$ and $d_{\mathbb{T}}(x',y')=d_{\mathbb{T}}(y',x')$, so the second term equals $|d_{\mathbb{T}}(y,x')^{2}-d_{\mathbb{T}}(y',x')^{2}|$, which that same inequality, applied to the points $y,y',x'$ in place of $x,x',y$, bounds by $\sqrt{d}\,\lVert y-y'\rVert$.

Now let $w,w'\in\mathbb{R}^{d+d}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $w=\iota(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$ and likewise for $w'$, so claim 2 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space gives $w-w'=\iota(\mathrm{pr}_{1}(w)-\mathrm{pr}_{1}(w'),\mathrm{pr}_{2}(w)-\mathrm{pr}_{2}(w'))$; the defining property of the projections in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections then gives $\mathrm{pr}_{k}(w-w')=\mathrm{pr}_{k}(w)-\mathrm{pr}_{k}(w')$, and the norm bound recorded there gives $\lVert\mathrm{pr}_{k}(w)-\mathrm{pr}_{k}(w')\rVert\le\lVert w-w'\rVert$, for $k=1,2$. Combining this with (E), using claim 5 of Elementary Arithmetic in an Ordered Field (as $0\le\sqrt{d}$) and distributivity,

$$|c_{\mathbb{T}}(w)-c_{\mathbb{T}}(w')|\le\sqrt{d}\,\lVert w-w'\rVert+\sqrt{d}\,\lVert w-w'\rVert=\Lambda\,\lVert w-w'\rVert .$$

Here $\lVert w-w'\rVert=d_{E}(w,w')$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $|s-t|$ is the metric of the real line of The Absolute Value Metric on the Real Line. Hence $c_{\mathbb{T}}$ is Lipschitz with constant $\Lambda$ from $(\mathbb{R}^{d+d},d_{E})$ to the real line, and therefore continuous on $\mathbb{R}^{d+d}$ by A Lipschitz Map is Uniformly Continuous.

(iii) The infimum. For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ let $c_{*}(\mu,\nu)$ be the greatest lower bound of $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$, a nonempty set of reals bounded below by $0$, as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance. By that clause $W_{\mathbb{T}}(\mu,\nu)$ is the nonnegative square root of the nonnegative real $c_{*}(\mu,\nu)$, so $0\le W_{\mathbb{T}}(\mu,\nu)$ and $W_{\mathbb{T}}(\mu,\nu)^{2}=c_{*}(\mu,\nu)$. As $c_{*}(\mu,\nu)$ is a lower bound, $W_{\mathbb{T}}(\mu,\nu)^{2}\le I_{\mathbb{T}}(\gamma)$ for every $\gamma\in\Pi(\mu,\nu)$, whence $W_{\mathbb{T}}(\mu,\nu)\le\sqrt{I_{\mathbb{T}}(\gamma)}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative. By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal, $\gamma\in\Pi(\mu,\nu)$ is optimal exactly when $I_{\mathbb{T}}(\gamma)=c_{*}(\mu,\nu)$.

(iv) Sets of full measure. Let $\rho$ be a probability measure on a measurable space $(Y,\mathcal{G})$ and let $E\in\mathcal{G}$ satisfy $\rho(E)=1$. Then $\rho(Y\setminus E)=1-1=0$ by claim 3 of Basic Properties of a Measure, and for every $A\in\mathcal{G}$ finite additivity (claim 1 there), applied to the disjoint sets $A\cap E$ and $A\setminus E$, together with $0\le\rho(A\setminus E)\le\rho(Y\setminus E)=0$ (claim 2 there), gives $\rho(A)=\rho(A\cap E)$. In particular, if $\mu\in\mathcal{P}(\mathbb{T}^{d})$, then $\mu(\mathbb{R}^{d}\setminus Q)=0$ and $\mu(A)=\mu(A\cap Q)$ for every $A\in\mathcal{B}(\mathbb{R}^{d})$, by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures; moreover $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $M_{2}(\mu)\le d$ by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §inclusion.

Claim 1. Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and write $c_{*}=c_{*}(\mu,\nu)$.

A minimising sequence. For $k\in\mathbb{N}$ let $t_{k}$ be the image of $k$ under the canonical map of $\mathbb{N}$ into $\mathbb{R}$; by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $t_{k}^{-1}$ exists and $0<t_{k}^{-1}$. Claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, applied to the set $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$ and to $t_{k}^{-1}$, provides a coupling with cost below $c_{*}+t_{k}^{-1}$; choose, for each $k\in\mathbb{N}$, such a $\gamma_{k}\in\Pi(\mu,\nu)$:

$$I_{\mathbb{T}}(\gamma_{k})<c_{*}+t_{k}^{-1}.$$

Extraction. By (iv), $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so by Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment §couplings, with $m=d$, the set $\Pi(\mu,\nu)$ is tight in $(\mathbb{R}^{d+d},d_{E})$. The set $\{\gamma_{k}:k\in\mathbb{N}\}$ is a subset of $\Pi(\mu,\nu)$, so for each tolerance the compact set witnessing the tightness of $\Pi(\mu,\nu)$ in Tight Family of Borel Measures on a Metric Space §tight witnesses it for this subset; hence the sequence $(\gamma_{k})_{k\in\mathbb{N}}$ is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence. Its terms belong to $\mathcal{P}(\mathbb{R}^{d+d})$, so Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence, with $m=d+d$, gives a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and $\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$ such that $(\gamma_{k_{j}})_{j\in\mathbb{N}}$ converges weakly to $\gamma$ on $(\mathbb{R}^{d+d},d_{E})$. By The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence, with $m=d$, applied to $(\gamma_{k_{j}})_{j\in\mathbb{N}}$, whose terms lie in $\Pi(\mu,\nu)$, we get $\gamma\in\Pi(\mu,\nu)$.

The cost of the limit. By (i) and (ii), $c_{\mathbb{T}}$ is bounded and continuous on $(\mathbb{R}^{d+d},d_{E})$, so by Weak Convergence of Finite Borel Measures on a Metric Space the real sequence $(\int c_{\mathbb{T}}\,d\gamma_{k_{j}})_{j\in\mathbb{N}}$, which is $(I_{\mathbb{T}}(\gamma_{k_{j}}))_{j\in\mathbb{N}}$ by (i), converges to $\int c_{\mathbb{T}}\,d\gamma=I_{\mathbb{T}}(\gamma)$. As $c_{*}$ is a lower bound, $c_{*}\le I_{\mathbb{T}}(\gamma)$. Suppose, for contradiction, that $c_{*}<I_{\mathbb{T}}(\gamma)$. Then $\eta=I_{\mathbb{T}}(\gamma)-c_{*}$ is positive by claim 1 of Elementary Order Arithmetic in an Ordered Field, and $\eta'=\eta\cdot2^{-1}$ is positive with $\eta'+\eta'=\eta$ by claim 8 there. We choose, in this order: first $N\in\mathbb{N}$ with $0<t_{N}^{-1}<\eta'$, by claim 3 of The Archimedean Property of the Real Numbers applied to $\eta'$; then $J\in\mathbb{N}$ with $|I_{\mathbb{T}}(\gamma_{k_{j}})-I_{\mathbb{T}}(\gamma)|<\eta'$ for every $j\ge J$, by the convergence just established; then $j\in\mathbb{N}$ with $N\le j$ and $J\le j$, namely the larger of $N$ and $J$, which exists by trichotomy (claim 3 of Properties of the Order on the Natural Numbers). By Strictly Increasing Sequences of Natural Numbers Dominate Their Index, $j\le k_{j}$, so $N\le k_{j}$ by claim 1 of Properties of the Order on the Natural Numbers. Hence $t_{N}\le t_{k_{j}}$: either $N=k_{j}$, or $N<k_{j}$ and then claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field gives $t_{N}<t_{k_{j}}$. Multiplying $t_{N}\le t_{k_{j}}$ by $t_{N}^{-1}t_{k_{j}}^{-1}$, which is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field, gives $t_{k_{j}}^{-1}\le t_{N}^{-1}$ by claim 5 of Elementary Arithmetic in an Ordered Field. By claim 9 of Properties of the Absolute Value in an Ordered Field, $-\eta'<I_{\mathbb{T}}(\gamma_{k_{j}})-I_{\mathbb{T}}(\gamma)$, that is, $I_{\mathbb{T}}(\gamma)<I_{\mathbb{T}}(\gamma_{k_{j}})+\eta'$ by claim 1 of Elementary Order Arithmetic in an Ordered Field. Combining these with the choice of $\gamma_{k_{j}}$ by claims 1, 2 and 3 of Elementary Order Arithmetic in an Ordered Field,

$$I_{\mathbb{T}}(\gamma)<I_{\mathbb{T}}(\gamma_{k_{j}})+\eta'<c_{*}+t_{k_{j}}^{-1}+\eta'<c_{*}+\eta'+\eta'=c_{*}+\eta=I_{\mathbb{T}}(\gamma),$$

which is impossible. The order being total, $I_{\mathbb{T}}(\gamma)\le c_{*}$, so $I_{\mathbb{T}}(\gamma)=c_{*}$ and $\gamma$ is an optimal coupling of $\mu$ and $\nu$ by (iii).

Claim 2. By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance, $W_{\mathbb{T}}$ is a map from $\mathcal{P}(\mathbb{T}^{d})\times\mathcal{P}(\mathbb{T}^{d})$ to $\mathbb{R}$. We verify the four conditions of Metric Space; let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$.

Nonnegativity. $0\le W_{\mathbb{T}}(\mu,\nu)$ by (iii).

Vanishing on the diagonal. Let $\mathrm{id}$ be the identity of $\mathbb{R}^{d}$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward, $\gamma_{0}=(\mathrm{id},\mathrm{id})_{\#}\mu\in\Pi(\mu,\mu)$. For $x\in\mathbb{R}^{d}$ the pairing takes the value $\iota(x,x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, whose two projections are $x$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; so $c_{\mathbb{T}}((\mathrm{id},\mathrm{id})(x))=d_{\mathbb{T}}(x,x)^{2}=0$, because $d_{\mathbb{T}}(x,x)=0$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry, $x-x$ being the zero vector, which lies in $\mathbb{Z}^{d}$. The change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives $I_{\mathbb{T}}(\gamma_{0})=\int c_{\mathbb{T}}\circ(\mathrm{id},\mathrm{id})\,d\mu=0$, the integral of the zero function being $0$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral (take the constant $0$ there). By (iii), $W_{\mathbb{T}}(\mu,\mu)^{2}\le0\le W_{\mathbb{T}}(\mu,\mu)^{2}$, so $W_{\mathbb{T}}(\mu,\mu)^{2}=0$ and $W_{\mathbb{T}}(\mu,\mu)=0$ by claim 3 of Zero Products and Elementary Identities in a Field.

Separation. Suppose $W_{\mathbb{T}}(\mu,\nu)=0$. By claim 1 there is an optimal $\gamma\in\Pi(\mu,\nu)$, so $\int c_{\mathbb{T}}\,d\gamma=I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2}=0$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, applied in the measure space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma)$ to the nonnegative Borel function $c_{\mathbb{T}}$, we have $c_{\mathbb{T}}(w)=0$ for $\gamma$-almost every $w$; by A Property Holding Almost Everywhere and Null Set of a Measure there is $N_{0}\in\mathcal{B}(\mathbb{R}^{d+d})$ with $\gamma(N_{0})=0$ and $c_{\mathbb{T}}(w)=0$ for every $w\notin N_{0}$. Put $N_{1}=\mathrm{pr}_{1}^{-1}(\mathbb{R}^{d}\setminus Q)$ and $N_{2}=\mathrm{pr}_{2}^{-1}(\mathbb{R}^{d}\setminus Q)$, Borel because the projections are Borel; by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling and (iv), $\gamma(N_{1})=\mu(\mathbb{R}^{d}\setminus Q)=0$ and $\gamma(N_{2})=\nu(\mathbb{R}^{d}\setminus Q)=0$. Countable subadditivity (claim 4 of Basic Properties of a Measure), applied to the sequence $N_{0},N_{1},N_{2},\emptyset,\emptyset,\dots$, gives $\gamma(N_{0}\cup N_{1}\cup N_{2})=0$, so $E=\mathbb{R}^{d+d}\setminus(N_{0}\cup N_{1}\cup N_{2})$ satisfies $\gamma(E)=1$ by claim 3 there. Let $w\in E$. Then $\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\in Q$ and $d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))^{2}=0$, so $d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))=0$ by claim 3 of Zero Products and Elementary Identities in a Field, and $\mathrm{pr}_{1}(w)=\mathrm{pr}_{2}(w)$ by condition 2 of Metric Space for the metric on $Q$ of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §metric. Consequently, for $B\in\mathcal{B}(\mathbb{R}^{d})$ the Borel sets $\mathrm{pr}_{1}^{-1}(B)\cap E$ and $\mathrm{pr}_{2}^{-1}(B)\cap E$ coincide, and (iv) applied to $\gamma$ and $E$ gives

$$\mu(B)=\gamma\bigl(\mathrm{pr}_{1}^{-1}(B)\bigr)=\gamma\bigl(\mathrm{pr}_{1}^{-1}(B)\cap E\bigr)=\gamma\bigl(\mathrm{pr}_{2}^{-1}(B)\cap E\bigr)=\gamma\bigl(\mathrm{pr}_{2}^{-1}(B)\bigr)=\nu(B),$$

the outer equalities by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling. Thus $\mu$ and $\nu$ take the same value on every Borel set, that is, $\mu=\nu$.

Symmetry. Let $\sigma$ be the swap of Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap, with $\sigma(w)=\iota(\mathrm{pr}_{2}(w),\mathrm{pr}_{1}(w))$ as in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing; by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathrm{pr}_{1}(\sigma(w))=\mathrm{pr}_{2}(w)$ and $\mathrm{pr}_{2}(\sigma(w))=\mathrm{pr}_{1}(w)$, so $c_{\mathbb{T}}(\sigma(w))=d_{\mathbb{T}}(\mathrm{pr}_{2}(w),\mathrm{pr}_{1}(w))^{2}=c_{\mathbb{T}}(w)$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry. Let $\gamma\in\Pi(\mu,\nu)$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap, $\sigma_{\#}\gamma\in\Pi(\nu,\mu)$, and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives $I_{\mathbb{T}}(\sigma_{\#}\gamma)=\int c_{\mathbb{T}}\circ\sigma\,d\gamma=I_{\mathbb{T}}(\gamma)$. So every member of $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$ belongs to $\{I_{\mathbb{T}}(\gamma'):\gamma'\in\Pi(\nu,\mu)\}$, and the same argument with $\mu$ and $\nu$ exchanged gives the reverse inclusion. The two sets are equal, hence have the same greatest lower bound, and $W_{\mathbb{T}}(\mu,\nu)=W_{\mathbb{T}}(\nu,\mu)$ by (iii) and the uniqueness of the nonnegative square root, Existence and Uniqueness of the Nonnegative Square Root.

Triangle inequality. Let $\mu_{1},\mu_{2},\mu_{3}\in\mathcal{P}(\mathbb{T}^{d})$. By claim 1 there are optimal couplings $\gamma_{12}\in\Pi(\mu_{1},\mu_{2})$ and $\gamma_{23}\in\Pi(\mu_{2},\mu_{3})$, and $\mu_{1},\mu_{2},\mu_{3}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by (iv). By Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued, with $\mu_{1},\mu_{2},\mu_{3}$ in place of $\mu,\rho,\nu$, there is $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$ with $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\gamma_{12}$ and $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\gamma_{23}$, where $\mathrm{q}_{1},\mathrm{q}_{2},\mathrm{q}_{3}$ are the coordinate maps named there; and by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite, $\gamma_{13}=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma\in\Pi(\mu_{1},\mu_{3})$. The triple $(\mathbb{R}^{3d},\mathcal{B}(\mathbb{R}^{3d}),\sigma)$ is a probability space. For $(a,b)\in\{(1,2),(2,3),(1,3)\}$ let $F_{ab}:\mathbb{R}^{3d}\to\mathbb{R}$ be $F_{ab}(s)=d_{\mathbb{T}}(\mathrm{q}_{a}(s),\mathrm{q}_{b}(s))$. The pairing $(\mathrm{q}_{a},\mathrm{q}_{b})$ is Borel by the preamble of Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling and takes the value $\iota(\mathrm{q}_{a}(s),\mathrm{q}_{b}(s))$ at $s$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, so by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections

$$F_{ab}=D\circ(\mathrm{q}_{a},\mathrm{q}_{b}),\qquad F_{ab}^{2}=c_{\mathbb{T}}\circ(\mathrm{q}_{a},\mathrm{q}_{b}),$$

where $D(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$ is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz. Hence $F_{ab}$ is Borel, as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), that is, a random variable on this probability space, and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward together with (i) gives

$$\int F_{12}^{2}\,d\sigma=I_{\mathbb{T}}(\gamma_{12})=W_{\mathbb{T}}(\mu_{1},\mu_{2})^{2},\qquad\int F_{23}^{2}\,d\sigma=I_{\mathbb{T}}(\gamma_{23})=W_{\mathbb{T}}(\mu_{2},\mu_{3})^{2},\qquad\int F_{13}^{2}\,d\sigma=I_{\mathbb{T}}(\gamma_{13}),$$

the first two by optimality. All three are finite, so $F_{12},F_{23},F_{13}$ are square-integrable, and so is $F_{12}+F_{23}$ by that definition. Their mean-square norms are the nonnegative square roots of these integrals, so $\lVert F_{12}\rVert_{2}=W_{\mathbb{T}}(\mu_{1},\mu_{2})$ and $\lVert F_{23}\rVert_{2}=W_{\mathbb{T}}(\mu_{2},\mu_{3})$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $W_{\mathbb{T}}(\mu_{1},\mu_{3})\le\lVert F_{13}\rVert_{2}$ by (iii). For every $s\in\mathbb{R}^{3d}$, The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §triangle gives $F_{13}(s)\le F_{12}(s)+F_{23}(s)$, both sides being nonnegative (claim 2 of Elementary Arithmetic in an Ordered Field), so $F_{13}(s)^{2}\le(F_{12}(s)+F_{23}(s))^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Monotonicity of the integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) gives $\int F_{13}^{2}\,d\sigma\le\int(F_{12}+F_{23})^{2}\,d\sigma$, hence $\lVert F_{13}\rVert_{2}\le\lVert F_{12}+F_{23}\rVert_{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and the triangle inequality, claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, gives $\lVert F_{12}+F_{23}\rVert_{2}\le\lVert F_{12}\rVert_{2}+\lVert F_{23}\rVert_{2}$. Chaining,

$$W_{\mathbb{T}}(\mu_{1},\mu_{3})\le\lVert F_{13}\rVert_{2}\le\lVert F_{12}+F_{23}\rVert_{2}\le W_{\mathbb{T}}(\mu_{1},\mu_{2})+W_{\mathbb{T}}(\mu_{2},\mu_{3}).$$

The four conditions hold, so $W_{\mathbb{T}}$ is a metric on $\mathcal{P}(\mathbb{T}^{d})$.

Claim 3. Let $(\mu_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{P}(\mathbb{T}^{d})$.

A bound on the cell. Let $D_{d}=\sum_{i=1}^{d}1$ and $K=D_{d}+1$. Summands $1$ being nonnegative (claim 1 of Elementary Arithmetic in an Ordered Field), $0\le D_{d}$ by claim 5 of Properties of Finite Sums, so $1\le K$ by claim 3 of Elementary Arithmetic in an Ordered Field and $0<K$ by claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field. Let $x\in Q$. By the definition of $Q$ in The Half-Open Unit Cell Tiles Euclidean Space, $0\le x_{i}<1$ for every $i\in[d]$, so $x_{i}^{2}=x_{i}x_{i}\le x_{i}\cdot1\le1$ by claim 5 of Elementary Arithmetic in an Ordered Field; thus $0\le1-x_{i}^{2}$ (claim 3 there), and claims 2, 3 and 5 of Properties of Finite Sums give $\sum_{i=1}^{d}x_{i}^{2}\le D_{d}$, that is, $\lVert x\rVert^{2}\le D_{d}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Since $D_{d}\le K$ (claim 3 of Elementary Arithmetic in an Ordered Field, as $K-D_{d}=1\ge0$) and $K=K\cdot1\le K\cdot K$ (claim 5 there, as $0\le K$ and $1\le K$), we get $\lVert x\rVert^{2}\le K^{2}$, hence $\lVert x\rVert\le K$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Extraction. By (iv), every $\mu_{j}$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $M_{2}(\mu_{j})\le d$, so the set $\{\mu_{j}:j\in\mathbb{N}\}$ is tight in $(\mathbb{R}^{d},d_{E})$ by Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment §moment, with $m=d$ and $R=d$; that is, the sequence is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence. By Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence, with $m=d$, there are a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and $\bar\mu\in\mathcal{P}(\mathbb{R}^{d})$ such that $(\mu_{n_{k}})_{k\in\mathbb{N}}$ converges weakly to $\bar\mu$ on $(\mathbb{R}^{d},d_{E})$.

Wasserstein convergence of the subsequence. We apply Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence, with $m=d$, to the sequence $(\mu_{n_{k}})_{k\in\mathbb{N}}$, whose terms lie in $\mathcal{P}_{2}(\mathbb{R}^{d})$, and to $\bar\mu$. Let $\varepsilon$ be a positive real; we take the positive number $K$ above, which does not depend on $\varepsilon$. The set $A_{K}=\{x\in\mathbb{R}^{d}:K<\lVert x\rVert\}$ is Borel, since $x\mapsto\lVert x\rVert$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable), and the integral in the hypothesis of that clause is the integral against $\mu_{n_{k}}$ of the nonnegative Borel function $f=\mathbf{1}_{A_{K}}\lVert\cdot\rVert^{2}$. By the bound on the cell, $x\notin A_{K}$ for $x\in Q$, so $f$ vanishes at every point outside $\mathbb{R}^{d}\setminus Q$, a set of $\mu_{n_{k}}$-measure $0$ by (iv). By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral, $\int f\,d\mu_{n_{k}}=0<\varepsilon$ for every $k\in\mathbb{N}$. The clause therefore gives $\bar\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and that the real sequence $(W_{2}(\mu_{n_{k}},\bar\mu))_{k\in\mathbb{N}}$ has limit $0$.

The limit on the torus. Put $\mu=\pi_{\#}\bar\mu$, which belongs to $\mathcal{P}(\mathbb{T}^{d})$ by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §wrap. For every $j$ we have $\pi_{\#}\mu_{j}=\mu_{j}$: the map $\pi$ is Borel by The Half-Open Unit Cell Tiles Euclidean Space §wrap, and for $B\in\mathcal{B}(\mathbb{R}^{d})$ one has $\pi^{-1}(B)\cap Q=B\cap Q$ because $\pi(x)=x$ for $x\in Q$ by the same clause, so (iv) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward give

$$\pi_{\#}\mu_{j}(B)=\mu_{j}\bigl(\pi^{-1}(B)\bigr)=\mu_{j}\bigl(\pi^{-1}(B)\cap Q\bigr)=\mu_{j}(B\cap Q)=\mu_{j}(B).$$

Since $\mu_{n_{k}}$ and $\bar\mu$ belong to $\mathcal{P}_{2}(\mathbb{R}^{d})$, the final assertion of The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §wrap gives, for every $k\in\mathbb{N}$,

$$W_{\mathbb{T}}(\mu_{n_{k}},\mu)=W_{\mathbb{T}}(\pi_{\#}\mu_{n_{k}},\pi_{\#}\bar\mu)\le W_{2}(\mu_{n_{k}},\bar\mu).$$

Let $\varepsilon$ be a positive real. By Limit of a Sequence of Real Numbers there is $L\in\mathbb{N}$ with $|W_{2}(\mu_{n_{k}},\bar\mu)-0|<\varepsilon$ for every $k\ge L$; for such $k$, claim 3 of Properties of the Absolute Value in an Ordered Field and claim 2 of Elementary Order Arithmetic in an Ordered Field give $W_{\mathbb{T}}(\mu_{n_{k}},\mu)\le W_{2}(\mu_{n_{k}},\bar\mu)\le|W_{2}(\mu_{n_{k}},\bar\mu)|<\varepsilon$. By claim 2, $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ is a metric space, and we have shown that the subsequence $(\mu_{n_{k}})_{k\in\mathbb{N}}$ of $(\mu_{j})_{j\in\mathbb{N}}$ converges to $\mu$ in it. This proves claim 3.