Proof of Tangency of Noise-Optimal Displacements out of a Measure with a Density Relative to a Diagonal Gaussian Measure, and the Noise Map Property

Each result cited is universally quantified over the data in its own statement. Elementary order and arithmetic of real numbers, square roots and limits of real sequences included, are those of The Real Numbers: Standing Notation and Background §background. The argument is the finite-dimensional approximation of Feyel and "Ust"unel: the noise-optimal displacement is approached by lifts of Euclidean optimal displacements between rescaled heads, which are tangent by the Euclidean theory.

Proof of claim 1. Fix $\mu\in\mathcal{Q}_{c}$, with a density $h$ with respect to $\gamma_{c}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities: $h:X\to\mathbb{R}$ is Borel, $h\ge0$, and $\mu(A)=\int_{X}\mathbf{1}_{A}\,h\,d\gamma_{c}$ for every $A\in\mathcal{B}(X)$. Fix $\nu\in\mathcal{P}^{a}_{\rho}$ and a noise-optimal map $T$ from $\mu$ to $\nu$. 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, $\mu,\nu\in\mathcal{P}_{2}(X)$, and 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 the pair $(\mu,\nu)$ is noise-connected. For $n\in\mathbb{N}$ let $r_{n}$ be the rescaled head map and $\Lambda_{n}$ the lift of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space, and put $\tilde{\mu}_{n}=(r_{n})_{\#}\mu$ and $\tilde{\nu}_{n}=(r_{n})_{\#}\nu$. By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head, applied to $\mu$ and to $\nu$, $r_{n}$ is continuous and Borel and $\tilde{\mu}_{n},\tilde{\nu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$. Write $\lambda_{n}$ for Lebesgue measure on $\mathcal{B}(\mathbb{R}^{n})$, the measure written $\lambda_{d}$ in Absolutely Continuous Probability Measure on Euclidean Space and Diagonal Gaussian Measures on Euclidean Space with $d=n$. Euclidean results below are read with $n$ in the role of the dimension $d$.

Step 1 (The heads of $\mu$ are absolutely continuous). Fix $n\in\mathbb{N}$. Let $R_{n},D_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the maps $u\mapsto Mu$ for the diagonal $n\times n$ matrices $M$ with diagonal entries $a_{k}^{-1/2}$, respectively $a_{k}^{1/2}$ ($k\le n$), and off-diagonal entries $0$. They are Borel by Lebesgue Measure under a Triangular or Symmetric Positive Definite Linear Map §borel, and by the matrix-vector product of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, $R_{n}(u)=(a_{1}^{-1/2}u_{1},\dots,a_{n}^{-1/2}u_{n})$ and $D_{n}(u)=(a_{1}^{1/2}u_{1},\dots,a_{n}^{1/2}u_{n})$. Hence, comparing with the formula for $r_{n}$ and with the coordinate map $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and using $a_{k}^{1/2}a_{k}^{-1/2}=1$,

$$r_{n}=R_{n}\circ p_{n},\qquad p_{n}=D_{n}\circ r_{n}.$$

The noise weights are positive by the definition of a weight sequence (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights), so the matrix of $R_{n}$ is lower triangular with positive diagonal entries $a_{k}^{-1/2}$. Let $B\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(B)=0$, and let $B'=R_{n}^{-1}(B)\in\mathcal{B}(\mathbb{R}^{n})$. Applying Lebesgue Measure under a Triangular or Symmetric Positive Definite Linear Map §triangular to $R_{n}$ and $f=\mathbf{1}_{B}$, for which $f\circ R_{n}=\mathbf{1}_{B'}$, and evaluating the integrals of indicators by The Integral of an Indicator Function is the Measure of the Set, gives $(\det M_{n})\,\lambda_{n}(B')=\lambda_{n}(B)=0$ with $0<\det M_{n}$, where $M_{n}$ is the matrix of $R_{n}$, so $\lambda_{n}(B')=0$ by the conventions of Measure Spaces and the Lebesgue Integral: Standing Notation §extended. The truncation $c^{(n)}$ is a variance vector by Variance Sequences and Their Truncations §truncations; write $g_{n}=\rho_{c^{(n)}}:\mathbb{R}^{n}\to\mathbb{R}$ for the diagonal Gaussian density with variances $c^{(n)}$, a density on $\mathbb{R}^{n}$ unrelated to the reference measure $\rho$. It is positive and Borel by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity, so $\mathbf{1}_{B'}g_{n}\ge0$ is Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and it vanishes off the $\lambda_{n}$-null set $B'$; so by Diagonal Gaussian Measures on Euclidean Space §measure and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral

$$\gamma_{c^{(n)}}(B')=\int_{\mathbb{R}^{n}}\mathbf{1}_{B'}\,g_{n}\,d\lambda_{n}=0 .$$

Let $A=p_{n}^{-1}(B')$, which belongs to $\mathcal{B}(X)$ since $p_{n}$ is Borel (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity) and equals $r_{n}^{-1}(B)$ since $r_{n}=R_{n}\circ p_{n}$. By Diagonal Gaussian Measures on a Hilbert Space §measure, $\gamma_{c}(A)=((p_{n})_{\#}\gamma_{c})(B')=\gamma_{c^{(n)}}(B')=0$. The nonnegative function $\mathbf{1}_{A}h$, Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, vanishes off the $\gamma_{c}$-null set $A$, so by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral

$$\tilde{\mu}_{n}(B)=\mu\bigl(r_{n}^{-1}(B)\bigr)=\mu(A)=\int_{X}\mathbf{1}_{A}\,h\,d\gamma_{c}=0 .$$

Hence $\tilde{\mu}_{n}$ is absolutely continuous.

Step 2 (Euclidean optimal maps and their lifts). For each $n\in\mathbb{N}$, Step 1 and Out of an Absolutely Continuous Measure the Optimal Map and the Optimal Displacement are Tangent §uniquely-mapped, applied to $\tilde{\mu}_{n},\tilde{\nu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$, give an optimal map from $\tilde{\mu}_{n}$ to $\tilde{\nu}_{n}$; for each $n\in\mathbb{N}$ fix one such map $S_{n}$ (one choice for each $n$). This fixes the sequences $(S_{n})_{n\in\mathbb{N}}$, $(\eta_{n})_{n\in\mathbb{N}}$ and $(\xi_{n})_{n\in\mathbb{N}}$ defined below, and with them the sequences $(\beta_{n})_{n\in\mathbb{N}}$ and $(\theta_{n})_{n\in\mathbb{N}}$ of Step 4 used in Steps 5 and 6. Now fix $n\in\mathbb{N}$. By the definition of an optimal map, $S_{n}$ is Borel, $(S_{n})_{\#}\tilde{\mu}_{n}=\tilde{\nu}_{n}$, and the coupling $(\mathrm{id},S_{n})_{\#}\tilde{\mu}_{n}$ is optimal, that is, by Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal, its quadratic cost is $w_{n}^{2}$, where $w_{n}=W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})$ is the quadratic Wasserstein distance. Let $\eta_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $\eta_{n}(u)=S_{n}(u)-u$. It is the composite of the Borel pairing $(\mathrm{id},S_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n+n}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing with the map $z\mapsto\mathrm{pr}_{2}(z)-\mathrm{pr}_{1}(z)$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, whose $k$-th component $z\mapsto z_{n+k}-z_{k}$ is continuous, since $|(z_{n+k}-z_{k})-(z'_{n+k}-z'_{k})|\le2\lVert z-z'\rVert$; so $\eta_{n}$ is Borel by claims 3(b) and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Since $\lVert\eta_{n}(u)\rVert^{2}=\lVert u-S_{n}(u)\rVert^{2}$ is the value at $(\mathrm{id},S_{n})(u)$ of $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$, the change-of-variables formula gives

$$\int_{\mathbb{R}^{n}}\lVert\eta_{n}\rVert^{2}\,d\tilde{\mu}_{n}=I\bigl((\mathrm{id},S_{n})_{\#}\tilde{\mu}_{n}\bigr)=w_{n}^{2}<\infty. \tag{2.1}$$

By Out of an Absolutely Continuous Measure the Optimal Map and the Optimal Displacement are Tangent §tangent, $\mathrm{id}-S_{n}\in T_{\tilde{\mu}_{n}}$; since $T_{\tilde{\mu}_{n}}$ is a linear subspace of $L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed, it contains $(-1)(\mathrm{id}-S_{n})$, which is the class of $\eta_{n}$. Let $\xi_{n}=\Lambda_{n}\eta_{n}:X\to X$. By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift, which applies by (2.1), $\xi_{n}$ is Borel from $X$ to $X$, $\xi_{n}(x)\in X^{a}$ and $|\xi_{n}(x)|_{a}^{2}=\lVert\eta_{n}(r_{n}(x))\rVert^{2}$ for every $x\in X$; and $\lVert\xi_{n}\rVert_{\mu}=\lVert\eta_{n}\rVert_{\tilde{\mu}_{n}}$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift, with both arguments equal to $\eta_{n}$ in its inner-product identity. As $n_{a}=|\cdot|_{a}^{2}$ on $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel), the change-of-variables formula, the norm of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu and (2.1) give

$$\int_{X}n_{a}\bigl(\xi_{n}(x)\bigr)\,\mu(dx)=\int_{X}\lVert\eta_{n}(r_{n}(x))\rVert^{2}\,\mu(dx)=w_{n}^{2},\qquad\lVert\xi_{n}\rVert_{\mu}=w_{n}. \tag{2.2}$$

Finally $\xi_{n}\in T^{a}_{\mu}$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §tangent.

Step 3 (The bound $w_{n}\le W_{a}(\mu,\nu)$). 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 §optimal there is a noise-optimal coupling $\pi\in\Pi^{a}(\mu,\nu)$, so $I^{a}(\pi)=W_{a}(\mu,\nu)^{2}$. Fix $n\in\mathbb{N}$. The maps $r_{n}\circ\pi_{1},r_{n}\circ\pi_{2}:X\times X\to\mathbb{R}^{n}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma and claim 4 of Borel Measurability and Bounded Integration on a Metric Space, so their pairing $\Phi_{n}=(r_{n}\circ\pi_{1},r_{n}\circ\pi_{2}):X\times X\to\mathbb{R}^{n+n}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing; let $\omega_{n}=(\Phi_{n})_{\#}\pi$. Since $\mathrm{pr}_{i}\circ\Phi_{n}=r_{n}\circ\pi_{i}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections), and the image measure under a composite $g\circ f$ is the image under $g$ of the image under $f$ (as $(g\circ f)^{-1}(E)=f^{-1}(g^{-1}(E))$, push-forwards), $(\mathrm{pr}_{1})_{\#}\omega_{n}=(r_{n})_{\#}(\pi_{1})_{\#}\pi=\tilde{\mu}_{n}$ and likewise $(\mathrm{pr}_{2})_{\#}\omega_{n}=\tilde{\nu}_{n}$; so $\omega_{n}\in\Pi(\tilde{\mu}_{n},\tilde{\nu}_{n})$ is a coupling. For $z\in X\times X$ the $k$-th component of $r_{n}(y)-r_{n}(x)$ is $a_{k}^{-1/2}(y_{k}-x_{k})$, and $y_{k}-x_{k}$ is the $k$-th coordinate of $y-x$; so by the definition of the Euclidean norm

$$\lVert r_{n}(x)-r_{n}(y)\rVert^{2}=\sum_{k=1}^{n}a_{k}^{-1}(y-x)_{k}^{2}=\Sigma_{n}(y-x),$$

where $\Sigma_{n}$ is the function written $S_{N}$, with $N=n$, in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability. The function $z\mapsto\lVert r_{n}(x)-r_{n}(y)\rVert^{2}$ on $X\times X$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to the Borel maps $r_{n}\circ\pi_{1}$ and $r_{n}\circ\pi_{2}$. For $z\in D_{a}$ one has $y-x\in X^{a}$, so The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums gives $\Sigma_{n}(y-x)\le|y-x|_{a}^{2}=c_{a}(z)$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs). As $\pi(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, the complement of $D_{a}$ is $\pi$-null, and the change-of-variables formula, The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and The Quadratic Wasserstein Distance on Euclidean Space §distance give

$$w_{n}^{2}\le I(\omega_{n})=\int_{X\times X}\lVert r_{n}(x)-r_{n}(y)\rVert^{2}\,\pi(dz)\le\int_{X\times X}c_{a}\,d\pi=I^{a}(\pi)=W_{a}(\mu,\nu)^{2}.$$

Taking nonnegative square roots, $w_{n}\le W_{a}(\mu,\nu)$ for every $n\in\mathbb{N}$.

Step 4 (Head-matched targets converge to $\nu$). Fix $n\in\mathbb{N}$. The map $\mathrm{id}+\xi_{n}$ is Borel, as the composite of the Borel pairing $(\mathrm{id},\xi_{n})$ (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing) with the continuous addition of $X$ (claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space). Let $\beta_{n}=(\mathrm{id}+\xi_{n})_{\#}\mu\in\mathcal{P}(X)$ and $\theta_{n}=(\mathrm{id},\mathrm{id}+\xi_{n})_{\#}\mu$. Since $(\mathrm{id}+\xi_{n})(x)-x=\xi_{n}(x)\in X^{a}$ and, by (2.2), $\int_{X}n_{a}(\xi_{n}(x))\,\mu(dx)=w_{n}^{2}<\infty$, Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement gives, with Step 3,

$$\theta_{n}\in\Pi^{a}(\mu,\beta_{n}),\qquad I^{a}(\theta_{n})=w_{n}^{2}\le W_{a}(\mu,\nu)^{2}. \tag{4.1}$$

As $\mu\in\mathcal{P}_{2}(X)$, Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound gives $\beta_{n}\in\mathcal{P}_{2}(X)$.

Heads: by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §shift, $r_{n}(x+\xi_{n}(x))=r_{n}(x)+\eta_{n}(r_{n}(x))=S_{n}(r_{n}(x))$ for every $x\in X$, so $r_{n}\circ(\mathrm{id}+\xi_{n})=S_{n}\circ r_{n}$ and, push-forwards under composites being iterated push-forwards as in Step 3, $(r_{n})_{\#}\beta_{n}=(S_{n})_{\#}\tilde{\mu}_{n}=\tilde{\nu}_{n}=(r_{n})_{\#}\nu$. Since $p_{n}=D_{n}\circ r_{n}$ (Step 1), $(p_{n})_{\#}\beta_{n}=(D_{n})_{\#}(r_{n})_{\#}\beta_{n}=(D_{n})_{\#}(r_{n})_{\#}\nu=(p_{n})_{\#}\nu$.

Tails: by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §shift, $Q_{n}(x+\xi_{n}(x))=Q_{n}x$, so by the change-of-variables formula $\int_{X}|Q_{n}y|^{2}\,\beta_{n}(dy)=\int_{X}|Q_{n}x|^{2}\,\mu(dx)$, the integrand being continuous, hence Borel, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. The sequence $(X_{m})_{m\in\mathbb{N}}$ is exhausting for $X$ and $P_{m}$ is the orthogonal projection onto $X_{m}$, by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates; so for every $x\in X$, $(Q_{m}x)_{m}$ converges to $0_{X}$ by Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections §tail, that is, $|Q_{m}x|\to0$, and hence $|Q_{m}x|^{2}\to0$. Moreover $|Q_{m}x|^{2}\le|x|^{2}$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and $x\mapsto|x|^{2}$ is integrable with respect to $\mu$ because $\mu\in\mathcal{P}_{2}(X)$ (The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, Measure Spaces and the Lebesgue Integral: Standing Notation §integral). By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated, with $f_{m}(x)=|Q_{m}x|^{2}$, $f=0$ and $g(x)=|x|^{2}$,

$$\lim_{n\to\infty}\int_{X}|Q_{n}y|^{2}\,\beta_{n}(dy)=\lim_{n\to\infty}\int_{X}|Q_{n}x|^{2}\,\mu(dx)=0 .$$

Hence Measures Whose Heads Agree with a Fixed Measure and Whose Tails Vanish Converge in the Quadratic Wasserstein Distance §convergence, applied to $\nu$ and to the sequence $(\beta_{n})_{n\in\mathbb{N}}$ in the role of $(\lambda_{n})_{n\in\mathbb{N}}$, gives

$$\lim_{n\to\infty}W_{2}(\beta_{n},\nu)=0 . \tag{4.2}$$

Step 5 (A cluster point of the couplings $\theta_{n}$ is the graph coupling of $T$). By (4.2) and Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness §weak, $\beta_{n}\Rightarrow\nu$, so the sequence $(\beta_{n})_{n\in\mathbb{N}}$, that is, the set $\{\beta_{n}:n\in\mathbb{N}\}$, is tight by Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §weakly-convergent and the definition of a tight sequence. The set $\{\mu\}$ is tight by Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight, $(X,d)$ being complete and separable by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight, with $\{\mu\}$ and $\{\beta_{n}:n\in\mathbb{N}\}$ as the two tight sets, the set of all couplings of $\mu$ with some $\beta_{n}$ is tight in $X\times X$; it contains every $\theta_{n}$ by (4.1), and a subset of a tight set is tight, the same compact sets serving (Tight Family of Borel Measures on a Metric Space §tight). So $(\theta_{n})_{n\in\mathbb{N}}$ is a tight sequence of Borel probability measures on the metric space $(X\times X,d)$, and Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence, applied to this whole sequence, gives a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and $\theta\in\mathcal{P}(X\times X)$ with $\theta_{n_{j}}\Rightarrow\theta$; the subsequence $(n_{j})_{j\in\mathbb{N}}$ and the limit coupling $\theta$ are thus chosen after all the couplings $\theta_{n}$, $n\in\mathbb{N}$.

By (4.2), read in the metric space $(\mathbb{R},d_{\mathbb{R}})$, and A Subsequence of a Convergent Sequence Has the Same Limit, $\lim_{j\to\infty}W_{2}(\beta_{n_{j}},\nu)=0$, so $\beta_{n_{j}}\Rightarrow\nu$ by Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness §weak applied to the sequence $(\beta_{n_{j}})_{j\in\mathbb{N}}$. By (4.1) the sequence $(I^{a}(\theta_{n_{j}}))_{j\in\mathbb{N}}$ lies in $[0,W_{a}(\mu,\nu)^{2}]$, so it is bounded, and Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §lsc, with $\mu_{j}=\mu$ (a constant sequence, which converges weakly to $\mu$ by Weak Convergence of Finite Borel Measures on a Metric Space, its integrals being constant), $\nu_{j}=\beta_{n_{j}}$ and $\pi_{j}=\theta_{n_{j}}$, gives $\theta\in\Pi^{a}(\mu,\nu)$ and $I^{a}(\theta)\le\liminf_{j}I^{a}(\theta_{n_{j}})$. Each term $I^{a}(\theta_{n_{j}})$ is at most $W_{a}(\mu,\nu)^{2}$, so their limit inferior is at most $W_{a}(\mu,\nu)^{2}$. Together with $W_{a}(\mu,\nu)^{2}\le I^{a}(\theta)$ from The Noise Wasserstein Distance §distance, this gives $I^{a}(\theta)=W_{a}(\mu,\nu)^{2}$: $\theta$ is noise-optimal. By Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map §uniquely-mapped, which applies since $\mu$ has a density with respect to $\gamma_{c}$, the pair $(\mu,\nu)$ is uniquely noise-mapped: 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 $(\mathrm{id},T_{0})_{\#}\mu$. The coupling $(\mathrm{id},T)_{\#}\mu$ is noise-optimal by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map; hence

$$\theta=(\mathrm{id},T_{0})_{\#}\mu=(\mathrm{id},T)_{\#}\mu .$$

Step 6 (Strong convergence of the lifted displacements, and tangency). Let $v=T-\mathrm{id}$ and $v_{j}=\xi_{n_{j}}$ for $j\in\mathbb{N}$. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map and its preamble, $v$ is Borel, $v(x)\in X^{a}$ for every $x$ and $\int_{X}n_{a}(v(x))\,\mu(dx)<\infty$; by Step 2 and (2.2) each $v_{j}$ is Borel with values in $X^{a}$ and $\int_{X}n_{a}(v_{j}(x))\,\mu(dx)=w_{n_{j}}^{2}<\infty$. So $v$ and every $v_{j}$ are noise displacements for $\mu$ in the sense of Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms. Since $\mathrm{id}+v=T$ and $\mathrm{id}+v_{j}=\mathrm{id}+\xi_{n_{j}}$, Step 5 reads

$$(\mathrm{id},\mathrm{id}+v_{j})_{\#}\mu=\theta_{n_{j}}\Rightarrow\theta=(\mathrm{id},\mathrm{id}+v)_{\#}\mu .$$

By (2.2), Step 3 and The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source §cost (taking nonnegative square roots), $\lVert v_{j}\rVert_{\mu}=w_{n_{j}}\le W_{a}(\mu,\nu)=\lVert v\rVert_{\mu}$ for every $j$, so the norm hypothesis holds with $J=1$ for every positive $\varepsilon$. Therefore Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms §strong-displacement gives $\lim_{j\to\infty}\lVert\xi_{n_{j}}-(T-\mathrm{id})\rVert_{\mu}=0$: the sequence $(\xi_{n_{j}})_{j\in\mathbb{N}}$, which lies in $T^{a}_{\mu}$ by Step 2, converges to $T-\mathrm{id}$ in the metric space $L^{2}(\mu;X^{a})$ (Real Hilbert Space §topology). As $T^{a}_{\mu}$ is closed in $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace, Sequential Characterization of Closed Subsets of a Metric Space gives $T-\mathrm{id}\in T^{a}_{\mu}$, proving claim 1.

Proof of claim 2. By definition $\mathcal{Q}_{c}\subseteq\mathcal{P}^{a}_{\rho}$. Let $\mu\in\mathcal{Q}_{c}$ and $\nu\in\mathcal{P}^{a}_{\rho}$. Since $\mu$ has a density with respect to $\gamma_{c}$, the pair $(\mu,\nu)$ is uniquely noise-mapped by Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map §uniquely-mapped; in particular, by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped, there is a noise-optimal map $T$ from $\mu$ to $\nu$, and every such $T$ satisfies $T-\mathrm{id}\in T^{a}_{\mu}$ by claim 1. Hence $\mathcal{Q}_{c}$ has the noise map property of The Noise Map Property of a Set of Probability Measures §map-property.