Proof of The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions

The setting adopted by the statement is in force: that of The Flat Torus: Standing Notation, used with $n=d$, and that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, with $\mathcal{P}(\mathbb{T}^{d})$, $I_{\mathbb{T}}$ and $W_{\mathbb{T}}$ as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures, Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost and Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance.

Preliminaries. Let $G:\mathbb{R}^{d+d}\to\mathbb{R}$, $G(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$. It is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, so $G^{2}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. It is nonnegative, being a norm, and $G(w)^{2}\le d/4$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range; since $0\le(G(w)-1)^{2}=G(w)^{2}-2G(w)+1$ we get $G(w)\le2G(w)\le G(w)^{2}+1\le d/4+1$, so $G$ and $G^{2}$ are bounded Borel functions, 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. By that clause a nonnegative Borel real function is integrated as a $[0,\infty]$-valued map, so $I_{\mathbb{T}}(\gamma)=\int G^{2}\,d\gamma$ may be compared with $[0,\infty]$-valued integrals by claim 1 of Linearity and Monotonicity of the Lebesgue Integral. For every $\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$ and every real $c\ge0$ the constant function $c$ equals $c\mathbf{1}_{\mathbb{R}^{d+d}}$, so by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set, $\int c\,d\gamma=c\,\gamma(\mathbb{R}^{d+d})=c$; the same holds on $\mathbb{R}^{d}$. For $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ the number $W_{\mathbb{T}}(\mu,\nu)^{2}$ is, by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance and the definition of the nonnegative square root, the greatest lower bound of $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$; likewise $W_{2}(\mu,\nu)^{2}$ is the greatest lower bound of $\{I(\gamma):\gamma\in\Pi(\mu,\nu)\}$ for $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, by The Quadratic Wasserstein Distance on Euclidean Space §distance.

Claim 1 (Inclusion). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$. For $x\in Q$ and $i\in[d]$, the definition of $Q$ gives $0\le x_{i}\le1$, so $x_{i}^{2}\le1$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers,

$$\lVert x\rVert^{2}=\sum_{i=1}^{d}x_{i}^{2}\le\sum_{i=1}^{d}1=d,$$

the real number $d$ being the image of $d$ under the canonical map. The set $Q$ is Borel by The Half-Open Unit Cell Tiles Euclidean Space §cell, and $\mu(\mathbb{R}^{d}\setminus Q)=\mu(\mathbb{R}^{d})-\mu(Q)=1-1=0$ by claim 3 of Basic Properties of a Measure; so $\lVert x\rVert^{2}\le d$ holds for every $x$ outside a null set, that is, almost everywhere. The map $x\mapsto\lVert x\rVert^{2}$ is nonnegative and Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs) and the constant $d$ is Borel (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and the preliminaries give

$$M_{2}(\mu)=\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,\mu(dx)\le\int_{\mathbb{R}^{d}}d\,d\mu=d<\infty .$$

Hence $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space.

Claim 2 (Comparison with the Euclidean cost). Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$. For every $w$, The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal gives $0\le G(w)\le\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert$, hence $G(w)^{2}\le\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Both sides are nonnegative Borel functions (the right one by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs), so monotonicity of the integral, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, gives $I_{\mathbb{T}}(\gamma)\le I(\gamma)$ (the latter as in Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost). By Claim 1, $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so $W_{2}(\mu,\nu)$ is defined. For every $\gamma\in\Pi(\mu,\nu)$ we have $W_{\mathbb{T}}(\mu,\nu)^{2}\le I_{\mathbb{T}}(\gamma)\le I(\gamma)$, so $W_{\mathbb{T}}(\mu,\nu)^{2}$ is a lower bound of $\{I(\gamma):\gamma\in\Pi(\mu,\nu)\}$ and therefore $W_{\mathbb{T}}(\mu,\nu)^{2}\le W_{2}(\mu,\nu)^{2}$. Both distances are nonnegative, so $W_{\mathbb{T}}(\mu,\nu)\le W_{2}(\mu,\nu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 3 (Wrapping). Let $\mu'\in\mathcal{P}(\mathbb{R}^{d})$. The map $\pi$ is Borel by The Half-Open Unit Cell Tiles Euclidean Space §wrap, so $\pi_{\#}\mu'\in\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward; since $\pi(x)\in Q$ for every $x$ (same clause), $\pi^{-1}(Q)=\mathbb{R}^{d}$ and $\pi_{\#}\mu'(Q)=\mu'(\mathbb{R}^{d})=1$, so $\pi_{\#}\mu'\in\mathcal{P}(\mathbb{T}^{d})$.

Let $\mu',\nu'\in\mathcal{P}(\mathbb{R}^{d})$, $\gamma'\in\Pi(\mu',\nu')$, and write $\Phi=\pi\times\pi=(\pi\circ\mathrm{pr}_{1},\pi\circ\mathrm{pr}_{2})$. The projections are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, the composites $\pi\circ\mathrm{pr}_{j}$ are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and $\Phi$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. Hence $\Phi_{\#}\gamma'\in\mathcal{P}(\mathbb{R}^{d+d})$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathrm{pr}_{1}(\Phi(w))=\pi(\mathrm{pr}_{1}(w))$ and $\mathrm{pr}_{2}(\Phi(w))=\pi(\mathrm{pr}_{2}(w))$. For $A\in\mathcal{B}(\mathbb{R}^{d})$ the set $\pi^{-1}(A)$ is Borel, and by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling,

$$\Phi_{\#}\gamma'\bigl(\mathrm{pr}_{1}^{-1}(A)\bigr)=\gamma'\bigl((\mathrm{pr}_{1}\circ\Phi)^{-1}(A)\bigr)=\gamma'\bigl(\mathrm{pr}_{1}^{-1}(\pi^{-1}(A))\bigr)=\mu'\bigl(\pi^{-1}(A)\bigr)=\pi_{\#}\mu'(A),$$

and in the same way $\Phi_{\#}\gamma'(\mathrm{pr}_{2}^{-1}(A))=\pi_{\#}\nu'(A)$. Thus $\Phi_{\#}\gamma'\in\Pi(\pi_{\#}\mu',\pi_{\#}\nu')$, and since both marginals lie in $\mathcal{P}(\mathbb{T}^{d})$ its torus cost is defined. By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to the nonnegative Borel $G^{2}$,

$$I_{\mathbb{T}}(\Phi_{\#}\gamma')=\int_{\mathbb{R}^{d+d}}G(\Phi(w))^{2}\,\gamma'(dw),$$

and $G(\Phi(w))=d_{\mathbb{T}}(\pi(\mathrm{pr}_{1}(w)),\pi(\mathrm{pr}_{2}(w)))=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))=G(w)$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry; this is the displayed identity. The inequality with $I(\gamma')$ follows from the pointwise bound $G^{2}\le\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert^{2}$ and monotonicity exactly as in Claim 2. Finally let $\mu',\nu'\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For every $\gamma'\in\Pi(\mu',\nu')$,

$$W_{\mathbb{T}}(\pi_{\#}\mu',\pi_{\#}\nu')^{2}\le I_{\mathbb{T}}(\Phi_{\#}\gamma')\le I(\gamma'),$$

so $W_{\mathbb{T}}(\pi_{\#}\mu',\pi_{\#}\nu')^{2}$ is a lower bound of $\{I(\gamma'):\gamma'\in\Pi(\mu',\nu')\}$, hence at most $W_{2}(\mu',\nu')^{2}$, and $W_{\mathbb{T}}(\pi_{\#}\mu',\pi_{\#}\nu')\le W_{2}(\mu',\nu')$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 4 (Integrals of periodic functions). Let $f\in C_{\mathrm{per}}$. By Elementary Properties of Lattice-Periodic Functions §bounded there is a real $M_{0}\ge0$ with $|f|\le M_{0}$; put $M=M_{0}+1$, so $0<M$ and $|f(x)|\le M$ for every $x$. By the definition of $C_{\mathrm{per}}$, $f$ is continuous for the Euclidean distance and the absolute-value metric, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures.

(P) For $x,y\in\mathbb{R}^{d}$, the point $k=y-x-\varpi(y-x)$ 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, and $y=(x+\varpi(y-x))+k$; so by periodicity $f(y)=f(x+\varpi(y-x))$, while $\lVert x-(x+\varpi(y-x))\rVert=\lVert\varpi(y-x)\rVert=d_{\mathbb{T}}(x,y)$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

(T) Let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$ and $\gamma\in\Pi(\mu,\nu)$, and put $h=f\circ\mathrm{pr}_{1}-f\circ\mathrm{pr}_{2}$. Since $(\mathrm{pr}_{1})_{\#}\gamma=\mu$ and $(\mathrm{pr}_{2})_{\#}\gamma=\nu$, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward shows that $f\circ\mathrm{pr}_{1}$ and $f\circ\mathrm{pr}_{2}$ are integrable with respect to $\gamma$, with integrals $\int f\,d\mu$ and $\int f\,d\nu$. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

$$\Bigl|\int f\,d\mu-\int f\,d\nu\Bigr|=\Bigl|\int h\,d\gamma\Bigr|\le\int|h|\,d\gamma ,$$

where $|h|$ is Borel (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and bounded by $2M$ (claims 2 and 5 of Properties of the Absolute Value in an Ordered Field). Writing $x=\mathrm{pr}_{1}(w)$ and $y=\mathrm{pr}_{2}(w)$, (P) gives $|h(w)|=|f(x)-f(x+\varpi(y-x))|$ with $\lVert x-(x+\varpi(y-x))\rVert=G(w)$.

Lipschitz case. Let $L\ge0$ and let $f$ be Lipschitz with constant $L$, that is (preamble 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), $|f(a)-f(b)|\le L\lVert a-b\rVert$ for all $a,b$. Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$. By (T), $|h(w)|\le L\,G(w)$ for every $w$ and every coupling $\gamma$, so by monotonicity and linearity (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), $\bigl|\int f\,d\mu-\int f\,d\nu\bigr|\le L\int G\,d\gamma$ for every $\gamma\in\Pi(\mu,\nu)$.

Let $\varepsilon>0$ be given and put $c=W_{\mathbb{T}}(\mu,\nu)+\varepsilon$, so $0\le W_{\mathbb{T}}(\mu,\nu)<c$ and $W_{\mathbb{T}}(\mu,\nu)^{2}<c^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Since $W_{\mathbb{T}}(\mu,\nu)^{2}$ is the greatest lower bound of the costs, claim 2 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ provides $\gamma\in\Pi(\mu,\nu)$ with $I_{\mathbb{T}}(\gamma)<c^{2}$. Pointwise $0\le(G(w)-c)^{2}$, that is, $2c\,G(w)\le G(w)^{2}+c^{2}$; integrating with claim 2 of Linearity and Monotonicity of the Lebesgue Integral and the preliminaries,

$$2c\int G\,d\gamma\le I_{\mathbb{T}}(\gamma)+c^{2}<2c^{2},$$

and multiplying by the positive inverse of $2c$ (claims 5, 7 and 10 of Elementary Order Arithmetic in an Ordered Field) gives $\int G\,d\gamma<c$. By claim 5 of Elementary Arithmetic in an Ordered Field,

$$\Bigl|\int f\,d\mu-\int f\,d\nu\Bigr|\le L\int G\,d\gamma\le L\,W_{\mathbb{T}}(\mu,\nu)+L\varepsilon .$$

Given any $\varepsilon'>0$, apply this with $\varepsilon=\varepsilon'/(L+1)>0$, for which $L\varepsilon\le(L+1)\varepsilon=\varepsilon'$; thus $\bigl|\int f\,d\mu-\int f\,d\nu\bigr|\le L\,W_{\mathbb{T}}(\mu,\nu)+\varepsilon'$ for every $\varepsilon'>0$, and Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $\bigl|\int f\,d\mu-\int f\,d\nu\bigr|\le L\,W_{\mathbb{T}}(\mu,\nu)$.

Convergence. Let $\mu_{j},\mu\in\mathcal{P}(\mathbb{T}^{d})$ with $W_{\mathbb{T}}(\mu_{j},\mu)\to0$. The quantities are chosen in the following order: first $\theta$, then $\eta$, then $r$, then $\beta$ and $\kappa$, then $N$, and finally, for each $j\ge N$, a coupling $\gamma$.

Let $\theta>0$. Put $\eta=\theta\cdot2^{-1}$, so $0<\eta$ and $\eta+\eta=\theta$ (claim 8 of Elementary Order Arithmetic in an Ordered Field). By A Continuous Lattice-Periodic Function is Uniformly Continuous §uniform choose a real $r>0$ with $|f(a)-f(b)|\le\eta$ for all $a,b\in\mathbb{R}^{d}$ with $\lVert a-b\rVert\le r$. Put $\beta=\eta r^{2}(2M)^{-1}$, which is positive by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field, and $\kappa=\min\{1,\beta\}$, the minimum; by claims 1 and 2 of Elementary Properties of the Minimum of Two Elements, $0<\kappa\le1$ and $\kappa\le\beta$. By the definition of convergence, choose $N$ such that $|W_{\mathbb{T}}(\mu_{j},\mu)-0|<\kappa$ for every $j\ge N$.

Fix $j\ge N$. Since $W_{\mathbb{T}}(\mu_{j},\mu)\ge0$ equals its absolute value, $W_{\mathbb{T}}(\mu_{j},\mu)<\kappa$, so $W_{\mathbb{T}}(\mu_{j},\mu)^{2}<\kappa^{2}\le\kappa\le\beta$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field (multiplying $\kappa\le1$ by $\kappa\ge0$). By claim 2 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ choose $\gamma\in\Pi(\mu_{j},\mu)$ with $I_{\mathbb{T}}(\gamma)<\beta$. Let $A=\{w\in\mathbb{R}^{d+d}:r^{2}\le G(w)^{2}\}$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov, applied to the measure space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma)$, to $G^{2}$ and to $t=r^{2}$, the set $A$ is Borel and $r^{2}\gamma(A)\le I_{\mathbb{T}}(\gamma)<\beta$; as $\gamma(A)\le1$ is real, multiplying by the positive inverse of $r^{2}$ gives $\gamma(A)<\eta(2M)^{-1}$.

Now let $w\in\mathbb{R}^{d+d}$, $x=\mathrm{pr}_{1}(w)$, $y=\mathrm{pr}_{2}(w)$. If $w\notin A$, then $G(w)^{2}<r^{2}$, so $G(w)<r$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and by (T) and the choice of $r$, $|h(w)|=|f(x)-f(x+\varpi(y-x))|\le\eta$. If $w\in A$, then $|h(w)|\le|f(x)|+|f(y)|\le2M$. In both cases $|h(w)|\le\eta+2M\mathbf{1}_{A}(w)$, the right side being bounded and Borel (claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). By (T), claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the preliminaries and The Integral of an Indicator Function is the Measure of the Set,

$$\Bigl|\int f\,d\mu_{j}-\int f\,d\mu\Bigr|\le\int|h|\,d\gamma\le\eta+2M\gamma(A)<\eta+2M\cdot\eta(2M)^{-1}=\eta+\eta=\theta .$$

This holds for every $j\ge N$; as $\theta>0$ was arbitrary, $\bigl(\int f\,d\mu_{j}\bigr)_{j}$ converges to $\int f\,d\mu$ by the definition of convergence. $\blacksquare$