Proof of The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source

Each result cited is universally quantified over the data in its own statement.

Throughout, $W=W_{a}(\mu,\nu)$; more generally $W_{a}(\lambda,\lambda')$ is defined for all $\lambda,\lambda'\in\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, and $W_{a}(\lambda,\lambda')^{2}\le I^{a}(\pi)$ for every $\pi\in\Pi^{a}(\lambda,\lambda')$ by The Noise Wasserstein Distance §distance. We record four facts used repeatedly.

(F1) The coordinate maps $\pi_{1},\pi_{2}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma; a pair of Borel maps into $X$ is a Borel map into $X\times X$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing; a composite of measurable maps is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; and continuous maps between metric spaces are Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space.

(F2) For a Borel map $\Phi$ defined on $X$, $X\times X$ or $X_{(3)}$ with values in $X$ or $X\times X$, and a Borel probability measure $\lambda$ on its domain, the push-forward $\Phi_{\#}\lambda$ is the image measure $B\mapsto\lambda(\Phi^{-1}(B))$, a probability measure, and $\int g\,d(\Phi_{\#}\lambda)=\int g\circ\Phi\,d\lambda$ for every nonnegative Borel $g$ on the target, while a real-valued Borel $g$ is integrable against $\Phi_{\#}\lambda$ exactly when $g\circ\Phi$ is integrable against $\lambda$, with the same identity; this is claims 1 and 2 of Image Measures, Measures with Densities, and Change of Variables, as recorded in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. In particular $(\Psi\circ\Phi)_{\#}\lambda=\Psi_{\#}(\Phi_{\#}\lambda)$ for Borel $\Psi$ on the target of $\Phi$, since $(\Psi\circ\Phi)^{-1}(B)=\Phi^{-1}(\Psi^{-1}(B))$.

(F3) $X^{a}$ is a linear subspace of $X$ and a real Hilbert space by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert. For $u,v\in X^{a}$, expanding with the symmetry and the linearity in the first argument of the inner product (Real Inner Product Space §inner-product) gives $|u\pm v|_{a}^{2}=|u|_{a}^{2}\pm2\langle u,v\rangle_{a}+|v|_{a}^{2}$, hence $|u+v|_{a}^{2}+|u-v|_{a}^{2}=2|u|_{a}^{2}+2|v|_{a}^{2}$ and $|u\pm v|_{a}^{2}\le2|u|_{a}^{2}+2|v|_{a}^{2}$.

(F4) If $\lambda,\lambda'\in\mathcal{P}(X)$ and $\pi\in\Pi^{a}(\lambda,\lambda')$, then $\pi\in\Pi(\lambda,\lambda')$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings, so $(\pi_{1})_{\#}\pi=\lambda$ and $(\pi_{2})_{\#}\pi=\lambda'$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling, and $\pi(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite. By the definitions of $n_{a}$ in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel and of $c_{a}$ in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, $c_{a}(z)=n_{a}(y-x)=|y-x|_{a}^{2}$ for $z\in D_{a}$.

Step 1 (claim 1). Let $T$ be a noise-optimal map from $\mu$ to $\nu$. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, $T$ is Borel, $T_{\#}\mu=\nu$, $T(x)-x\in X^{a}$ for every $x$, $\int_{X}n_{a}(T(x)-x)\,\mu(dx)<\infty$, the map $h:X\to X^{a}$, $h(x)=T(x)-x$, is a representative of $T-\mathrm{id}$ with $|h(x)|_{a}^{2}=n_{a}(T(x)-x)$, and $\gamma_{T}=(\mathrm{id},T)_{\#}\mu$ is noise-optimal. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement (applied with $S=T$) and Noise-Optimal Couplings §optimal,

$$\lVert T-\mathrm{id}\rVert_{\mu}^{2}=\int_{X}n_{a}\bigl(T(x)-x\bigr)\,\mu(dx)=I^{a}(\gamma_{T})=W^{2}.$$

Step 2 (claim 2: the optimal coupling and the pairing along it). Assume the hypotheses of claim 2, and let $h$ and $\gamma_{T}$ be as in Step 1. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped there is a noise-optimal map $T_{0}$ from $\mu$ to $\nu$ such that every noise-optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T_{0})_{\#}\mu$. Since $\gamma_{T}$ is noise-optimal, $\gamma_{T}=(\mathrm{id},T_{0})_{\#}\mu$; hence every noise-optimal coupling of $\mu$ and $\nu$ equals $\gamma_{T}$. Moreover, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement, applied with this $\mu$, the representative $h$ and $t=1$, has $S_{1}(x)=x+h(x)=T(x)$, so its coupling $\pi_{1}$ is $\gamma_{T}$, and it gives $I^{a}(\gamma_{T})=\lVert h\rVert_{\mu}^{2}$ and $\mathcal{J}^{a}(\eta,\gamma_{T})=\langle\eta,h\rangle_{\mu}$ for every $\eta\in L^{2}(\mu;X^{a})$; in particular $\mathcal{J}^{a}(h,\gamma_{T})=\lVert h\rVert_{\mu}^{2}$, the norm of $L^{2}(\mu;X^{a})$ being that of its inner product by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert.

Step 3 (claim 2: an expansion). For $n\in\mathbb{N}$ let $A_{n}=\int_{X\times X}n_{a}(T(x)-y)\,\gamma_{n}(dz)$, a nonnegative real number as shown in the statement, and let $\delta_{n}$ be the displacement field of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement-field for $\gamma_{n}$. For $z\in D_{a}$ one has $\delta_{n}(z)=y-x$ and $T(x)-y=h(x)-\delta_{n}(z)\in X^{a}$, so by (F3) and (F4)

$$n_{a}\bigl(T(x)-y\bigr)=|h(x)-\delta_{n}(z)|_{a}^{2}=|h(x)|_{a}^{2}-2\langle h(x),\delta_{n}(z)\rangle_{a}+c_{a}(z).$$

Call the right-hand side $G_{n}(z)$, defined for every $z$. The function $z\mapsto|h(\pi_{1}z)|_{a}^{2}$ is nonnegative and Borel by (F1) and Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations, and its integral against $\gamma_{n}$ is $\int_{X}|h|_{a}^{2}\,d\mu=\lVert h\rVert_{\mu}^{2}<\infty$ by (F2) and (F4), so it is integrable by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral; the function $z\mapsto\langle h(x),\delta_{n}(z)\rangle_{a}$ is Borel and integrable with integral $\mathcal{J}^{a}(h,\gamma_{n})$ by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing; and $c_{a}$ is nonnegative and Borel with integral $I^{a}(\gamma_{n})<\infty$ by Couplings of Finite Noise Cost and Their Noise Cost §cost. By Linearity and Monotonicity of the Lebesgue Integral §integrable, $G_{n}$ is integrable with $\int G_{n}\,d\gamma_{n}=\lVert h\rVert_{\mu}^{2}-2\mathcal{J}^{a}(h,\gamma_{n})+I^{a}(\gamma_{n})$. Since $\gamma_{n}(D_{a})=1$ by (F4), the two Borel functions $z\mapsto n_{a}(T(x)-y)$ and $G_{n}$ agree $\gamma_{n}$-almost everywhere, and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison together with Step 1 gives

$$A_{n}=W^{2}-2\mathcal{J}^{a}(h,\gamma_{n})+I^{a}(\gamma_{n})\qquad(n\in\mathbb{N}).$$

Step 4 (claim 2: convergence of the pairing). We show: if $(n_{j})_{j\in\mathbb{N}}$ is strictly increasing in $\mathbb{N}$ and $\gamma_{n_{j}}\Rightarrow\gamma_{T}$, then $\lim_{j\to\infty}\mathcal{J}^{a}(h,\gamma_{n_{j}})=\lVert h\rVert_{\mu}^{2}$. The convergent real sequence $(I^{a}(\gamma_{n}))_{n\in\mathbb{N}}$ is bounded, so there is a real $\beta$ with $I^{a}(\gamma_{n})\le\beta$ for every $n$; then $W^{2}\le I^{a}(\gamma_{n})\le\beta$, and $I^{a}(\gamma_{T})=W^{2}\le\beta$ by Step 1. Let $\varepsilon'>0$. The order of choices is: first $\varepsilon'$, then $\zeta$, then $K$, then $\varphi_{1},\dots,\varphi_{K}$ (and with them $g$, $B$ and $C$), then $M$, then $j_{0}$.

Step 4a (an approximating field). Let $\zeta=\varepsilon'/(3(\beta^{1/2}+\lVert h\rVert_{\mu}+1))>0$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, applied with $E=X^{a}$ and the basis $(f_{k})_{k\in\mathbb{N}}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, the coordinates $h_{k}\in L^{2}(\mu)$ of $h$ satisfy $\lVert h\rVert_{\mu}^{2}=\sum_{k=1}^{\infty}\lVert h_{k}\rVert_{L^{2}(\mu)}^{2}$, a convergent series; choose $K\in\mathbb{N}$ with $\sum_{k>K}\lVert h_{k}\rVert_{L^{2}(\mu)}^{2}<\zeta^{2}/2$. For each $k\le K$, Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density provides $\varphi_{k}\in\mathcal{F}C^{1}_{b}(X)$ whose class satisfies $\lVert h_{k}-\varphi_{k}\rVert_{L^{2}(\mu)}^{2}<\zeta^{2}/(2K)$, because the open ball of radius $\zeta/\sqrt{2K}$ about $h_{k}$ is a nonempty open set and therefore meets the dense set of classes of bounded $C^{1}$ cylindrical functions (dense in the sense of Real Hilbert Space §topology). By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel each $\varphi_{k}$ is Borel and bounded; fix a real $B\ge0$ with $|\varphi_{k}(x)|\le B$ for all $k\le K$ and $x\in X$. Put $g_{k}=\varphi_{k}$ for $k\le K$ and $g_{k}=0$ for $k>K$; for every $x$ the series $\sum_{k}g_{k}(x)^{2}$ has only finitely many nonzero terms and converges. By Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §synthesis the map $g:X\to X^{a}$, $g(x)=\sum_{k=1}^{K}\varphi_{k}(x)f_{k}$, is measurable with coordinates $\langle g(x),f_{k}\rangle_{a}=g_{k}(x)$, and by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations $|g(x)|_{a}^{2}=\sum_{k=1}^{K}\varphi_{k}(x)^{2}\le KB^{2}$, so $g$ is square-integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, the latter giving $\int_{X}KB^{2}\,d\mu=KB^{2}\mu(X)=KB^{2}$. Its class, again written $g$, has coordinates $g_{k}$ (the class of $\varphi_{k}$ for $k\le K$, and $0$ for $k>K$). By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, $(h-g)_{k}=h_{k}-g_{k}$ and

$$\lVert h-g\rVert_{\mu}^{2}=\sum_{k=1}^{K}\lVert h_{k}-\varphi_{k}\rVert_{L^{2}(\mu)}^{2}+\sum_{k>K}\lVert h_{k}\rVert_{L^{2}(\mu)}^{2}<\frac{\zeta^{2}}{2}+\frac{\zeta^{2}}{2}=\zeta^{2}.$$

Step 4b (replacing $h$ by $g$). By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §linear and Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound, for every $n$,

$$\bigl|\mathcal{J}^{a}(h,\gamma_{n})-\mathcal{J}^{a}(g,\gamma_{n})\bigr|=\bigl|\mathcal{J}^{a}(h-g,\gamma_{n})\bigr|\le\lVert h-g\rVert_{\mu}\sqrt{I^{a}(\gamma_{n})}\le\zeta\beta^{1/2},$$

and likewise, by Step 2, $\bigl|\mathcal{J}^{a}(g,\gamma_{T})-\lVert h\rVert_{\mu}^{2}\bigr|=\bigl|\mathcal{J}^{a}(g-h,\gamma_{T})\bigr|\le\lVert g-h\rVert_{\mu}\sqrt{I^{a}(\gamma_{T})}\le\zeta\lVert h\rVert_{\mu}$.

Step 4c (a continuous integrand). For $k\le K$, by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical there are $m_{k}\in\mathbb{N}$ and $\psi_{k}\in C^{1}_{b}(\mathbb{R}^{m_{k}})$ with $\varphi_{k}=\psi_{k}\circ p_{m_{k}}$. Define $F:X\times X\to\mathbb{R}$ by

$$F(z)=\sum_{k=1}^{K}a_{k}^{-1/2}\,\varphi_{k}(x)\,(y_{k}-x_{k}).$$

$F$ is continuous on $(X\times X,d)$: each $\psi_{k}$ is continuous by claims 1 and 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, the class $C^{1}_{b}$ consisting of $C^{1}$ functions by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded; the maps $p_{m_{k}}$ and the coordinate functions $x\mapsto x_{k}$ are Lipschitz by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and $\pi_{1},\pi_{2}$ are Lipschitz with constant $1$, being linear with $|\pi_{i}z|\le|z|$ by Properties of the Product of Two Real Inner Product Spaces §coordinates, so all of these are continuous by A Lipschitz Map is Uniformly Continuous; composites of continuous maps are continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map; and sums, products and real multiples of continuous real functions are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set. Hence $F$ is Borel by (F1).

Let $\pi\in\Pi^{a}(\mu,\nu)$ with displacement field $\delta$. For $z\in D_{a}$, $\delta(z)=y-x\in X^{a}$ and, by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis and the linearity of $u\mapsto\langle u,e_{k}\rangle$, $\langle f_{k},y-x\rangle_{a}=a_{k}^{-1/2}(y_{k}-x_{k})$; so the linearity and symmetry of $\langle\cdot,\cdot\rangle_{a}$ give $\langle g(x),\delta(z)\rangle_{a}=\sum_{k=1}^{K}\varphi_{k}(x)\langle f_{k},y-x\rangle_{a}=F(z)$. As $\pi(D_{a})=1$ by (F4), the integrand of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing for $g$ and $\pi$ agrees with $F$ $\pi$-almost everywhere, and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison shows that $F$ is integrable against $\pi$ with $\mathcal{J}^{a}(g,\pi)=\int F\,d\pi$. Further, $|y_{k}-x_{k}|\le|y-x|$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so with $C=B\sum_{k=1}^{K}a_{k}^{-1/2}$ one has $F(z)^{2}\le C^{2}|\pi_{1}(z)-\pi_{2}(z)|^{2}$ for every $z$, and Linearity and Monotonicity of the Lebesgue Integral §nonnegative, Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §cost and Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound give

$$\int_{X\times X}F^{2}\,d\pi\le C^{2}I(\pi)\le C^{2}\bar{a}\,I^{a}(\pi).$$

For $\pi=\gamma_{n}$ and for $\pi=\gamma_{T}$ the right-hand side is at most $C^{2}\bar{a}\beta$.

Step 4d (truncation). For a real $M>0$ let $\mathrm{cl}_{M}(s)=\max\{-M,\min\{M,s\}\}$ for $s\in\mathbb{R}$. Elementary order arithmetic gives $|\mathrm{cl}_{M}(s)-\mathrm{cl}_{M}(s')|\le|s-s'|$, $|\mathrm{cl}_{M}(s)|\le M$ and $|s-\mathrm{cl}_{M}(s)|\le s^{2}/M$ for all $s,s'$ (the last because the left side is $0$ when $|s|\le M$ and is $|s|-M<|s|<s^{2}/M$ otherwise). Choose $M>0$ with $C^{2}\bar{a}\beta/M\le\varepsilon'/12$. The function $F_{M}=\mathrm{cl}_{M}\circ F$ is continuous by A Lipschitz Map is Uniformly Continuous and claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, bounded by $M$, and Borel by (F1), hence integrable against every Borel probability measure on $X\times X$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. For $\pi\in\{\gamma_{n}:n\in\mathbb{N}\}\cup\{\gamma_{T}\}$, Linearity and Monotonicity of the Lebesgue Integral §integrable and Linearity and Monotonicity of the Lebesgue Integral §nonnegative together with Step 4c give

$$\Bigl|\int F\,d\pi-\int F_{M}\,d\pi\Bigr|\le\int|F-F_{M}|\,d\pi\le\frac{1}{M}\int F^{2}\,d\pi\le\frac{C^{2}\bar{a}\beta}{M}\le\frac{\varepsilon'}{12}.$$

Step 4e (weak convergence). Since $F_{M}$ is bounded and continuous and $\gamma_{n_{j}}\Rightarrow\gamma_{T}$, Weak Convergence of Finite Borel Measures on a Metric Space gives $\lim_{j}\int F_{M}\,d\gamma_{n_{j}}=\int F_{M}\,d\gamma_{T}$; choose $j_{0}$ with $|\int F_{M}\,d\gamma_{n_{j}}-\int F_{M}\,d\gamma_{T}|<\varepsilon'/6$ for every $j\ge j_{0}$. By Steps 4c and 4d, for $j\ge j_{0}$,

$$\bigl|\mathcal{J}^{a}(g,\gamma_{n_{j}})-\mathcal{J}^{a}(g,\gamma_{T})\bigr|\le\frac{\varepsilon'}{12}+\frac{\varepsilon'}{6}+\frac{\varepsilon'}{12}=\frac{\varepsilon'}{3}.$$

Combining with Step 4b, for $j\ge j_{0}$,

$$\bigl|\mathcal{J}^{a}(h,\gamma_{n_{j}})-\lVert h\rVert_{\mu}^{2}\bigr|\le\zeta\beta^{1/2}+\frac{\varepsilon'}{3}+\zeta\lVert h\rVert_{\mu}<\frac{\varepsilon'}{3}+\frac{\varepsilon'}{3}<\varepsilon',$$

by the choice of $\zeta$. As $\varepsilon'>0$ was arbitrary, the assertion of Step 4 follows.

Step 5 (claim 2: conclusion). Suppose, for a contradiction, that $(A_{n})_{n\in\mathbb{N}}$ does not converge to $0$. Since $A_{n}\ge0$, there are a real $\varepsilon>0$ and a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ with $A_{n_{j}}\ge\varepsilon$ for every $j$. The set $\Pi(\mu,\nu)$ is tight in $X\times X$ by Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight, hence so is its subset $\{\gamma_{n_{j}}:j\in\mathbb{N}\}$, directly from Tight Family of Borel Measures on a Metric Space §tight; thus the sequence $(\gamma_{n_{j}})_{j}$ is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence. By Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence, applied on the metric space $(X\times X,d)$, there are a strictly increasing $(j_{i})_{i\in\mathbb{N}}$ and $\gamma\in\mathcal{P}(X\times X)$ with $\gamma_{m_{i}}\Rightarrow\gamma$, where $m_{i}=n_{j_{i}}$ is strictly increasing. The constant sequences with terms $\mu$ and $\nu$ converge weakly to $\mu$ and $\nu$ directly from Weak Convergence of Finite Borel Measures on a Metric Space, each $\gamma_{m_{i}}$ lies in $\Pi^{a}(\mu,\nu)$, and $(I^{a}(\gamma_{m_{i}}))_{i}$ converges to $W^{2}$, being a subsequence of a sequence converging to $W^{2}$, so it is bounded, and its limit inferior, in the sense of Limit Inferior of a Bounded Sequence of Real Numbers, equals its limit $W^{2}$ by claim 5 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence. Hence Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §lsc gives $\gamma\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\gamma)\le W^{2}$, while $W^{2}\le I^{a}(\gamma)$ by The Noise Wasserstein Distance §distance. So $\gamma$ is noise-optimal by Noise-Optimal Couplings §optimal, and $\gamma=\gamma_{T}$ by Step 2. Step 4, applied to $(m_{i})_{i}$, gives $\lim_{i}\mathcal{J}^{a}(h,\gamma_{m_{i}})=\lVert h\rVert_{\mu}^{2}=W^{2}$ (Step 1), and Step 3 gives

$$\lim_{i\to\infty}A_{m_{i}}=W^{2}-2W^{2}+W^{2}=0,$$

contradicting $A_{m_{i}}\ge\varepsilon>0$ for every $i$. Hence $\lim_{n}A_{n}=0$, which is claim 2.

Step 6 (claim 3: a gluing). Assume the hypotheses of claim 3. By Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings, applied with $\mu$ in place of its $\nu$ and $(\mu_{n})$ in place of its $(\nu_{n})$, $\gamma_{n}\in\Pi^{a}(\mu_{n},\mu)$ for every $n$ and $\lim_{n}I^{a}(\gamma_{n})=0$. Fix $n$. Let $\tau:X\times X\to X\times X$, $\tau(x,y)=(y,x)$, be the swap map of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap (there written $\sigma$); it is Borel by (F1), and that clause gives $\gamma'_{n}=\tau_{\#}\gamma_{n}\in\Pi^{a}(\mu,\mu_{n})$ with $I^{a}(\gamma'_{n})=I^{a}(\gamma_{n})$. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, applied to $\mu_{n},\nu\in\mathcal{P}^{a}_{\rho}$, the coupling $\theta_{n}=(\mathrm{id},S_{n})_{\#}\mu_{n}$ belongs to $\Pi^{a}(\mu_{n},\nu)$ and is noise-optimal, so $I^{a}(\theta_{n})=W_{a}(\mu_{n},\nu)^{2}$ by Noise-Optimal Couplings §optimal. The measures $\mu,\mu_{n},\nu$ belong to $\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments, so Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued, applied with $\lambda=\mu_{n}$, $\pi_{12}=\gamma'_{n}$ and $\pi_{23}=\theta_{n}$, provides $\sigma_{n}\in\mathcal{P}(X_{(3)})$ with $(q_{1},q_{2})_{\#}\sigma_{n}=\gamma'_{n}$ and $(q_{2},q_{3})_{\#}\sigma_{n}=\theta_{n}$. Let $\kappa_{n}=(q_{1},q_{3})_{\#}\sigma_{n}$. By Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle, $\kappa_{n}\in\Pi^{a}(\mu,\nu)$ and $\sqrt{I^{a}(\kappa_{n})}\le\sqrt{I^{a}(\gamma_{n})}+W_{a}(\mu_{n},\nu)$, using $\sqrt{W_{a}(\mu_{n},\nu)^{2}}=W_{a}(\mu_{n},\nu)$ by Existence and Uniqueness of the Nonnegative Square Root.

Step 7 (claim 3: the composite couplings are nearly optimal). Let $r_{n}=\sqrt{I^{a}(\gamma_{n})}$; since $I^{a}(\gamma_{n})\to0$, also $r_{n}\to0$ (given $\varepsilon>0$, eventually $I^{a}(\gamma_{n})<\varepsilon^{2}$, hence $r_{n}<\varepsilon$). By The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §triangle, applied to $\mu_{n},\mu,\nu$, and by $W_{a}(\mu_{n},\mu)^{2}\le I^{a}(\gamma_{n})$, one has $W_{a}(\mu_{n},\nu)\le W_{a}(\mu_{n},\mu)+W\le r_{n}+W$. With Step 6 and $W^{2}\le I^{a}(\kappa_{n})$,

$$W^{2}\le I^{a}(\kappa_{n})\le(W+2r_{n})^{2}=W^{2}+4r_{n}W+4r_{n}^{2},$$

so $\lim_{n}I^{a}(\kappa_{n})=W^{2}$. Since $(\mu,\nu)$ is uniquely noise-mapped and $S$ is a noise-optimal map from $\mu$ to $\nu$, claim 2 (Steps 2 to 5), applied with $T=S$ and the sequence $(\kappa_{n})_{n}$, gives

$$\lim_{n\to\infty}\int_{X\times X}n_{a}\bigl(S(x)-y\bigr)\,\kappa_{n}(dz)=0 .$$

By (F2), with the nonnegative Borel function $z\mapsto n_{a}(S(x)-y)$ of claim 2 and the Borel map $(q_{1},q_{3})$, this integral equals $\int_{X_{(3)}}n_{a}(S(q_{1}w)-q_{3}w)\,\sigma_{n}(dw)$.

Step 8 (claim 3: a set of full measure). Let $E_{n}=\{z\in X\times X:y=S_{n}(x)\}$, the preimage of the closed set $\{0_{X}\}$ under the Borel map $z\mapsto y-S_{n}(x)$ (Borel by (F1) and Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations applied with $E=X$), so $E_{n}$ is Borel by claim 1 of Borel Measurability and Bounded Integration on a Metric Space. Since $(\mathrm{id},S_{n})^{-1}(E_{n})=X$, (F2) gives $\sigma_{n}((q_{2},q_{3})^{-1}(E_{n}))=\theta_{n}(E_{n})=\mu_{n}(X)=1$. Also $\sigma_{n}((q_{1},q_{2})^{-1}(D_{a}))=\gamma'_{n}(D_{a})=1$ by (F4). Let $N_{n}=(q_{2},q_{3})^{-1}(E_{n})\cap(q_{1},q_{2})^{-1}(D_{a})$; then $\sigma_{n}(N_{n})=1$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union. For $w\in N_{n}$ one has $q_{3}w=S_{n}(q_{2}w)$ and $v:=q_{2}w-q_{1}w\in X^{a}$; further $S(q_{1}w)-q_{1}w\in X^{a}$ and $S_{n}(q_{2}w)-q_{2}w\in X^{a}$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, so $u:=S(q_{1}w)-q_{3}w=(S(q_{1}w)-q_{1}w)-v-(S_{n}(q_{2}w)-q_{2}w)\in X^{a}$ by (F3).

Step 9 (claim 3: the discrepancy). Let $q_{n}^{\ast}(x)=S_{n}(x)-x$ and $q^{\ast}(y)=S(y)-y$ be the representatives of $S_{n}-\mathrm{id}\in L^{2}(\mu_{n};X^{a})$ and $S-\mathrm{id}\in L^{2}(\mu;X^{a})$ given by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\mu_{n}$ and $\mu$ in place of its $\nu$ and $\mu$ and with $\gamma_{n}\in\Pi(\mu_{n},\mu)$, the function $\Delta_{n}(z)=|q_{n}^{\ast}(x)-q^{\ast}(y)|_{a}^{2}$ is nonnegative, Borel and $\gamma_{n}$-integrable, and its integral is the discrepancy occurring in Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong. Since $\tau\circ(q_{1},q_{2})=(q_{2},q_{1})$, (F2) gives $\gamma_{n}=\tau_{\#}\gamma'_{n}=(q_{2},q_{1})_{\#}\sigma_{n}$ (as $\tau\circ\tau$ is the identity), and hence

$$\int_{X\times X}\Delta_{n}\,d\gamma_{n}=\int_{X_{(3)}}\bigl|(S_{n}(q_{2}w)-q_{2}w)-(S(q_{1}w)-q_{1}w)\bigr|_{a}^{2}\,\sigma_{n}(dw).$$

For $w\in N_{n}$, with $u$ and $v$ as in Step 8, the vector inside the norm is $-(u+v)$, whose norm is $|u+v|_{a}$ by Elementary Identities in a Real Inner Product Space §homogeneity in $X^{a}$, so by (F3) and (F4) the integrand is at most $2|u|_{a}^{2}+2|v|_{a}^{2}=2n_{a}(S(q_{1}w)-q_{3}w)+2c_{a}((q_{1},q_{2})(w))$. As $\sigma_{n}(N_{n})=1$, The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Linearity and Monotonicity of the Lebesgue Integral §nonnegative and (F2) (with $(q_{1},q_{2})_{\#}\sigma_{n}=\gamma'_{n}$) give

$$0\le\int_{X\times X}\Delta_{n}\,d\gamma_{n}\le2\int_{X_{(3)}}n_{a}\bigl(S(q_{1}w)-q_{3}w\bigr)\,\sigma_{n}(dw)+2I^{a}(\gamma'_{n}).$$

The first term on the right tends to $0$ by Step 7 and the second equals $2I^{a}(\gamma_{n})\to0$ by Step 6. Hence $\lim_{n}\int\Delta_{n}\,d\gamma_{n}=0$, which is, by Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, the strong convergence of $(S_{n}-\mathrm{id})_{n\in\mathbb{N}}$ to $S-\mathrm{id}$ along $(\gamma_{n})_{n\in\mathbb{N}}$.