Proof of A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings

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

Throughout, the notation is that of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, layered on Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, whose clause Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background puts Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation in force; elementary order and arithmetic of real numbers is carried by The Real Numbers: Standing Notation and Background §background, in force through Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §background. For a point $z$ of $X\times X$ we write $z=(x,y)$ with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$. We record three 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 $X$ defined on $X$, $X\times X$ or $X_{(3)}$ 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, and a composite of measurable maps between spaces among $X$, $X\times X$, $X_{(3)}$ and $X^{a}$ is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. The identity map $\mathrm{id}$ of $X$ is Lipschitz with constant $1$, hence continuous by A Lipschitz Map is Uniformly Continuous and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

(F2) Let $\Phi$ be a Borel map between two of the spaces $X$, $X\times X$, $X_{(3)}$, and let $\kappa$ be a Borel probability measure on its domain. Then $\Phi_{\#}\kappa$ is a probability measure by claim 1 of Image Measures, Measures with Densities, and Change of Variables, with $(\Phi_{\#}\kappa)(B)=\kappa(\Phi^{-1}(B))$ for every Borel set $B$, and by claim 2 of that lemma (the change-of-variables formula of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward) one has $\int g\,d(\Phi_{\#}\kappa)=\int g\circ\Phi\,d\kappa$ in $[0,\infty]$ for every nonnegative Borel $g$ on the target. Moreover $(\Psi\circ\Phi)_{\#}\kappa=\Psi_{\#}(\Phi_{\#}\kappa)$ for every Borel $\Psi$ defined on the target of $\Phi$, because $(\Psi\circ\Phi)^{-1}(B)=\Phi^{-1}(\Psi^{-1}(B))$ for every Borel set $B$.

(F3) Let $\Phi$ and $\kappa$ be as in (F2) with $\Phi$ taking values in $X$, let $\lambda=\Phi_{\#}\kappa$, and let $v:X\to X^{a}$ be measurable with $\int_{X}|v|_{a}^{2}\,d\lambda<\infty$. Then $v\circ\Phi$ is measurable into $X^{a}$ by (F1), the function $|v\circ\Phi|_{a}^{2}=|v|_{a}^{2}\circ\Phi$ is nonnegative and Borel 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, and (F2) gives $\int|v\circ\Phi|_{a}^{2}\,d\kappa=\int_{X}|v|_{a}^{2}\,d\lambda<\infty$.

Step 1 (claim 1). Let $\mu\in Q$; then $\mu\in\mathcal{P}^{a}_{\rho}$. We verify properties (a), (b) and (c) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test for $\varphi_{c}$.

(a) By Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants, applied in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of 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 §metric with $A=\mathcal{P}^{a}_{\rho}$ and $b=c$, the function $\varphi_{c}$ is continuous at every point of $\mathcal{P}^{a}_{\rho}$, that is, continuous on $\mathcal{P}^{a}_{\rho}$ in the sense of Continuous Map Between Metric Spaces.

(b) The space $L^{2}(\mu;X^{a})$ is a real Hilbert space whose norm $\lVert\cdot\rVert_{\mu}$ is the norm of its inner product, by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert, so $\lVert0_{\mu}\rVert_{\mu}=0$ by Elementary Identities in a Real Inner Product Space §zero. Let $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound, $|\mathcal{J}^{a}(0_{\mu},\pi)|\le\lVert0_{\mu}\rVert_{\mu}\sqrt{I^{a}(\pi)}=0$, so $\mathcal{J}^{a}(0_{\mu},\pi)=0$. Hence, for every positive $\varepsilon\in\mathbb{R}$, with $\theta=1$,

$$\bigl|\varphi_{c}(\nu)-\varphi_{c}(\mu)-\mathcal{J}^{a}(0_{\mu},\pi)\bigr|=|c-c-0|=0\le\varepsilon\,\sqrt{I^{a}(\pi)}$$

for every such $\nu$ and $\pi$, in particular for those with $I^{a}(\pi)<\theta^{2}$. By Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable, $\varphi_{c}$ is differentiable along noise couplings at $\mu$ with gradient $0_{\mu}$, and by the uniqueness in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient, $\nabla\varphi_{c}(\mu)=0_{\mu}$. Finally $T^{a}_{\mu}$ is a linear subspace of $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace, so it contains the zero vector $0_{\mu}$ by Linear Subspace; thus $\nabla\varphi_{c}(\mu)\in T^{a}_{\mu}$. As $\mu\in Q$ was arbitrary, (b) holds, and $\nabla\varphi_{c}(\mu)=0_{\mu}$ for every $\mu\in Q$, which is the second assertion of claim 1.

(c) Let $\mu\in Q$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{n})_{n\in\mathbb{N}}$ to $\mu$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings). By (b), $\nabla\varphi_{c}(\mu_{n})=0_{\mu_{n}}$ for every $n$ and $\nabla\varphi_{c}(\mu)=0_{\mu}$. Computing the discrepancy of $0_{\mu_{n}}$ and $0_{\mu}$ along $\pi_{n}$ with the representatives given by the constant map with value $0_{X}$, which is allowed because it does not depend on the representatives by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, the integrand is $|0_{X}-0_{X}|_{a}^{2}=0$ at every $z$ (Elementary Identities in a Real Inner Product Space §zero in the inner product space $X^{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert), so the discrepancy is the integral against $\pi_{n}$ of the constant function with value $0$ on $X\times X$. By claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, applied to the finite Borel measure $\pi_{n}$ with $c=0$, this function is integrable, in particular measurable (Integrable Function and the Lebesgue Integral), with integral $0\cdot\pi_{n}(X\times X)=0$; it is nonnegative and bounded in absolute value by $M=0$, so claims 6(b) and 6(c) of that lemma, with $f$ identically $0$ and $M=0$, show that its integral as a nonnegative measurable function is the same number $0$. Hence the discrepancy is $0$. The sequence of discrepancies is therefore the constant sequence $0$, which converges to $0$ by Constant Sequences and Index-Shifted Sequences of Real Numbers §constant; so $(\nabla\varphi_{c}(\mu_{n}))_{n\in\mathbb{N}}$ converges strongly to $\nabla\varphi_{c}(\mu)$ along $(\pi_{n})_{n\in\mathbb{N}}$ by Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong.

Hence $\varphi_{c}$ is a noise intrinsic test function on $Q$.

Step 2 (claim 2). Let $Q$, $\varphi$, $\psi$, $s$ and $t$ be as in claim 2, and write $\chi=s\varphi+t\psi$. We verify properties (a), (b) and (c) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test for $\chi$.

(a) By property (a) for $\varphi$ and $\psi$, both are continuous on $\mathcal{P}^{a}_{\rho}$. By Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, applied in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of 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 §metric with $A=\mathcal{P}^{a}_{\rho}$, first with $f=g=\varphi$ and $c=s$, then with $f=g=\psi$ and $c=t$, the functions $s\varphi$ and $t\psi$ are continuous on $\mathcal{P}^{a}_{\rho}$; applied once more with $f=s\varphi$ and $g=t\psi$, it shows that their sum $\chi$ is continuous on $\mathcal{P}^{a}_{\rho}$.

(b) Let $\mu\in Q$. By property (b) for $\varphi$ and $\psi$, both are differentiable along noise couplings at $\mu$, and by Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient the fields $\eta=\nabla\varphi(\mu)$ and $\xi=\nabla\psi(\mu)$ of $L^{2}(\mu;X^{a})$ have the property of Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable for $\varphi$ and for $\psi$ respectively. Put $\zeta=s\,\eta+t\,\xi\in L^{2}(\mu;X^{a})$ and $c=|s|+|t|+1$, a positive real number with $(|s|+|t|)\,c^{-1}\le1$. Let $\varepsilon\in\mathbb{R}$ be positive; then $\varepsilon c^{-1}$ is positive. The order of choice is: first a positive $\theta_{1}$ as in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable for $\varphi$ and $\eta$, and a positive $\theta_{2}$ as there for $\psi$ and $\xi$, both with $\varepsilon c^{-1}$ in place of $\varepsilon$; then $\theta=\min\{\theta_{1},\theta_{2}\}$, the minimum of the two radii, which is positive because it equals $\theta_{1}$ or $\theta_{2}$ by claim 2 of Elementary Properties of the Minimum of Two Elements. Let $\nu\in\mathcal{P}^{a}_{\rho}$ and $\pi\in\Pi^{a}(\mu,\nu)$ satisfy $I^{a}(\pi)<\theta^{2}$. Since $0<\theta\le\theta_{1}$ and $\theta\le\theta_{2}$ by claim 1 of Elementary Properties of the Minimum of Two Elements, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\theta^{2}\le\theta_{1}^{2}$ and $\theta^{2}\le\theta_{2}^{2}$, whence $I^{a}(\pi)<\theta_{1}^{2}$ and $I^{a}(\pi)<\theta_{2}^{2}$ by Elementary Order Arithmetic in an Ordered Field §mixed-transitivity. Put $Y_{1}=\varphi(\nu)-\varphi(\mu)-\mathcal{J}^{a}(\eta,\pi)$ and $Y_{2}=\psi(\nu)-\psi(\mu)-\mathcal{J}^{a}(\xi,\pi)$; by the choice of $\theta_{1}$ and $\theta_{2}$, $|Y_{1}|\le\varepsilon c^{-1}\sqrt{I^{a}(\pi)}$ and $|Y_{2}|\le\varepsilon c^{-1}\sqrt{I^{a}(\pi)}$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §linear, $\mathcal{J}^{a}(\zeta,\pi)=s\,\mathcal{J}^{a}(\eta,\pi)+t\,\mathcal{J}^{a}(\xi,\pi)$, so $\chi(\nu)-\chi(\mu)-\mathcal{J}^{a}(\zeta,\pi)=sY_{1}+tY_{2}$, and the triangle inequality and multiplicativity of the absolute value (Properties of the Absolute Value in an Ordered Field §triangle, Properties of the Absolute Value in an Ordered Field §multiplicative) give

$$\bigl|\chi(\nu)-\chi(\mu)-\mathcal{J}^{a}(\zeta,\pi)\bigr|\le|s|\,|Y_{1}|+|t|\,|Y_{2}|\le(|s|+|t|)\,\varepsilon c^{-1}\sqrt{I^{a}(\pi)}\le\varepsilon\sqrt{I^{a}(\pi)} .$$

Hence $\chi$ is differentiable along noise couplings at $\mu$ with gradient $\zeta$ (Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable), and $\nabla\chi(\mu)=\zeta=s\,\nabla\varphi(\mu)+t\,\nabla\psi(\mu)$ by the uniqueness in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient. The fields $\eta$ and $\xi$ lie in $T^{a}_{\mu}$ by property (b), and $T^{a}_{\mu}$ is a linear subspace of $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace, so $\zeta\in T^{a}_{\mu}$ by Linear Subspace. As $\mu\in Q$ was arbitrary, (b) holds, and the gradient formula of claim 2 holds at every $\mu\in Q$.

(c) Let $\mu\in Q$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{n})_{n\in\mathbb{N}}$ to $\mu$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings); thus $\pi_{n}\in\Pi^{a}(\mu_{n},\mu)\subseteq\Pi(\mu_{n},\mu)$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings. For $n\in\mathbb{N}$ let $A_{n}$, $B_{n}$ and $C_{n}$ be the discrepancies along $\pi_{n}$ (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 $\mu$) of $\nabla\varphi(\mu_{n})$ and $\nabla\varphi(\mu)$, of $\nabla\psi(\mu_{n})$ and $\nabla\psi(\mu)$, and of $\nabla\chi(\mu_{n})$ and $\nabla\chi(\mu)$. By property (c) for $\varphi$ and for $\psi$ and Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, $(A_{n})_{n\in\mathbb{N}}$ and $(B_{n})_{n\in\mathbb{N}}$ converge to $0$.

Fix $n$, and fix representatives $f_{n},f,g_{n},g$, measurable maps into $X^{a}$, of $\nabla\varphi(\mu_{n})$, $\nabla\varphi(\mu)$, $\nabla\psi(\mu_{n})$, $\nabla\psi(\mu)$. By (b), applied at $\mu_{n}\in Q$ and at $\mu\in Q$, and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, the pointwise combinations $sf_{n}+tg_{n}$ and $sf+tg$ represent $\nabla\chi(\mu_{n})$ and $\nabla\chi(\mu)$; the discrepancies do not depend on the representatives by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, so $A_{n}$, $B_{n}$ and $C_{n}$ are the integrals against $\pi_{n}$ of the nonnegative Borel functions $z\mapsto|f_{n}(x)-f(y)|_{a}^{2}$, $z\mapsto|g_{n}(x)-g(y)|_{a}^{2}$ and $z\mapsto|(sf_{n}(x)+tg_{n}(x))-(sf(y)+tg(y))|_{a}^{2}$ of that clause. Let $z\in X\times X$ and put $u=f_{n}(x)-f(y)$ and $v=g_{n}(x)-g(y)$, elements of the real inner product space $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert); by the vector space axioms of $X^{a}$, $(sf_{n}(x)+tg_{n}(x))-(sf(y)+tg(y))=su+tv$. The triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and homogeneity Elementary Identities in a Real Inner Product Space §homogeneity in $X^{a}$ give $|su+tv|_{a}\le|s|\,|u|_{a}+|t|\,|v|_{a}$, both sides being nonnegative; so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, the real inequality $(p+r)^{2}\le2p^{2}+2r^{2}$ (which holds because $2p^{2}+2r^{2}-(p+r)^{2}=(p-r)^{2}\ge0$ by claim 2 of Nonnegativity of Squares in an Ordered Field), and $|s|^{2}=s^{2}$, $|t|^{2}=t^{2}$ (claim 1 of that lemma) give

$$\bigl|(sf_{n}(x)+tg_{n}(x))-(sf(y)+tg(y))\bigr|_{a}^{2}\le2s^{2}\,|f_{n}(x)-f(y)|_{a}^{2}+2t^{2}\,|g_{n}(x)-g(y)|_{a}^{2}.$$

Integrating against $\pi_{n}$, Linearity and Monotonicity of the Lebesgue Integral §nonnegative (monotonicity, additivity and nonnegative multiples) gives $0\le C_{n}\le2s^{2}A_{n}+2t^{2}B_{n}$ for every $n$. The right-hand side converges to $2s^{2}\cdot0+2t^{2}\cdot0=0$ by Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar, and the constant sequence $0$ converges to $0$ by Constant Sequences and Index-Shifted Sequences of Real Numbers §constant; so $(C_{n})_{n\in\mathbb{N}}$ converges to $0$ by claim 2 (squeeze) of Order Properties of Limits of Real Sequences. By Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, $(\nabla\chi(\mu_{n}))_{n\in\mathbb{N}}$ converges strongly to $\nabla\chi(\mu)$ along $(\pi_{n})_{n\in\mathbb{N}}$, which is (c).

Hence $\chi=s\varphi+t\psi$ is a noise intrinsic test function on $Q$, which with (b) proves claim 2.

Step 3 (claim 3). Let $\mu\in\mathcal{P}^{a}_{\rho}$.

The identity is a noise-optimal map. The map $\mathrm{id}$ is Borel by (F1), and $\mathrm{id}_{\#}\mu=\mu$ because $\mathrm{id}^{-1}(B)=B$ for every Borel set $B$. For $x\in X$, $\mathrm{id}(x)-x=0_{X}\in X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and $n_{a}(0_{X})=|0_{X}|_{a}^{2}=0$ by the definition 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 Elementary Identities in a Real Inner Product Space §zero; so the function $x\mapsto n_{a}(\mathrm{id}(x)-x)$ is the constant function with value $0$ on $X$. By claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, applied to the finite Borel measure $\mu$ with $c=0$, this function is integrable, in particular measurable (Integrable Function and the Lebesgue Integral), with integral $0\cdot\mu(X)=0$; it is nonnegative and bounded in absolute value by $M=0$, so claims 6(b) and 6(c) of that lemma, with $f$ identically $0$ and $M=0$, show that its integral as a nonnegative measurable function is the same number $0$. Thus $\int_{X}n_{a}(\mathrm{id}(x)-x)\,\mu(dx)=0<\infty$. By the last sentence of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement, $\Delta_{\mu}=(\mathrm{id},\mathrm{id})_{\#}\mu\in\Pi^{a}(\mu,\mu)$ and $I^{a}(\Delta_{\mu})=0$. Since $\mu\in\mathcal{P}^{a}_{\rho}$, 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 §separation gives $W_{a}(\mu,\mu)=0$, which is the last assertion of claim 3. Thus $I^{a}(\Delta_{\mu})=0=W_{a}(\mu,\mu)^{2}$, so $\Delta_{\mu}$ is noise-optimal by Noise-Optimal Couplings §optimal, and $\mathrm{id}$ is a noise-optimal map from $\mu$ to $\mu$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map. Its displacement $\mathrm{id}-\mathrm{id}$ is, by that clause, the class of the map $x\mapsto x-x=0_{X}$, the constant map with value $0_{X}$, which is $0_{\mu}$.

Uniqueness. Let $\pi$ be a noise-optimal coupling of $\mu$ and $\mu$. Then $\pi\in\Pi^{a}(\mu,\mu)$ and $\int_{X\times X}c_{a}\,d\pi=I^{a}(\pi)=W_{a}(\mu,\mu)^{2}=0$ by Noise-Optimal Couplings §optimal and Couplings of Finite Noise Cost and Their Noise Cost §cost. The function $c_{a}$ is Borel and nonnegative by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, in the measure space $(X\times X,\mathcal{B}(X\times X),\pi)$, gives $c_{a}(z)=0$ for $\pi$-almost every $z$: by A Property Holding Almost Everywhere the Borel set $N_{1}=\{z:c_{a}(z)\ne0\}$ is $\pi$-null. The set $D_{a}$ is Borel by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs and $\pi(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, so $N_{2}=(X\times X)\setminus D_{a}$ is Borel with $\pi(N_{2})=\pi(X\times X)-\pi(D_{a})=0$, by the additivity of the measure $\pi$ (Measure, Measure Space, and Probability Measure). Hence $N=N_{1}\cup N_{2}$ is a Borel set which is $\pi$-null by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union, and as it is Borel, $\pi(N)=0$ by Null Set of a Measure and the additivity of $\pi$. For $z\notin N$ one has $z\in D_{a}$ and $0=c_{a}(z)=n_{a}(y-x)=|y-x|_{a}^{2}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel; hence $|y-x|_{a}=0$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $y-x=0_{X}$ by Elementary Identities in a Real Inner Product Space §vanishing in the inner product space $X^{a}$, whose zero vector is $0_{X}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert); that is, $y=x$.

Let $A\in\mathcal{B}(X\times X)$, and let $A'=(\mathrm{id},\mathrm{id})^{-1}(A)=\{x\in X:(x,x)\in A\}$, a Borel subset of $X$ by (F1). For $z\notin N$ we have $z=(x,x)$, so $z\in A$ if and only if $x\in A'$, that is, if and only if $z\in\pi_{1}^{-1}(A')$. Hence $A\setminus N=\pi_{1}^{-1}(A')\setminus N$. For every Borel set $B\subseteq X\times X$, additivity of $\pi$ gives $\pi(B)=\pi(B\setminus N)+\pi(B\cap N)$ and $\pi(N)=\pi(B\cap N)+\pi(N\setminus B)$, so $\pi(B\cap N)=0$ and $\pi(B)=\pi(B\setminus N)$. Applying this to $B=A$ and to $B=\pi_{1}^{-1}(A')$, which is Borel by (F1),

$$\pi(A)=\pi(A\setminus N)=\pi\bigl(\pi_{1}^{-1}(A')\setminus N\bigr)=\pi\bigl(\pi_{1}^{-1}(A')\bigr)=\mu(A')=\bigl((\mathrm{id},\mathrm{id})_{\#}\mu\bigr)(A),$$

where $\pi(\pi_{1}^{-1}(A'))=((\pi_{1})_{\#}\pi)(A')=\mu(A')$ because $\pi\in\Pi(\mu,\mu)$ has first marginal $\mu$ (Couplings of Finite Noise Cost and Their Noise Cost §couplings and Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling), and the last equality is the definition of the push-forward in (F2). As $A$ was arbitrary, $\pi=(\mathrm{id},\mathrm{id})_{\#}\mu$.

Uniquely noise-mapped. The map $T=\mathrm{id}$ is a noise-optimal map from $\mu$ to $\mu$, and every noise-optimal coupling of $\mu$ and $\mu$ equals $(\mathrm{id},T)_{\#}\mu$; so $(\mu,\mu)$ is uniquely noise-mapped by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped.

Step 4 (claim 4). Let $\nu$, $q$, $\eta$ and $\Delta$ be as in claim 4. By the last sentence of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement, applied with $\nu$ in place of its $\mu$, $\Delta\in\Pi^{a}(\nu,\nu)$ and $I^{a}(\Delta)=0$; and $\Pi^{a}(\nu,\nu)\subseteq\Pi(\nu,\nu)$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings. Fix representatives $q,\eta:X\to X^{a}$, measurable and square-integrable with respect to $\nu$ by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\nu,\nu$ and $\Delta$, the function $g:X\times X\to\mathbb{R}$, $g(z)=|q(x)-\eta(y)|_{a}^{2}$, is Borel and nonnegative, and its integral against $\Delta$ is the discrepancy. Since $\pi_{1}\circ(\mathrm{id},\mathrm{id})=\pi_{2}\circ(\mathrm{id},\mathrm{id})=\mathrm{id}$, one has $g\circ(\mathrm{id},\mathrm{id})=|q-\eta|_{a}^{2}$, the function $x\mapsto|q(x)-\eta(x)|_{a}^{2}$, where $q-\eta=q+(-1)\eta$ is the pointwise difference. By (F2),

$$\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\Delta(dz)=\int_{X}|q-\eta|_{a}^{2}\,d\nu=\lVert q-\eta\rVert_{\nu}^{2},$$

the last equality by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, the pointwise difference being a representative of the class $q-\eta$ by that clause.

Step 5 (claim 5). Let the data be as in claim 5.

Marginals of the gluing. By Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued, $(q_{1},q_{2})_{\#}\sigma=\pi_{12}$ and $(q_{2},q_{3})_{\#}\sigma=\pi_{23}$, and $\sigma\in\mathcal{P}(X_{(3)})$. The maps $q_{1},q_{2},q_{3}$ and the pairs $(q_{1},q_{2})$, $(q_{2},q_{3})$, $(q_{1},q_{3})$ are Borel, as recorded in the statement of Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs. Since $q_{1}=\pi_{1}\circ(q_{1},q_{2})$, $q_{2}=\pi_{1}\circ(q_{2},q_{3})$ and $q_{3}=\pi_{2}\circ(q_{2},q_{3})$, (F2) and the marginals of $\pi_{12}\in\Pi(\nu,\lambda)$ and $\pi_{23}\in\Pi(\lambda,\mu)$ (Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling) give

$$(q_{1})_{\#}\sigma=(\pi_{1})_{\#}\pi_{12}=\nu,\qquad(q_{2})_{\#}\sigma=(\pi_{1})_{\#}\pi_{23}=\lambda,\qquad(q_{3})_{\#}\sigma=(\pi_{2})_{\#}\pi_{23}=\mu .$$

The three fields over the gluing. Let $L^{2}(\sigma;X^{a})$ be the space of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes for the measure space $(X_{(3)},\mathcal{B}(X_{(3)}),\sigma)$, with $E=X^{a}$ and the orthonormal 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, with the norm $\lVert\cdot\rVert_{\sigma}$ of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations; it is a real Hilbert space whose norm is the norm of its inner product, by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert. Fix representatives $q,\eta,\theta:X\to X^{a}$ of the three fields. By (F3), applied with $\kappa=\sigma$ and $\Phi=q_{1},q_{2},q_{3}$ respectively, the maps $\hat{q}=q\circ q_{1}$, $\hat{\eta}=\eta\circ q_{2}$ and $\hat{\theta}=\theta\circ q_{3}$ are measurable into $X^{a}$ and square-integrable with respect to $\sigma$; their classes in $L^{2}(\sigma;X^{a})$ are written with the same symbols.

Discrepancies as distances. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\pi_{12}$, the function $g_{12}(z)=|q(x)-\eta(y)|_{a}^{2}$ on $X\times X$ is Borel and nonnegative, and its integral against $\pi_{12}$ is the corresponding discrepancy. Its composite with $(q_{1},q_{2})$ is $|\hat{q}-\hat{\eta}|_{a}^{2}$, where $\hat{q}-\hat{\eta}=\hat{q}+(-1)\hat{\eta}$ is the pointwise difference, a representative of the class $\hat{q}-\hat{\eta}$ by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations. By (F2) through $(q_{1},q_{2})$ and the definition of $\lVert\cdot\rVert_{\sigma}$ in that clause,

$$\lVert\hat{q}-\hat{\eta}\rVert_{\sigma}^{2}=\int_{X_{(3)}}|\hat{q}-\hat{\eta}|_{a}^{2}\,d\sigma=\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi_{12}(dz).$$

In the same way, through $(q_{2},q_{3})$ with $(q_{2},q_{3})_{\#}\sigma=\pi_{23}$, and through $(q_{1},q_{3})$ with $(q_{1},q_{3})_{\#}\sigma=\pi_{13}$, using Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy for $\pi_{23}$ and for $\pi_{13}$,

$$\lVert\hat{\eta}-\hat{\theta}\rVert_{\sigma}^{2}=\int_{X\times X}|\eta(x)-\theta(y)|_{a}^{2}\,\pi_{23}(dz),\qquad\lVert\hat{q}-\hat{\theta}\rVert_{\sigma}^{2}=\int_{X\times X}|q(x)-\theta(y)|_{a}^{2}\,\pi_{13}(dz).$$

Conclusion. In the vector space $L^{2}(\sigma;X^{a})$, $\hat{q}-\hat{\theta}=(\hat{q}-\hat{\eta})+(\hat{\eta}-\hat{\theta})$, so the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives $\lVert\hat{q}-\hat{\theta}\rVert_{\sigma}\le\lVert\hat{q}-\hat{\eta}\rVert_{\sigma}+\lVert\hat{\eta}-\hat{\theta}\rVert_{\sigma}$. Each norm is nonnegative and its square is the corresponding discrepancy, so it is the nonnegative square root of that discrepancy by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root; this is the claimed inequality. $\blacksquare$