Proof of The Gaussian Entropy Pair Has Closed Score Along Noise Couplings

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

Write $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ for the Gaussian entropy pair with temperature $\beta$ of the statement. By Noise Penalty Pairs with Closed Score Along Noise Couplings §closed, with sequences of couplings of vanishing noise cost and weak convergence along them those of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings and Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §weak, we must show the following. Let $R\in\mathbb{R}$ be nonnegative, let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$ for every $n\in\mathbb{N}$, let $\nu\in\mathcal{D}$, and let $(\pi_{n})_{n\in\mathbb{N}}$ satisfy $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$ for every $n$ and $\lim_{n\to\infty}I^{a}(\pi_{n})=0$. Then $\nu\in\mathcal{D}_{\Sigma}$, and $\lim_{n\to\infty}\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta,\pi_{n})=\langle\Sigma(\nu),\eta\rangle_{\nu}$ for every $\eta\in L^{2}(\nu;X^{a})$.

Notation. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain, $\nu$ and every $\nu_{n}$ lie in $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and $\mathcal{D}\subseteq\mathcal{P}_{2}(X)$. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, each $\nu_{n}$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$; let $Z_{n}=Z^{a}_{\nu_{n}}\in L^{2}(\nu_{n};X^{a})$ be its noise score field of The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, so that $\Sigma(\nu_{n})=\beta Z_{n}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair. Put $R'=R/\beta$. Since $\lVert\cdot\rVert_{\nu_{n}}$ is the norm of the real Hilbert space $L^{2}(\nu_{n};X^{a})$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, $\beta\lVert Z_{n}\rVert_{\nu_{n}}=\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$, so $\lVert Z_{n}\rVert_{\nu_{n}}\le R'$. As in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, a point $z\in X\times X$ is written $z=(x,y)$ with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$. For $\sigma\in\mathcal{P}(X)$, $v\in L^{2}(\sigma;X^{a})$ and $i\in\mathbb{N}$, $v_{i}\in L^{2}(\sigma)$ is the coordinate of $v$ along $f_{i}$ of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates. Limits are taken as $n\to\infty$.

Step 1 (integrals of functions of quadratic growth). We show: if $h:X\to\mathbb{R}$ is continuous and there is $A\in\mathbb{R}$ with $|h(x)|\le A(1+|x|^{2})$ for every $x\in X$, then $h$ is integrable with respect to $\nu$ and every $\nu_{n}$, and $\lim\int_{X}h\,d\nu_{n}=\int_{X}h\,d\nu$. Since $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$, the pair $(\nu_{n},\nu)$ is noise-connected by Couplings of Finite Noise Cost and Their Noise Cost §connected, and $W_{a}(\nu_{n},\nu)^{2}\le I^{a}(\pi_{n})$ by The Noise Wasserstein Distance §distance. As $\nu_{n},\nu\in\mathcal{P}^{a}_{\rho}$, 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 §comparison gives $W_{2}(\nu_{n},\nu)\le\sqrt{\bar{a}}\,W_{a}(\nu_{n},\nu)$, hence $0\le W_{2}(\nu_{n},\nu)^{2}\le\bar{a}\,I^{a}(\pi_{n})$. Given a positive $\varepsilon\in\mathbb{R}$, $\bar{a}\,I^{a}(\pi_{n})<\varepsilon^{2}$ for all large $n$, and then $W_{2}(\nu_{n},\nu)<\varepsilon$; so $\lim W_{2}(\nu_{n},\nu)=0$, and the claim is 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 §quadratic.

Step 2 (cylindrical functions). Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$, the set of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with a representation $(m,\psi)$ in the sense of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation, and let $k\in\mathbb{N}$.

(a) Lipschitz bound. By Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, $\psi$ is of class $C^{1}$ on $\mathbb{R}^{m}$ and there are nonnegative reals $B,B_{1},\dots,B_{m}$ with $|\psi|\le B$ and $|\partial_{i}\psi|\le B_{i}$ on $\mathbb{R}^{m}$ for $i\in[m]$; put $L=\sum_{i=1}^{m}B_{i}$. Let $u,w\in\mathbb{R}^{m}$ and $h=u-w$. The set $\mathbb{R}^{m}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so claims 2 and 3 of Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set, applied with $\mathbb{R}^{m}$, $\psi$, the point $w$, the increment $h$ and the interval $[0,1]$, show that $F(\tau)=\psi(w+\tau h)$ is continuous on $[0,1]$ and differentiable at every $\tau\in(0,1)$ with $F'(\tau)=\sum_{i=1}^{m}\partial_{i}\psi(w+\tau h)\,h_{i}$. By Mean Value Theorem on a Closed Real Interval there is $\tau_{0}\in(0,1)$ with $\psi(u)-\psi(w)=F(1)-F(0)=F'(\tau_{0})$, so, by the coordinate bound $|h_{i}|\le\lVert h\rVert$,

$$|\psi(u)-\psi(w)|\le\sum_{i=1}^{m}B_{i}\,|h_{i}|\le L\,\lVert u-w\rVert .$$

Taking $u=p_{m}(x)$ and $w=p_{m}(y)$ and using $\lVert p_{m}(x)-p_{m}(y)\rVert\le|x-y|$ from Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, we get $|\varphi(x)-\varphi(y)|\le L\,|x-y|$ for all $x,y\in X$. Thus $\varphi$ is Lipschitz, hence continuous by A Lipschitz Map is Uniformly Continuous, and $|\varphi|\le B$.

(b) The field $\varphi f_{k}$. The map $\varphi f_{k}:X\to X^{a}$, $y\mapsto\varphi(y)f_{k}$, takes values in $X^{a}$, a linear subspace of $X$ containing $f_{k}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis. Its coordinate along $e_{i}$ is $\langle\varphi(y)f_{k},e_{i}\rangle=a_{k}^{1/2}\varphi(y)\langle e_{k},e_{i}\rangle$, which is $a_{k}^{1/2}\varphi(y)$ for $i=k$ and $0$ otherwise, a Borel function of $y$ 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; so $\varphi f_{k}$ is measurable into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable. By orthonormality of $(f_{i})_{i\in\mathbb{N}}$ in $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis), $\langle\varphi(y)f_{k},f_{i}\rangle_{a}$ is $\varphi(y)$ for $i=k$ and $0$ otherwise, and $|\varphi(y)f_{k}|_{a}^{2}=\varphi(y)^{2}\le B^{2}$. Hence, by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\varphi f_{k}$ is square-integrable with respect to every $\sigma\in\mathcal{P}(X)$; its class in $L^{2}(\sigma;X^{a})$, again written $\varphi f_{k}$, has coordinate $(\varphi f_{k})_{k}=\varphi$ and $(\varphi f_{k})_{i}=0$ for $i\ne k$.

(c) Discrepancy bound. Let $n\in\mathbb{N}$. Since $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)\subseteq\Pi(\nu_{n},\nu)$, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\nu_{n}$ and $\nu$ in place of its $\nu$ and $\mu$ and with $\varphi f_{k}$ in both slots, shows that the function $z\mapsto|\varphi(x)f_{k}-\varphi(y)f_{k}|_{a}^{2}=(\varphi(x)-\varphi(y))^{2}$ is Borel, nonnegative and integrable with respect to $\pi_{n}$. By (a) it is at most $L^{2}|x-y|^{2}$, so by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, the quadratic cost $I(\pi_{n})$ and the support bound $I(\pi_{n})\le\bar{a}\,I^{a}(\pi_{n})$ of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound,

$$D_{n}(\varphi):=\int_{X\times X}(\varphi(x)-\varphi(y))^{2}\,\pi_{n}(dz)\le L^{2}\,I(\pi_{n})\le L^{2}\,\bar{a}\,I^{a}(\pi_{n}) .$$

(d) Two functions of quadratic growth. Let $h_{1},h_{2}:X\to\mathbb{R}$ be $h_{1}(x)=x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)$ and $h_{2}=\varphi^{2}$, where $\partial_{k}\varphi$ is as in Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial. By that claim, $\partial_{k}\varphi=(\partial_{k}\psi)\circ p_{m}$ if $k\le m$ and $\partial_{k}\varphi=0$ if $k>m$. For $k\le m$, $\partial_{k}\psi$ is continuous at every point in the sense of clause 1 of C^k Maps on a Euclidean Open Set and Continuity at a Point for Maps Between Euclidean Spaces; since $\sum_{i=1}^{m}(y_{i}-y'_{i})^{2}<\delta^{2}$ is equivalent to $d_{E}(y,y')<\delta$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance, and $(s-s')^{2}<\varepsilon^{2}$ to $|s-s'|<\varepsilon$, it is continuous from $(\mathbb{R}^{m},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$; $p_{m}$ is Lipschitz by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous by A Lipschitz Map is Uniformly Continuous; so $\partial_{k}\varphi$ is continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. For $k>m$, $\partial_{k}\varphi=0$ is constant, hence continuous. The coordinate function $x\mapsto x_{k}$ is Lipschitz with constant $1$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous, and $|x_{k}|\le|x|$, its value at $0_{X}$ being $0$. Since $\varphi$ is continuous by (a), every sequence converging to a point $x$ of $X$ is carried by $x\mapsto x_{k}$, $\varphi$ and $\partial_{k}\varphi$ to sequences converging to their values at $x$ (Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential); by the limit laws of The Real Numbers: Standing Notation and Background §background the same holds for $h_{1}$ and $h_{2}$, which are therefore continuous by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset. 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 there is $B'\ge0$ with $|\partial_{k}\varphi|\le B'$; with $|x|\le1+|x|^{2}$ this gives $|h_{1}(x)|\le(B/c_{k}+B')(1+|x|^{2})$ ($c_{k}$ being positive by Variance Sequences and Their Truncations §variances) and $|h_{2}(x)|\le B^{2}(1+|x|^{2})$. So $h_{1}$ and $h_{2}$ satisfy the hypotheses of Step 1.

Step 3 ($\nu\in\mathcal{D}_{\Sigma}$). Let $m\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{m})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=m$. For $k\in[m]$, $\partial_{k}g$ is of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set and bounded with bounded partial derivatives by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, so $\partial_{k}g\in C^{1}_{b}(\mathbb{R}^{m})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded and $\varphi_{k}=(\partial_{k}g)\circ p_{m}\in\mathcal{F}C^{1}_{b}(X)$ has the representation $(m,\partial_{k}g)$; by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, $\partial_{k}\varphi_{k}=(\partial_{k}\partial_{k}g)\circ p_{m}$. Let $h^{(k)}_{1}$ and $h^{(k)}_{2}=\varphi_{k}^{2}$ be the functions of Step 2(d) for $\varphi_{k}$ and $k$. For $\mu\in\mathcal{P}_{2}(X)$, The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional reads

$$L^{a}_{\mu}(g)=\sum_{k=1}^{m}a_{k}\int_{X}h^{(k)}_{1}\,d\mu .$$

Also $g\in C^{1}_{b}(\mathbb{R}^{m})$ as recorded in The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §pairing-functional, $g\circ p_{m}$ has the representation $(m,g)$, and $\partial_{k}(g\circ p_{m})=\varphi_{k}$ for $k\le m$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial; so by the formula for $|\nabla_{a}(g\circ p_{m})|_{a}^{2}$ in The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, the norm of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and Linearity and Monotonicity of the Lebesgue Integral §integrable ($\varphi_{k}^{2}=h^{(k)}_{2}$ being continuous by Step 2(d), hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, and bounded, hence integrable by claim 6(b) of that lemma),

$$N(\mu)^{2}:=\lVert\nabla_{a}(g\circ p_{m})\rVert_{\mu}^{2}=\sum_{k=1}^{m}a_{k}\int_{X}h^{(k)}_{2}\,d\mu .$$

By Steps 1 and 2(d) and the limit laws, $\lim L^{a}_{\nu_{n}}(g)=L^{a}_{\nu}(g)$ and $\lim N(\nu_{n})^{2}=N(\nu)^{2}$. For each $n$, The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §pairing-functional applied to $\nu_{n}$ and The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu_{n};X^{a})$ give

$$L^{a}_{\nu_{n}}(g)^{2}=\langle Z_{n},\nabla_{a}(g\circ p_{m})\rangle_{\nu_{n}}^{2}\le\lVert Z_{n}\rVert_{\nu_{n}}^{2}\,N(\nu_{n})^{2}\le R'^{2}\,N(\nu_{n})^{2}.$$

Passing to the limit with the order of limits of The Real Numbers: Standing Notation and Background §background, $L^{a}_{\nu}(g)^{2}\le R'^{2}N(\nu)^{2}$, hence $|L^{a}_{\nu}(g)|\le R'\,\lVert\nabla_{a}(g\circ p_{m})\rVert_{\nu}$, both sides being nonnegative. As $m$ and $g$ were arbitrary and $\nu\in\mathcal{P}_{2}(X)$, A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information §fisher, applied with $\nu$ and $R'$ in place of its $\mu$ and $R$, shows that $\nu$ has a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, with $\mathcal{I}_{a}(\nu\,|\,\gamma_{c})\le R'^{2}$. Since $\nu\in\mathcal{D}$, The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain gives $\nu\in\mathcal{D}_{\Sigma}$. Let $Z=Z^{a}_{\nu}$ be its noise score field, so that $\Sigma(\nu)=\beta Z$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair; by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, $\lVert Z\rVert_{\nu}^{2}=\mathcal{I}_{a}(\nu\,|\,\gamma_{c})\le R'^{2}$, so $\lVert\Sigma(\nu)\rVert_{\nu}\le R$.

Step 4 (pairing a score field with $\varphi f_{k}$). Let $\sigma\in\mathcal{D}_{\Sigma}$ have relative score $(\zeta^{\sigma}_{i})_{i\in\mathbb{N}}$ and noise score field $Z^{a}_{\sigma}$, let $\varphi\in\mathcal{F}C^{1}_{b}(X)$, $k\in\mathbb{N}$, and let $h_{1}$ be as in Step 2(d). By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the coordinates of $\varphi f_{k}$ found in Step 2(b), and The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, which gives $(Z^{a}_{\sigma})_{k}=a_{k}^{1/2}\zeta^{\sigma}_{k}$, the series for $\langle Z^{a}_{\sigma},\varphi f_{k}\rangle_{\sigma}$ has the single nonzero term with index $k$, and by The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score

$$\langle Z^{a}_{\sigma},\varphi f_{k}\rangle_{\sigma}=a_{k}^{1/2}\langle\zeta^{\sigma}_{k},\varphi\rangle_{L^{2}(\sigma)}=a_{k}^{1/2}\int_{X}h_{1}\,d\sigma .$$

Step 5 (convergence on the fields $\varphi f_{k}$). Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, with $L$ as in Step 2(a). We show $\lim\mathcal{K}^{a}(\Sigma(\nu_{n}),\varphi f_{k},\pi_{n})=\langle\Sigma(\nu),\varphi f_{k}\rangle_{\nu}$. Fix $n$, and on $(X\times X,\mathcal{B}(X\times X),\pi_{n})$ consider the maps $V(z)=Z_{n}(x)$ and $W(z)=(\varphi(y)-\varphi(x))f_{k}$ into $X^{a}$, for a fixed representative of $Z_{n}$. They are measurable: $V=Z_{n}\circ\pi_{1}$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, $\pi_{1}$ and $\pi_{2}$ being Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma, and $W=(\varphi f_{k})\circ\pi_{2}-(\varphi f_{k})\circ\pi_{1}$ by the same claim 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. Since $(\pi_{1})_{\#}\pi_{n}=\nu_{n}$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling, claim 2 of Image Measures, Measures with Densities, and Change of Variables gives $\int|V|_{a}^{2}\,d\pi_{n}=\int_{X}|Z_{n}|_{a}^{2}\,d\nu_{n}=\lVert Z_{n}\rVert_{\nu_{n}}^{2}\le R'^{2}$, and it shows that $z\mapsto\langle Z_{n}(x),\varphi(x)f_{k}\rangle_{a}$ is integrable with respect to $\pi_{n}$ with integral $\langle Z_{n},\varphi f_{k}\rangle_{\nu_{n}}$, the function $x\mapsto\langle Z_{n}(x),\varphi(x)f_{k}\rangle_{a}$ being integrable with respect to $\nu_{n}$ with that integral by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations. Moreover $|W(z)|_{a}^{2}=(\varphi(y)-\varphi(x))^{2}$, so $\int|W|_{a}^{2}\,d\pi_{n}=D_{n}(\varphi)$ of Step 2(c). 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 §square-integrable, $\langle V,W\rangle_{a}$ is integrable with respect to $\pi_{n}$ and

$$|\Delta_{n}|\le R'\,D_{n}(\varphi)^{1/2}\le R'\,L\,\bigl(\bar{a}\,I^{a}(\pi_{n})\bigr)^{1/2},\qquad\Delta_{n}:=\int_{X\times X}\langle V,W\rangle_{a}\,d\pi_{n},$$

using Step 2(c). Since $\langle V(z),W(z)\rangle_{a}=\langle Z_{n}(x),\varphi(y)f_{k}\rangle_{a}-\langle Z_{n}(x),\varphi(x)f_{k}\rangle_{a}$ and the first term is integrable with integral $\mathcal{K}^{a}(Z_{n},\varphi f_{k},\pi_{n})$ by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, Linearity and Monotonicity of the Lebesgue Integral §integrable and Step 4 for $\sigma=\nu_{n}$ give

$$\mathcal{K}^{a}(Z_{n},\varphi f_{k},\pi_{n})=\langle Z_{n},\varphi f_{k}\rangle_{\nu_{n}}+\Delta_{n}=a_{k}^{1/2}\int_{X}h_{1}\,d\nu_{n}+\Delta_{n}.$$

Now $\lim\Delta_{n}=0$, because $\Delta_{n}^{2}\le R'^{2}L^{2}\bar{a}\,I^{a}(\pi_{n})$ and $\lim I^{a}(\pi_{n})=0$; and $\lim\int h_{1}\,d\nu_{n}=\int h_{1}\,d\nu$ by Steps 1 and 2(d). Since $\beta Z_{n}$ is a representative of $\Sigma(\nu_{n})$, the integrand of $\mathcal{K}^{a}(\Sigma(\nu_{n}),\varphi f_{k},\pi_{n})$ is $\beta$ times that of $\mathcal{K}^{a}(Z_{n},\varphi f_{k},\pi_{n})$; so by Linearity and Monotonicity of the Lebesgue Integral §integrable, the limit laws, Step 4 for $\sigma=\nu$ (legitimate by Step 3) and the linearity of the inner product of $L^{2}(\nu;X^{a})$,

$$\lim\mathcal{K}^{a}(\Sigma(\nu_{n}),\varphi f_{k},\pi_{n})=\beta\,a_{k}^{1/2}\int_{X}h_{1}\,d\nu=\beta\,\langle Z,\varphi f_{k}\rangle_{\nu}=\langle\Sigma(\nu),\varphi f_{k}\rangle_{\nu}.$$

Step 6 (approximation by finite sums). Let $\mathcal{S}$ be the set of the elements $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}$ of $L^{2}(\nu;X^{a})$ with $N\in\mathbb{N}$ and $\varphi_{1},\dots,\varphi_{N}\in\mathcal{F}C^{1}_{b}(X)$, each $\varphi_{i}f_{i}$ being the class of Step 2(b). We show: for every $\eta\in L^{2}(\nu;X^{a})$ and every positive $\varepsilon\in\mathbb{R}$ there is $\xi\in\mathcal{S}$ with $\lVert\eta-\xi\rVert_{\nu}<\varepsilon$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the series $\sum_{i=1}^{\infty}\lVert\eta_{i}\rVert_{L^{2}(\nu)}^{2}$ converges with sum $\lVert\eta\rVert_{\nu}^{2}$; with $s_{M}$ its partial sums, Series of Real Numbers §convergent gives $N\in\mathbb{N}$ with $\lVert\eta\rVert_{\nu}^{2}-s_{N}<\varepsilon^{2}/2$. For $i\in[N]$, the classes of the members of $\mathcal{F}C^{1}_{b}(X)$ are dense in $L^{2}(\nu)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density, so by the meaning of density and claim 3 of Characterization of the Closure in a Metric Space by Open Balls there is $\varphi_{i}\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert\eta_{i}-\varphi_{i}\rVert_{L^{2}(\nu)}^{2}<\varepsilon^{2}/(2N)$. Let $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}\in\mathcal{S}$. By Step 2(b) and the linearity of coordinates in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, $(\eta-\xi)_{l}=\eta_{l}-\varphi_{l}$ for $l\le N$ and $(\eta-\xi)_{l}=\eta_{l}$ for $l>N$. So for $M\ge N$ the $M$-th partial sum of the series $\sum_{l}\lVert(\eta-\xi)_{l}\rVert_{L^{2}(\nu)}^{2}$ equals $\sum_{l=1}^{N}\lVert\eta_{l}-\varphi_{l}\rVert_{L^{2}(\nu)}^{2}+s_{M}-s_{N}$, and letting $M\to\infty$, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates gives

$$\lVert\eta-\xi\rVert_{\nu}^{2}=\sum_{l=1}^{N}\lVert\eta_{l}-\varphi_{l}\rVert_{L^{2}(\nu)}^{2}+\lVert\eta\rVert_{\nu}^{2}-s_{N}<\frac{\varepsilon^{2}}{2}+\frac{\varepsilon^{2}}{2}=\varepsilon^{2}.$$

Step 7 (weak convergence). For $q\in L^{2}(\nu_{n};X^{a})$ the cross pairing $\mathcal{K}^{a}(q,\cdot,\pi_{n})$ is linear on $L^{2}(\nu;X^{a})$: representatives of sums and real multiples may be taken pointwise by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $\langle\cdot,\cdot\rangle_{a}$ is bilinear, all integrands are integrable by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, and the integral is linear by Linearity and Monotonicity of the Lebesgue Integral §integrable. Hence for $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}\in\mathcal{S}$, Step 5, the limit laws and the linearity of $\langle\Sigma(\nu),\cdot\rangle_{\nu}$ give $\lim\mathcal{K}^{a}(\Sigma(\nu_{n}),\xi,\pi_{n})=\sum_{i=1}^{N}\langle\Sigma(\nu),\varphi_{i}f_{i}\rangle_{\nu}=\langle\Sigma(\nu),\xi\rangle_{\nu}$.

Now let $\eta\in L^{2}(\nu;X^{a})$ and let $\varepsilon\in\mathbb{R}$ be positive. First choose, by Step 6, $\xi\in\mathcal{S}$ with $\lVert\eta-\xi\rVert_{\nu}<\varepsilon/(3(R+1))$; then, for this $\xi$, choose $n_{0}\in\mathbb{N}$ with $|\mathcal{K}^{a}(\Sigma(\nu_{n}),\xi,\pi_{n})-\langle\Sigma(\nu),\xi\rangle_{\nu}|<\varepsilon/3$ for every $n\ge n_{0}$. For $n\ge n_{0}$, linearity in the second slot, the choice of $n_{0}$ for the middle term, the bound Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross-bound with $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$, and The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu;X^{a})$ with $\lVert\Sigma(\nu)\rVert_{\nu}\le R$ from Step 3 give

$$\begin{aligned} \bigl|\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta,\pi_{n})-\langle\Sigma(\nu),\eta\rangle_{\nu}\bigr|&\le\bigl|\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta-\xi,\pi_{n})\bigr|+\bigl|\mathcal{K}^{a}(\Sigma(\nu_{n}),\xi,\pi_{n})-\langle\Sigma(\nu),\xi\rangle_{\nu}\bigr|+\bigl|\langle\Sigma(\nu),\xi-\eta\rangle_{\nu}\bigr|\\ &\le\frac{R\,\varepsilon}{3(R+1)}+\frac{\varepsilon}{3}+\frac{R\,\varepsilon}{3(R+1)}<\varepsilon . \end{aligned}$$

Hence $\lim\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta,\pi_{n})=\langle\Sigma(\nu),\eta\rangle_{\nu}$ for every $\eta\in L^{2}(\nu;X^{a})$: by Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §weak, $(\Sigma(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$. Together with $\nu\in\mathcal{D}_{\Sigma}$ from Step 3, this is the closed-score property of Noise Penalty Pairs with Closed Score Along Noise Couplings §closed.