Proof of Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Let $\iota$ be the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. For a Borel map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ write

$$D_{S}(z)=\bigl\lVert S(\mathrm{pr}_{1}(z))-\mathrm{pr}_{2}(z)\bigr\rVert^{2}\qquad(z\in\mathbb{R}^{d+d}),$$

Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), in agreement with the notation $D_{T}$ of the statement. For every $\pi'\in\Pi(\mu,\nu)$ the change-of-variables formula, with $(\mathrm{pr}_{1})_{\#}\pi'=\mu$ and $(\mathrm{pr}_{2})_{\#}\pi'=\nu$, gives

$$\int_{\mathbb{R}^{d+d}}\lVert S(\mathrm{pr}_{1}(z))-T(\mathrm{pr}_{1}(z))\rVert^{2}\,\pi'(dz)=\int_{\mathbb{R}^{d}}\lVert S-T\rVert^{2}\,d\mu,\qquad \int_{\mathbb{R}^{d+d}}\lVert\mathrm{pr}_{2}(z)\rVert^{2}\,\pi'(dz)=M_{2}(\nu). \tag{$*$}$$

Step 1 (Comparison of two maps). Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel with $\int\lVert S\rVert^{2}\,d\mu<\infty$ and let $\pi'\in\Pi(\mu,\nu)$. Writing $a=T(\mathrm{pr}_{1}(z))$, $b=S(\mathrm{pr}_{1}(z))$ and $c=\mathrm{pr}_{2}(z)$, the inequality $\lVert x-y\rVert^{2}\le2\lVert x\rVert^{2}+2\lVert y\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), applied to $x=a-b$ and $y=c-b$, gives $D_{T}(z)\le2\lVert a-b\rVert^{2}+2D_{S}(z)$ pointwise. Integrating against $\pi'$ and using claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and $(*)$,

$$\int_{\mathbb{R}^{d+d}}D_{T}\,d\pi'\le2\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu+2\int_{\mathbb{R}^{d+d}}D_{S}\,d\pi' . \tag{$**$}$$

Step 2 (A uniform tail bound). For $j\in\mathbb{N}$ put $R_{j}=\iota(j)$ and $h_{j}(y)=\lVert y\rVert^{2}$ if $R_{j}<\lVert y\rVert^{2}$ and $h_{j}(y)=0$ otherwise; each $h_{j}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), satisfies $0\le h_{j}\le\lVert\cdot\rVert^{2}$, and converges pointwise to $0$, since for fixed $y$ claim 1 of The Archimedean Property of the Real Numbers gives $j$ with $\lVert y\rVert^{2}<\iota(j)$ and claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field makes $h_{j'}(y)=0$ for every $j'\ge j$. As $\lVert\cdot\rVert^{2}$ is integrable with respect to $\nu$, claim 7 of The Lebesgue Integral and Null Sets gives that $\bigl(\int_{\mathbb{R}^{d}}h_{j}\,d\nu\bigr)_{j\in\mathbb{N}}$ converges to $0$.

Step 3 (The limit of a weakly convergent subsequence is the coupling induced by $T$). Let $(\sigma_{k})_{k\in\mathbb{N}}$ be a sequence in $\Pi(\mu,\nu)$ with $(I(\sigma_{k}))_{k}$ converging to $W_{2}(\mu,\nu)^{2}$. By Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment (ii) the set $\Pi(\mu,\nu)$ is a tight family, so Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence provides a subsequence $(\sigma_{k_{j}})_{j}$ converging weakly to some $\pi\in\mathcal{P}(\mathbb{R}^{d+d})$, and $\pi\in\Pi(\mu,\nu)$ by The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence.

Since $(I(\sigma_{k_{j}}))_{j}$ converges to $W_{2}(\mu,\nu)^{2}$ by A Subsequence of a Convergent Sequence Has the Same Limit, for every positive real $\eta$ there is $j_{0}$ with $I(\sigma_{k_{j}})\le W_{2}(\mu,\nu)^{2}+\eta$ for every $j\ge j_{0}$, so Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit (iii), applied with $c=W_{2}(\mu,\nu)^{2}$, gives $I(\pi)\le W_{2}(\mu,\nu)^{2}$; the reverse inequality holds because $W_{2}(\mu,\nu)^{2}$ is the greatest lower bound of the costs of couplings of $\mu$ and $\nu$, by The Quadratic Wasserstein Distance on Euclidean Space (i). Hence $I(\pi)=W_{2}(\mu,\nu)^{2}$ and $\pi$ is optimal. The coupling $(\mathrm{id},T)_{\#}\mu$ is optimal as well, $T$ being an optimal map, so both equal the coupling $(\mathrm{id},T_{0})_{\#}\mu$ induced by the optimal map $T_{0}$ witnessing unique mapping; hence $\pi=(\mathrm{id},T)_{\#}\mu$.

Step 4 (The subsequence has vanishing discrepancy). Keep the notation of Step 3 and let $\eta$ be a positive real number. The map $T$ is Borel with $\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu=M_{2}(\nu)<\infty$ by the change-of-variables formula, $T_{\#}\mu=\nu$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so its class lies in the space $L^{2}(\mu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). Put $\eta'=\min(\eta,1)$, a positive real number. By The Bounded Lipschitz Vector Fields are Dense in the Square-Integrable Vector Fields Against a Probability Measure (i), applied with slack $\eta'$, there is a bounded Lipschitz $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$, say with $\lVert S\rVert\le M_{S}$, such that $\lVert T-S\rVert_{\mu}\le\eta'$; by the formula for $\lVert\cdot\rVert_{\mu}$ in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), and as $0<\eta'\le1$,

$$\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu=\lVert T-S\rVert_{\mu}^{2}\le\eta'^{2}\le\eta'\le\eta .$$

The function $D_{S}$ is continuous: $S$ is continuous by A Lipschitz Map is Uniformly Continuous and Continuous, the projections are continuous by claim 1 of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit, and $D_{S}$ is obtained from them by composition and by the continuous map $v\mapsto\lVert v\rVert^{2}$, continuous by claim 3 of Euclidean Space Is Open in Itself; claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map gives the composition. Moreover

$$D_{S}(z)\le2M_{S}^{2}+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}$$

pointwise, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv).

By Step 2 choose $j_{1}\in\mathbb{N}$ with $\int_{\mathbb{R}^{d}}h_{j_{1}}\,d\nu\le\eta$ and with $2M_{S}^{2}\le R_{j_{1}}$; the order of choice is $\eta$, then $S$ and $M_{S}$, then $j_{1}$. Put $R=R_{j_{1}}$ and $K=4R$, and let $g=\min(D_{S},K)$, which is continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space and satisfies $0\le g\le K$, hence is bounded.

Let $\pi'\in\Pi(\mu,\nu)$ and let $E=\{z:K<D_{S}(z)\}$, a Borel set. For $z\in E$ one has $4R<D_{S}(z)\le2M_{S}^{2}+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}\le R+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, so $R<\lVert\mathrm{pr}_{2}(z)\rVert^{2}$ and therefore $2M_{S}^{2}\le R\le\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, whence $D_{S}(z)\le3\lVert\mathrm{pr}_{2}(z)\rVert^{2}=3h_{j_{1}}(\mathrm{pr}_{2}(z))$. Since $D_{S}=g$ off $E$ and $D_{S}\le3\,h_{j_{1}}\circ\mathrm{pr}_{2}$ on $E$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, together with the change-of-variables formula and $(\mathrm{pr}_{2})_{\#}\pi'=\nu$, gives

$$\int_{\mathbb{R}^{d+d}}D_{S}\,d\pi'\le\int_{\mathbb{R}^{d+d}}g\,d\pi'+3\int_{\mathbb{R}^{d}}h_{j_{1}}\,d\nu\le\int_{\mathbb{R}^{d+d}}g\,d\pi'+3\eta . \tag{$\dagger$}$$

Using $\pi=(\mathrm{id},T)_{\#}\mu$ together with $g\le D_{S}$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and the change-of-variables formula,

$$\int_{\mathbb{R}^{d+d}}g\,d\pi\le\int_{\mathbb{R}^{d+d}}D_{S}\,d\pi=\int_{\mathbb{R}^{d}}\lVert S-T\rVert^{2}\,d\mu\le\eta .$$

Since $g$ is bounded and continuous and $(\sigma_{k_{j}})_{j}$ converges weakly to $\pi$, the sequence $\bigl(\int g\,d\sigma_{k_{j}}\bigr)_{j}$ converges to $\int g\,d\pi\le\eta$, so there is $j_{2}$ with $\int g\,d\sigma_{k_{j}}\le2\eta$ for every $j\ge j_{2}$. Combining this with $(\dagger)$ applied to $\sigma_{k_{j}}$ and then with $(**)$,

$$\int_{\mathbb{R}^{d+d}}D_{T}\,d\sigma_{k_{j}}\le2\eta+2\bigl(2\eta+3\eta\bigr)=12\eta\qquad\text{for every }j\ge j_{2}.$$

Step 5 (Claim 1). Suppose the sequence $\bigl(\int D_{T}\,d\pi_{n}\bigr)_{n\in\mathbb{N}}$ does not converge to $0$. Its terms are nonnegative, so there are a positive real $\varepsilon$ and a subsequence $(\pi_{n_{k}})_{k\in\mathbb{N}}$ with $\varepsilon\le\int D_{T}\,d\pi_{n_{k}}$ for every $k$. By A Subsequence of a Convergent Sequence Has the Same Limit the costs $I(\pi_{n_{k}})$ converge to $W_{2}(\mu,\nu)^{2}$, so Steps 3 and 4 apply to $\sigma_{k}=\pi_{n_{k}}$ and yield, for every positive real $\eta$, a further subsequence along which $\int D_{T}\,d\sigma_{k_{j}}\le12\eta$ eventually. Taking $\eta=\tfrac{\varepsilon}{24}$ produces an index $j$ with $\int D_{T}\,d\pi_{n_{k_{j}}}\le\tfrac{\varepsilon}{2}<\varepsilon$, contradicting the choice of the subsequence. Hence $\bigl(\int D_{T}\,d\pi_{n}\bigr)_{n}$ converges to $0$, which is claim 1.