Proof of McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential

Each result cited below is universally quantified over the data in its own statement. Write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$, which is Borel, and $F:\mathbb{R}^{d+d}\to\mathbb{R}$ for the nonnegative Borel function $F(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))^{2}$ of Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost, so that $I_{\mathbb{T}}(\gamma)=\int_{\mathbb{R}^{d+d}}F\,d\gamma$ for every coupling $\gamma$ of two members of $\mathcal{P}(\mathbb{T}^{d})$. Since $0\in\mathbb{Z}$, $1\in\mathbb{Z}$ and $\mathbb{Z}$ is closed under negation and addition (claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers), the lattice $\mathbb{Z}^{d}$ of Lattice-Periodic Functions and the Periodic Function Classes §lattice is closed under negation and addition, points of $\mathbb{R}^{d}$ being added coordinatewise (Sum of Points of $\mathbb{R}^n$, Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$). Note that $\mathcal{P}(\mathbb{T}^{d})\subseteq\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures, so results stated for members of $\mathcal{P}(\mathbb{R}^{d})$ apply to $\mu$ and $\nu$.

Step 0. Three general facts.

(a) Supports. Let $\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$; by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures it is a Borel measure on $(\mathbb{R}^{d+d},d_{E})$, and let $\Gamma=\operatorname{supp}\gamma$ be its support. By The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space §closed, $\Gamma$ is closed and belongs to $\mathcal{B}(\mathbb{R}^{d+d})$. The metric space $(\mathbb{R}^{d+d},d_{E})$ is separable by Euclidean Space is a Separable Metric Space §separable, so The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space §full gives $\gamma(\mathbb{R}^{d+d}\setminus\Gamma)=0$, whence $\gamma(\Gamma)=1$ by claim 3 of Basic Properties of a Measure. In particular $\Gamma\ne\emptyset$, because a measure vanishes on the empty set (Measure, Measure Space, and Probability Measure).

(b) Graphs. For a Borel $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ let $\Gamma_{S}\in\mathcal{B}(\mathbb{R}^{d+d})$ be its graph, as in A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map. For Borel $T,S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the point $(\mathrm{id},T)(x)=\iota(x,T(x))$, with $\iota$ the concatenation map of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, has projections $x$ and $T(x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, so it lies in $\Gamma_{S}$ exactly when $T(x)=S(x)$. Hence $(\mathrm{id},T)^{-1}(\Gamma_{S})=\{x\in\mathbb{R}^{d}:T(x)=S(x)\}$, and in particular $(\mathrm{id},S)^{-1}(\Gamma_{S})=\mathbb{R}^{d}$, so that $(\mathrm{id},S)_{\#}\rho(\Gamma_{S})=\rho(\mathbb{R}^{d})=1$ for every $\rho\in\mathcal{P}(\mathbb{R}^{d})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward).

(c) Intersections of sets of full measure. Let $\rho$ be a probability measure on a measurable space $(E,\mathcal{E})$ and let $B_{1},B_{2},B_{3}\in\mathcal{E}$ satisfy $\rho(B_{j})=1$. Then $\rho(E\setminus B_{j})=0$ by claim 3 of Basic Properties of a Measure; the complement of $B_{1}\cap B_{2}\cap B_{3}$ is the union of the three complements, and claim 4 there, applied to the sequence consisting of these three complements followed by empty sets, all of measure $0$, shows that this union has measure $0$. By claim 3 there, $\rho(B_{1}\cap B_{2}\cap B_{3})=1$. Taking $B_{3}=E$ covers the case of two sets.

Step 1. The potential of an optimal coupling and its gradient. Let $\gamma\in\Pi(\mu,\nu)$ be optimal and let $\Gamma=\operatorname{supp}\gamma$. By Step 0(a), $\Gamma$ is nonempty, closed, Borel and $\gamma(\Gamma)=1$; by The Support of an Optimal Coupling on the Torus is Torus-Cyclically Monotone §monotone it is torus-cyclically monotone. Hence A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients provides functions $\varphi$ and $\psi(x)=\tfrac12\lVert x\rVert^{2}-\varphi(x)$ satisfying claims 1 to 4 there for this set $\Gamma$; in particular $\psi$ is convex on $\mathbb{R}^{d}$ by A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients §convex.

The set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and convex, since every point $t\,x+(1-t)\,y$ lies in it. Every $x\in\mathbb{R}^{d}$ is therefore an interior point of $\mathbb{R}^{d}$, the open set $\mathbb{R}^{d}$ itself containing $x$, and A Convex Function is Lipschitz on a Ball around an Interior Point, applied to the convex set $\mathbb{R}^{d}$ and the convex function $\psi$, gives a positive real $r$ and a nonnegative real $L'$ with $\bar{B}(x,r)\subseteq\mathbb{R}^{d}$ and $|\psi(y)-\psi(y')|\le L'\lVert y-y'\rVert$ for $y,y'\in\bar{B}(x,r)$. Thus $\psi$ is locally Lipschitz on $\mathbb{R}^{d}$, and Rademacher's Theorem in $\mathbb{R}^n$ §locally-lipschitz, read with $m=1$ and $U=\mathbb{R}^{d}$, shows that the set of points at which $\psi$ is not differentiable is null; choose $N_{0}\in\mathcal{B}(\mathbb{R}^{d})$ with $\lambda_{d}(N_{0})=0$ containing it, as that notion of nullity provides, and put $A=\mathbb{R}^{d}\setminus N_{0}\in\mathcal{B}(\mathbb{R}^{d})$. Then $\psi$ is differentiable at every point of $A$; $\mu(N_{0})=0$ because $\mu$ is absolutely continuous; and $\mu(A)=1$ by claim 3 of Basic Properties of a Measure.

For $x\in A$, Elementary Calculus of the Subdifferential of a Convex Function §gradient, applied to the open convex set $U=\mathbb{R}^{d}$ and the convex function $\psi$, gives a point $G(x)\in\mathbb{R}^{d}$, read off from the derivative matrix of $\psi$ at $x$ as described there, with

$$\partial_{\mathbb{R}^{d}}\psi(x)=\{G(x)\}.$$

The map $G:A\to\mathbb{R}^{d}$ is continuous on $A$: given $x\in A$ and a positive real $\varepsilon$, Elementary Calculus of the Subdifferential of a Convex Function §continuity supplies a positive $\delta$ such that every $x'\in\mathbb{R}^{d}$ with $\lVert x'-x\rVert<\delta$ and every $q\in\partial_{\mathbb{R}^{d}}\psi(x')$ satisfy $\lVert q-G(x)\rVert<\varepsilon$; taking $x'\in A$ and $q=G(x')$ gives the requirement of Continuous Map Between Metric Spaces. By The Metric Subspace: Continuity, the Borel Sigma-Algebra, and the Restriction of a Borel Measure §continuity-map, $G$ is continuous from the metric space $(A,d_{E})$ to $(\mathbb{R}^{d},d_{E})$, hence measurable with respect to $\mathcal{B}(A)$ and $\mathcal{B}(\mathbb{R}^{d})$ by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; and $\mathcal{B}(A)=\{B\in\mathcal{B}(\mathbb{R}^{d}):B\subseteq A\}$ by The Metric Subspace: Continuity, the Borel Sigma-Algebra, and the Restriction of a Borel Measure §borel-subset, $A$ being Borel. Extend $G$ to $\mathbb{R}^{d}$ by $G(x)=0_{\mathbb{R}^{d}}$ for $x\in N_{0}$. For $B\in\mathcal{B}(\mathbb{R}^{d})$ the preimage of $B$ under the extended map is $(G|_{A})^{-1}(B)$ if $0_{\mathbb{R}^{d}}\notin B$ and $(G|_{A})^{-1}(B)\cup N_{0}$ otherwise, in both cases a member of $\mathcal{B}(\mathbb{R}^{d})$; so $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel.

Step 2. Every optimal coupling is induced by a $Q$-valued map. Keep the data of Step 1 for an optimal $\gamma\in\Pi(\mu,\nu)$ and let $\pi$ be the wrapping map. Put $T_{0}=\pi\circ G$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $\pi$ is measurable and takes values in $Q$, so $T_{0}$ is Borel as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps) and takes values in $Q$.

Let $\Gamma'=\Gamma\cap\mathrm{pr}_{1}^{-1}(A)\cap\mathrm{pr}_{2}^{-1}(Q)$; the projections are Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections) and $Q\in\mathcal{B}(\mathbb{R}^{d})$ by The Half-Open Unit Cell Tiles Euclidean Space §cell, so $\Gamma'$ is Borel. Since $\gamma$ is a coupling of $\mu$ and $\nu$, $\gamma(\mathrm{pr}_{1}^{-1}(A))=\mu(A)=1$ and $\gamma(\mathrm{pr}_{2}^{-1}(Q))=\nu(Q)=1$, the latter by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures; with $\gamma(\Gamma)=1$, Step 0(c) gives $\gamma(\Gamma')=1$.

Let $w\in\Gamma'$, $x=\mathrm{pr}_{1}(w)\in A$ and $y=\mathrm{pr}_{2}(w)\in Q$. The point $k_{0}=y-x-\varpi(y-x)$ lies in $\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, so $\tilde y=x+\varpi(y-x)=y+(-k_{0})$ lies in $y+\mathbb{Z}^{d}$, and $\lVert\tilde y-x\rVert=\lVert\varpi(y-x)\rVert=d_{\mathbb{T}}(x,y)$ by The Wrapped Displacement and the Flat Torus Distance §distance. As $w\in\Gamma$, A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients §subgradient gives $\tilde y\in\partial_{\mathbb{R}^{d}}\psi(x)=\{G(x)\}$, so $G(x)=y+(-k_{0})$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $\pi(y+(-k_{0}))=\pi(y)$, and $\pi(y)=y$ because $y\in Q$. Hence $T_{0}(\mathrm{pr}_{1}(w))=\mathrm{pr}_{2}(w)$, that is $w\in\Gamma_{T_{0}}$. So $\Gamma'\subseteq\Gamma_{T_{0}}$, and claim 2 of Basic Properties of a Measure gives $\gamma(\Gamma_{T_{0}})=1$. By A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map §graph,

$$\gamma=(\mathrm{id},T_{0})_{\#}\mu,\qquad(T_{0})_{\#}\mu=\nu .$$

Step 3. There is exactly one optimal coupling. By The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §optimal there is an optimal coupling $\gamma^{*}$ of $\mu$ and $\nu$. Let $\gamma_{1},\gamma_{2}\in\Pi(\mu,\nu)$ be optimal. By Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination, applied to $E=\mathbb{R}^{d+d}$, $\mathcal{E}=\mathcal{B}(\mathbb{R}^{d+d})$, the finite measures $\alpha=\gamma_{1}$, $\beta=\gamma_{2}$ and $s=t=\tfrac12$, the set function $\hat\gamma(B)=\tfrac12\gamma_{1}(B)+\tfrac12\gamma_{2}(B)$ is a measure on $\mathcal{B}(\mathbb{R}^{d+d})$, with

$$\int_{\mathbb{R}^{d+d}}F\,d\hat\gamma=\tfrac12\int_{\mathbb{R}^{d+d}}F\,d\gamma_{1}+\tfrac12\int_{\mathbb{R}^{d+d}}F\,d\gamma_{2}$$

for the nonnegative Borel function $F$. Now $\hat\gamma(\mathbb{R}^{d+d})=\tfrac12+\tfrac12=1$, so $\hat\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$, and for $B\in\mathcal{B}(\mathbb{R}^{d})$ one has $\hat\gamma(\mathrm{pr}_{1}^{-1}(B))=\tfrac12\mu(B)+\tfrac12\mu(B)=\mu(B)$ and likewise $\hat\gamma(\mathrm{pr}_{2}^{-1}(B))=\nu(B)$, so $\hat\gamma\in\Pi(\mu,\nu)$. The displayed identity reads $I_{\mathbb{T}}(\hat\gamma)=\tfrac12W_{\mathbb{T}}(\mu,\nu)^{2}+\tfrac12W_{\mathbb{T}}(\mu,\nu)^{2}=W_{\mathbb{T}}(\mu,\nu)^{2}$, so $\hat\gamma$ is optimal. By Step 2 there is a Borel $R:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\hat\gamma=(\mathrm{id},R)_{\#}\mu$, and Step 0(b) gives $\hat\gamma(\Gamma_{R})=1$, hence $\hat\gamma(\mathbb{R}^{d+d}\setminus\Gamma_{R})=0$ by claim 3 of Basic Properties of a Measure. Since $\gamma_{j}(B)\le2\hat\gamma(B)$ for every Borel $B$ and $j\in\{1,2\}$, directly from the formula for $\hat\gamma$, we get $\gamma_{j}(\mathbb{R}^{d+d}\setminus\Gamma_{R})=0$, so $\gamma_{j}(\Gamma_{R})=1$, and A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map §graph gives $\gamma_{j}=(\mathrm{id},R)_{\#}\mu$ for $j=1,2$. Hence $\gamma_{1}=\gamma_{2}$: every optimal coupling of $\mu$ and $\nu$ equals $\gamma^{*}$.

Step 4. Claim 1. Step 2 applied to $\gamma^{*}$ gives a Borel map $T_{0}$ with values in $Q$, $(T_{0})_{\#}\mu=\nu$ and $(\mathrm{id},T_{0})_{\#}\mu=\gamma^{*}$ optimal; so $T_{0}$ is an optimal map on the torus from $\mu$ to $\nu$ with values in $Q$. By Step 3 every optimal coupling of $\mu$ and $\nu$ equals $\gamma^{*}=(\mathrm{id},T_{0})_{\#}\mu$, so $(\mu,\nu)$ is uniquely mapped on the torus. Let $T,T'$ be optimal maps on the torus from $\mu$ to $\nu$. Then $(\mathrm{id},T)_{\#}\mu$ and $(\mathrm{id},T')_{\#}\mu$ are optimal couplings of $\mu$ and $\nu$ (Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map), so both equal $\gamma^{*}$ by Step 3. By Step 0(b), $(\mathrm{id},T')_{\#}\mu(\Gamma_{T'})=1$, while

$$(\mathrm{id},T)_{\#}\mu(\Gamma_{T'})=\mu\bigl((\mathrm{id},T)^{-1}(\Gamma_{T'})\bigr)=\mu\bigl(\{x\in\mathbb{R}^{d}:T(x)=T'(x)\}\bigr).$$

Hence the set $\{x:T(x)=T'(x)\}$ has $\mu$-measure $1$, and its complement $\{x:T(x)\ne T'(x)\}$ has $\mu$-measure $0$ by claim 3 of Basic Properties of a Measure.

Step 5. Claim 2. Let $T$ be an optimal map on the torus from $\mu$ to $\nu$. Then $\gamma=(\mathrm{id},T)_{\#}\mu$ belongs to $\Pi(\mu,\nu)$ (preamble of Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map) and is optimal. Apply Step 1 to this $\gamma$, obtaining $\Gamma=\operatorname{supp}\gamma$, $\varphi$, $\psi$, $A$ and $G$. By A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients §periodic, $\varphi$ is $\mathbb{Z}^{d}$-periodic and Lipschitz with constant $\sqrt{d}/2$, and by A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients §convex, $\psi$ is convex on $\mathbb{R}^{d}$. Put

$$D=A\cap(\mathrm{id},T)^{-1}(\Gamma).$$

The pairing $(\mathrm{id},T)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing and $\Gamma$ is Borel, so $D\in\mathcal{B}(\mathbb{R}^{d})$. Moreover $\mu((\mathrm{id},T)^{-1}(\Gamma))=\gamma(\Gamma)=1$ and $\mu(A)=1$, so $\mu(D)=1$ by Step 0(c).

Let $x\in D$. Then $\psi$ is differentiable at $x$ and $\partial_{\mathbb{R}^{d}}\psi(x)=\{G(x)\}$, as $x\in A$. The point $w=(\mathrm{id},T)(x)$ lies in $\Gamma$ and has $\mathrm{pr}_{1}(w)=x$, $\mathrm{pr}_{2}(w)=T(x)$ (Step 0(b)). By A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients §subgradient, every $\tilde y\in T(x)+\mathbb{Z}^{d}$ with $\lVert\tilde y-x\rVert=d_{\mathbb{T}}(x,T(x))$ belongs to $\{G(x)\}$, that is, equals $G(x)$. Put $z=T(x)-x$ and $k=z-\varpi(z)$, which lies in $\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range. The point $\tilde y_{1}=x+\varpi(z)=T(x)+(-k)$ lies in $T(x)+\mathbb{Z}^{d}$ and $\lVert\tilde y_{1}-x\rVert=\lVert\varpi(z)\rVert=d_{\mathbb{T}}(x,T(x))$ by The Wrapped Displacement and the Flat Torus Distance §distance; hence $G(x)=\tilde y_{1}$.

$z$ is regular. Suppose not. Then, by the definition of regular points, there is $i\in[d]$ with $z_{i}-\tfrac12\in\mathbb{Z}$, and then $z_{i}+\tfrac12=(z_{i}-\tfrac12)+1\in\mathbb{Z}$ by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers. The integer $n=z_{i}+\tfrac12$ satisfies $n\le z_{i}+\tfrac12<n+1$, so by the uniqueness in Existence and Uniqueness of the Integer Part of a Real Number it is $\lfloor z_{i}+\tfrac12\rfloor$, and The Wrapped Displacement and the Flat Torus Distance §displacement gives $\varpi(z)_{i}=z_{i}-(z_{i}+\tfrac12)=-\tfrac12$. Let $e\in\mathbb{R}^{d}$ be the point with $e_{i}=1$ and $e_{j}=0$ for $j\ne i$ (Euclidean Points as Tuples of Real Numbers); it lies in $\mathbb{Z}^{d}$. Put $\tilde y_{2}=\tilde y_{1}+e=T(x)+(e-k)$, a point of $T(x)+\mathbb{Z}^{d}$. The coordinates of $\tilde y_{2}-x=\varpi(z)+e$ are $\varpi(z)_{j}$ for $j\ne i$ and $-\tfrac12+1=\tfrac12$ for $j=i$; since $(\tfrac12)^{2}=(-\tfrac12)^{2}$, the squares of the coordinates of $\varpi(z)+e$ and of $\varpi(z)$ agree index by index, so the two sums of squares coincide. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ the norm of a point is the unique nonnegative real whose square is the sum of the squares of its coordinates; hence $\lVert\tilde y_{2}-x\rVert=\lVert\varpi(z)\rVert=d_{\mathbb{T}}(x,T(x))$. By the previous paragraph $\tilde y_{2}=G(x)=\tilde y_{1}$, which is impossible because the $i$th coordinates of $\tilde y_{2}$ and $\tilde y_{1}$ differ by $1\ne0$. Therefore $z=T(x)-x$ is regular.

Finally, with the displacement field $v_{T}(x)=\varpi(T(x)-x)=\varpi(z)$,

$$\partial_{\mathbb{R}^{d}}\psi(x)=\{G(x)\}=\{\tilde y_{1}\}=\{x+v_{T}(x)\}.$$

So $\varphi$, $\psi$ and $D$ have all the properties required in claim 2.

Step 6. Claim 3. Assume that $\nu$ is also absolutely continuous, and let $T$ and $S$ be as in claim 3. Let $\gamma=(\mathrm{id},T)_{\#}\mu$, an optimal coupling of $\mu$ and $\nu$, and let $\sigma:\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$ be the swap of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, $\sigma(w)=\iota(\mathrm{pr}_{2}(w),\mathrm{pr}_{1}(w))$, which is Borel. 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 $F\circ\sigma=F$ by the symmetry $d_{\mathbb{T}}(x,y)=d_{\mathbb{T}}(y,x)$ of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry.

The swapped coupling is optimal. 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, $\eta\mapsto\sigma_{\#}\eta$ is a bijection of $\Pi(\mu,\nu)$ onto $\Pi(\nu,\mu)$. For $\eta\in\Pi(\mu,\nu)$ the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to the nonnegative Borel function $F$, gives

$$I_{\mathbb{T}}(\sigma_{\#}\eta)=\int_{\mathbb{R}^{d+d}}F\,d(\sigma_{\#}\eta)=\int_{\mathbb{R}^{d+d}}F\circ\sigma\,d\eta=I_{\mathbb{T}}(\eta).$$

Hence the sets $\{I_{\mathbb{T}}(\eta'):\eta'\in\Pi(\nu,\mu)\}$ and $\{I_{\mathbb{T}}(\eta):\eta\in\Pi(\mu,\nu)\}$ coincide, so they have the same greatest lower bound and $W_{\mathbb{T}}(\nu,\mu)^{2}=W_{\mathbb{T}}(\mu,\nu)^{2}$ (Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance). Therefore $I_{\mathbb{T}}(\sigma_{\#}\gamma)=I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\nu,\mu)^{2}$, and $\sigma_{\#}\gamma$ is an optimal coupling of $\nu$ and $\mu$.

Inverse relation. Since $\nu$ is absolutely continuous, claim 1, already proved and applied to the pair $(\nu,\mu)$, shows that $(\nu,\mu)$ is uniquely mapped on the torus: there is an optimal map $R$ from $\nu$ to $\mu$ such that every optimal coupling of $\nu$ and $\mu$ equals $(\mathrm{id},R)_{\#}\nu$. Both $\sigma_{\#}\gamma$ and $(\mathrm{id},S)_{\#}\nu$ are optimal couplings of $\nu$ and $\mu$, the latter because $S$ is an optimal map (Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map); hence $\sigma_{\#}\gamma=(\mathrm{id},R)_{\#}\nu=(\mathrm{id},S)_{\#}\nu$, and Step 0(b) gives $\sigma_{\#}\gamma(\Gamma_{S})=1$. On the other hand, by the definition of push-forwards,

$$\sigma_{\#}\gamma(\Gamma_{S})=\gamma\bigl(\sigma^{-1}(\Gamma_{S})\bigr)=\mu\bigl((\sigma\circ(\mathrm{id},T))^{-1}(\Gamma_{S})\bigr),$$

and $\sigma((\mathrm{id},T)(x))=\iota(T(x),x)$ has projections $T(x)$ and $x$, so it lies in $\Gamma_{S}$ exactly when $x=S(T(x))$. Hence $E=\{x\in\mathbb{R}^{d}:S(T(x))=x\}=(\sigma\circ(\mathrm{id},T))^{-1}(\Gamma_{S})$ belongs to $\mathcal{B}(\mathbb{R}^{d})$, as the preimage of a Borel set under a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and $\mu(E)=1$.

Let $D$ be the set constructed in Step 5 for the optimal map $T$, and put $D'=D\cap E\in\mathcal{B}(\mathbb{R}^{d})$; then $\mu(D')=1$ by Step 0(c). Let $x\in D'$. Then $S(T(x))=x$, and $z=T(x)-x$ is regular by Step 5. Hence, using The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular for the regular point $z$,

$$v_{S}(T(x))=\varpi\bigl(S(T(x))-T(x)\bigr)=\varpi(x-T(x))=\varpi(-z)=-\varpi(z)=-v_{T}(x).$$

This proves claim 3.