Proof of The Gibbs Sobolev Space: Linearity, Closedness, the Chain Rule, and the Score of a Measure of Finite Intrinsic Fisher Information

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

We write $\nu=\gamma^{V}_{\beta}$ and $\mathbb{D}=\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$. By Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space §measure, $\nu$ is a Borel probability measure in $\mathcal{P}_{2}(X)$; by The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §gibbs it is the measure with density $Z_{V,\beta}^{-1}w_{V,\beta}$ with respect to $\gamma_{c}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables, so for a $\gamma_{c}$-null Borel set $N$ we have $\nu(N)=Z_{V,\beta}^{-1}\int_{X}\mathbf{1}_{N}w_{V,\beta}\,d\gamma_{c}=0$ by the homogeneity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral; that is, $\gamma_{c}$-null Borel sets are $\nu$-null. We fix a Borel representative of each $\partial_{k}V$ (Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights §partial); by what was just said (and, for $\mu$ below, by the same argument applied to its density relative to $\nu$) nothing depends on this choice. A function square-integrable with respect to a probability measure is integrable, by Hoelder's Inequality, for Two and for Finitely Many Factors §holder with exponents $2,2$ against the constant $1$; and the product of two members of $L^{2}(\nu)$ is $\nu$-integrable with integral their inner product, by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ the functions $\varphi$ and $\partial_{k}\varphi$ are Borel and bounded (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), and the $k$-th coordinate $\langle\nabla_{a}\varphi(x),e_{k}\rangle$ of $\nabla_{a}\varphi(x)$ is $a_{k}\partial_{k}\varphi(x)$ for every $k$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial). A map $v:X\to X^{a}$ is measurable if and only if every $x\mapsto\langle v(x),e_{k}\rangle$ is Borel (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable). (CB) For $y\in X^{a}$ and $k\in\mathbb{N}$, $(a_{k}^{-1}\langle y,e_{k}\rangle)^{2}=a_{k}^{-1}\cdot a_{k}^{-1}\langle y,e_{k}\rangle^{2}\le a_{k}^{-1}|y|_{a}^{2}$, as $a_{k}^{-1}\langle y,e_{k}\rangle^{2}$ is a term of the series of nonnegative terms with sum $|y|_{a}^{2}$ (The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates).

Step 0 (calculus on cylindrical functions). (P) Let $\varphi,\chi\in\mathcal{F}C^{1}_{b}(X)$ have representations $(n,\psi)$, $(m,\vartheta)$, say $n\le m$, and let $s\in\mathbb{R}$. Let $\pi:\mathbb{R}^{m}\to\mathbb{R}^{n}$ keep the first $n$ coordinates; its coordinate functions are smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), hence of class $C^{1}$ (Smooth Map on a Euclidean Open Set), so it is of class $C^{1}$ (clauses 3 and 1 of C^k Maps on a Euclidean Open Set), with $\partial_{i}\pi_{l}=1$ if $i=l$ and $0$ otherwise, its difference quotients in Partial Derivative on a Euclidean Open Set being constant. By claims 2 and 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, $\psi\circ\pi\in C^{1}_{b}(\mathbb{R}^{m})$ with $\partial_{i}(\psi\circ\pi)=(\partial_{i}\psi)\circ\pi$ for $i\le n$ and $0$ for $i>n$, and $(m,\psi\circ\pi)$ represents $\varphi$ since $p_{n}=\pi\circ p_{m}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates). By claims 3 and 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\psi\circ\pi+\vartheta$, $s\psi$ and $(\psi\circ\pi)\vartheta$ are of class $C^{1}$, and bounded, as $\psi$ and $\vartheta$ are, with the bounded partial derivatives $\partial_{i}(\psi\circ\pi)+\partial_{i}\vartheta$, $s\partial_{i}\psi$ and $\partial_{i}(\psi\circ\pi)\vartheta+(\psi\circ\pi)\partial_{i}\vartheta$. Hence $\varphi+\chi$, $s\varphi$, $\varphi\chi\in\mathcal{F}C^{1}_{b}(X)$, and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial applied to these representations gives $\partial_{k}(\varphi+\chi)=\partial_{k}\varphi+\partial_{k}\chi$, $\partial_{k}(s\varphi)=s\partial_{k}\varphi$ and $\partial_{k}(\varphi\chi)=\partial_{k}\varphi\,\chi+\varphi\,\partial_{k}\chi$ for all $k$; by the formula of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, $\nabla_{a}(\varphi+\chi)=\nabla_{a}\varphi+\nabla_{a}\chi$ and $\nabla_{a}(s\varphi)=s\nabla_{a}\varphi$. (C) Let $h:\mathbb{R}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{1}$ with $|h'|\le L$, $h'=\partial_{1}h$. Then $h$ and $h'$ are continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, clause 1 of C^k Maps on a Euclidean Open Set, Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §distance and Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §equivalent), hence Borel (claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space), and $h$ is differentiable at every point with derivative $h'$ (claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line); so Mean Value Theorem on a Closed Real Interval gives $|h(t)-h(r)|\le L|t-r|$ for all real $r,t$. If $\varphi\in\mathcal{F}C^{1}_{b}(X)$ has a representation $(n,\psi)$ with $|\psi|\le M$, then by claims 2 and 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ $h\circ\psi$ is of class $C^{1}$ with $\partial_{i}(h\circ\psi)=h'(\psi)\partial_{i}\psi$, bounded, and $|h\circ\psi|\le|h(0)|+LM$; so $h\circ\varphi\in\mathcal{F}C^{1}_{b}(X)$, $\partial_{k}(h\circ\varphi)=h'(\varphi)\partial_{k}\varphi$ 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, and $\nabla_{a}(h\circ\varphi)=h'(\varphi)\nabla_{a}\varphi$ pointwise.

Step 1 (claim 1). For real $t$ the constant function $t$ lies in $\mathcal{F}C^{1}_{b}(X)$ (Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear) with the representation $(1,\psi_{t})$, $\psi_{t}$ constant on $\mathbb{R}^{1}$, which is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), hence of class $C^{1}$ (Smooth Map on a Euclidean Open Set), and bounded, with $\partial_{1}\psi_{t}=0$, all its difference quotients vanishing (Partial Derivative on a Euclidean Open Set). 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, $\partial_{k}t=0$ for all $k$, so $\nabla_{a}t=0$. The constant sequence approximates $t$, so $t\in\mathbb{D}$ and $\nabla^{V}_{a}t=\nabla_{a}t$ is the zero vector of $L^{2}(\nu;X^{a})$ (The Gibbs Sobolev Space of the Noise Gradient §gradient, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert). Let $F,F'\in\mathbb{D}$ have approximating sequences $(F_{j})$, $(F'_{j})$, and let $s,s'\in\mathbb{R}$. By (P), $sF_{j}+s'F'_{j}\in\mathcal{F}C^{1}_{b}(X)$ and $\nabla_{a}(sF_{j}+s'F'_{j})=s\nabla_{a}F_{j}+s'\nabla_{a}F'_{j}$, which by The Gibbs Sobolev Space of the Noise Gradient §gradient and the triangle inequality and homogeneity of the norm converges to $s\nabla^{V}_{a}F+s'\nabla^{V}_{a}F'$, hence is Cauchy; and $\lVert sF_{j}+s'F'_{j}-(sF+s'F')\rVert_{L^{2}(\nu)}\le|s|\lVert F_{j}-F\rVert_{L^{2}(\nu)}+|s'|\lVert F'_{j}-F'\rVert_{L^{2}(\nu)}\to0$. By The Gibbs Sobolev Space of the Noise Gradient §space and The Gibbs Sobolev Space of the Noise Gradient §gradient, $sF+s'F'\in\mathbb{D}$ and $\nabla^{V}_{a}(sF+s'F')=s\nabla^{V}_{a}F+s'\nabla^{V}_{a}F'$.

Step 2 (claim 2). For each $j$, an approximating sequence of $F_{j}$ converges to $F_{j}$ in $L^{2}(\nu)$ and its noise gradients converge to $\nabla^{V}_{a}F_{j}$ (The Gibbs Sobolev Space of the Noise Gradient §space, The Gibbs Sobolev Space of the Noise Gradient §gradient); so we may choose $\varphi_{j}\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert\varphi_{j}-F_{j}\rVert_{L^{2}(\nu)}<1/j$ and $\lVert\nabla_{a}\varphi_{j}-\nabla^{V}_{a}F_{j}\rVert_{\nu}<1/j$. Then $\lVert\varphi_{j}-F\rVert_{L^{2}(\nu)}<1/j+\lVert F_{j}-F\rVert_{L^{2}(\nu)}\to0$ and $\lVert\nabla_{a}\varphi_{j}-G\rVert_{\nu}<1/j+\lVert\nabla^{V}_{a}F_{j}-G\rVert_{\nu}\to0$. A convergent sequence is Cauchy, so $(\varphi_{j})$ approximates $F$; thus $F\in\mathbb{D}$ and $\nabla^{V}_{a}F=G$.

Step 3 (claim 3). Let $L$ bound $|h'|$ and let $(\varphi_{j})$ approximate $F$. By (C), $h\circ f$ is Borel (claim 4 of Borel Measurability and Bounded Integration on a Metric Space) with $|h\circ f|\le|h(0)|+L|f|$, so $(h\circ f)^{2}\le2h(0)^{2}+2L^{2}f^{2}$ is $\nu$-integrable; $h\circ\varphi_{j}\in\mathcal{F}C^{1}_{b}(X)$ with $\nabla_{a}(h\circ\varphi_{j})=h'(\varphi_{j})\nabla_{a}\varphi_{j}$; and $\lVert h\circ\varphi_{j}-h\circ f\rVert_{L^{2}(\nu)}\le L\lVert\varphi_{j}-f\rVert_{L^{2}(\nu)}\to0$. By The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §subsequence (with $p=2$, measure $\nu$) we may pass to a subsequence along which $\varphi_{j}\to f$ off a $\nu$-null set. The maps $h'(f)g$, $h'(\varphi_{j})(\nabla_{a}\varphi_{j}-g)$ and $(h'(\varphi_{j})-h'(f))g$ are measurable into $X^{a}$, their coordinates being products of Borel functions, and square-integrable, as $|t\,y|_{a}^{2}=t^{2}|y|_{a}^{2}$. By the triangle inequality of $\lVert\cdot\rVert_{\nu}$ and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations,

$$\lVert h'(\varphi_{j})\nabla_{a}\varphi_{j}-h'(f)g\rVert_{\nu}\le L\lVert\nabla_{a}\varphi_{j}-g\rVert_{\nu}+\Bigl(\int_{X}(h'(\varphi_{j})-h'(f))^{2}|g|_{a}^{2}\,d\nu\Bigr)^{1/2}.$$

The first term tends to $0$ by The Gibbs Sobolev Space of the Noise Gradient §gradient; the integrand of the second tends to $0$ $\nu$-almost everywhere by the continuity of $h'$ (Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §sequential) and is bounded by the integrable $4L^{2}|g|_{a}^{2}$, so the second term tends to $0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated. Thus $(h\circ\varphi_{j})$, along the subsequence, approximates the class of $h\circ f$ and its noise gradients converge to the class of $h'(f)g$, which proves claim 3.

Step 4 (claim 4). The constant function $1$ is a Borel density of $\nu$ relative to $\nu$, since $\nu(A)=\int_{X}\mathbf{1}_{A}\cdot1\,d\nu$, and $\sqrt{1}=1$ lies in $\mathbb{D}$ with $\nabla^{V}_{a}1=0$ by Step 1. So $\nu$ has finite intrinsic Fisher information by The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential §information, and $\mathcal{I}^{V}_{a}(\nu)=4\mathcal{E}^{V}_{a}(1,1)=4\langle0,0\rangle_{\nu}=0$ by The Gibbs Dirichlet Form of the Noise Gradient §form.

Step 5 (the gradient of $\sqrt{f}$ vanishes where $f$ does). Let $f,g$ be as in claims 5 to 7, put $u=\sqrt{f}\ge0$ (Borel by The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential §information) and $U$ its class in $\mathbb{D}$. For $m\in\mathbb{N}$ let $h_{m}=m\phi_{m}$, where $\phi_{m}$ is the function of A Bounded-Derivative Truncation of the Cube Map on the Real Line §defined with $\varepsilon=m$, so $h_{m}(s)=s^{3}/(1/m+s^{2})$. By A Bounded-Derivative Truncation of the Cube Map on the Real Line §derivative and claims 3 and 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $h_{m}$ is of class $C^{1}$ on $\mathbb{R}^{1}$ with $h_{m}'(s)=m(3s^{2}+ms^{4})(1+ms^{2})^{-2}$, which lies in $[0,4]$ by A Bounded-Derivative Truncation of the Cube Map on the Real Line §derivative-bound. Moreover $h_{m}'(0)=0$; for $s\ne0$, $h_{m}'(s)=(3s^{2}/m+s^{4})(1/m+s^{2})^{-2}\to1$ as $m\to\infty$; and $|h_{m}(s)|\le|s|$ and $|h_{m}(s)-s|=|s|\,m^{-1}(1/m+s^{2})^{-1}\to0$ for every $s$. By claim 3 (Step 3) with $u$ in place of its $f$, the class of $h_{m}\circ u$ lies in $\mathbb{D}$ with gradient the class of $h_{m}'(u)g$. As $m\to\infty$: $(h_{m}(u)-u)^{2}\le4u^{2}$, with $\int_{X}u^{2}\,d\nu=1$, and $(h_{m}(u)-u)^{2}\to0$ pointwise; and with $g^{+}=\mathbf{1}_{\{f>0\}}g$ (measurable and square-integrable), $|h_{m}'(u)g-g^{+}|_{a}^{2}=(h_{m}'(u)-\mathbf{1}_{\{u>0\}})^{2}|g|_{a}^{2}\le16|g|_{a}^{2}$ tends to $0$ pointwise, as $\{u>0\}=\{f>0\}$. By Dominated Convergence Theorem the classes of $h_{m}\circ u$ converge to $U$ in $L^{2}(\nu)$ and their gradients to the class of $g^{+}$; by claim 2 (Step 2), $\nabla^{V}_{a}U$ is the class of $g^{+}$, so $g=g^{+}$ off a $\nu$-null set (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes), that is, $g(x)=0$ for $\nu$-almost every $x$ with $f(x)=0$.

Step 6 (claims 5 and 6). By The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential §information, $\mu$ is the measure with density $f$ relative to $\nu$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables; so $\int_{X}\varrho\,d\mu=\int_{X}\varrho f\,d\nu$ for Borel $\varrho\ge0$, and a Borel $\varrho$ is $\mu$-integrable if and only if $\varrho f$ is $\nu$-integrable, with the same identity. Let $r(x)=1/\sqrt{f(x)}$ if $f(x)>0$ and $r(x)=0$ otherwise; for real $t$ the set of $x$ with $r(x)>t$ is $X$ if $t<0$, is $\{f>0\}$ if $t=0$, and is $\{0<f<t^{-2}\}$ if $t>0$, so $r$ is Borel by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable. Then $\zeta=2rg$, its coordinates $2r\langle g,e_{k}\rangle$ are Borel, and its values lie in the linear subspace $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert), so $\zeta$ is measurable into $X^{a}$. Since $r^{2}f=\mathbf{1}_{\{f>0\}}$, we get $|\zeta|_{a}^{2}f=4\cdot\mathbf{1}_{\{f>0\}}|g|_{a}^{2}$, and by Step 5 and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison,

$$\int_{X}|\zeta|_{a}^{2}\,d\mu=4\int_{X}\mathbf{1}_{\{f>0\}}|g|_{a}^{2}\,d\nu=4\int_{X}|g|_{a}^{2}\,d\nu=4\mathcal{E}^{V}_{a}(U,U)=\mathcal{I}^{V}_{a}(\mu)<\infty,$$

by The Gibbs Dirichlet Form of the Noise Gradient §form and The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential §information. So the class of $\zeta$ lies in $L^{2}(\mu;X^{a})$ and claim 6 holds. If $\tilde f,\tilde g$ are other admissible choices, then $\{f\ne\tilde f\}$ is $\nu$-null by The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §uniqueness, so $\sqrt{\tilde f}$ has the class $U$, and $\tilde g=g$ off a $\nu$-null set (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes), the set $\{g\ne\tilde g\}=\bigcup_{k\in\mathbb{N}}\{\langle g,e_{k}\rangle\ne\langle\tilde g,e_{k}\rangle\}$ being Borel since the coordinates of $g$ and $\tilde g$ are Borel by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable; hence the corresponding $\tilde\zeta$ equals $\zeta$ off a $\nu$-null Borel set $M$, and $\mu(M)=\int_{X}\mathbf{1}_{M}f\,d\nu=0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral, so $\tilde\zeta$ and $\zeta$ have the same class in $L^{2}(\mu;X^{a})$. This proves claim 5.

Step 7 (integration by parts on $\mathbb{D}$). Let $F\in\mathbb{D}$ with a Borel representative $v$, let $G$ be a map in the class $\nabla^{V}_{a}F$, let $\chi\in\mathcal{F}C^{1}_{b}(X)$, $k\in\mathbb{N}$, and $\rho(x)=\chi(x)(x_{k}/c_{k}+\partial_{k}V(x)/\beta)-\partial_{k}\chi(x)$, which lies in $L^{2}(\nu)$ by Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space §square-integrable and the boundedness of $\chi$, $\partial_{k}\chi$. We show

$$\int_{X}a_{k}^{-1}\langle G(x),e_{k}\rangle\,\chi(x)\,\nu(dx)=\int_{X}v\,\rho\,d\nu,$$

both integrands being integrable as products of members of $L^{2}(\nu)$, $a_{k}^{-1}\langle G,e_{k}\rangle$ being Borel and in $L^{2}(\nu)$ by (CB). For $\varphi\in\mathcal{F}C^{1}_{b}(X)$, (P) and Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space §ibp applied to $\varphi\chi$, together with Linearity and Monotonicity of the Lebesgue Integral §integrable, give $\int_{X}\partial_{k}\varphi\,\chi\,d\nu=\int_{X}\varphi\rho\,d\nu$. Let $(\varphi_{j})$ approximate $F$. By (CB) applied to $\nabla_{a}\varphi_{j}(x)-G(x)$, whose $k$-th coordinate is $a_{k}\partial_{k}\varphi_{j}(x)-\langle G(x),e_{k}\rangle$, we get $\lVert\partial_{k}\varphi_{j}-a_{k}^{-1}\langle G,e_{k}\rangle\rVert_{L^{2}(\nu)}^{2}\le a_{k}^{-1}\lVert\nabla_{a}\varphi_{j}-\nabla^{V}_{a}F\rVert_{\nu}^{2}\to0$, and $\lVert\varphi_{j}-v\rVert_{L^{2}(\nu)}\to0$; so the identity follows from that for $\varphi_{j}$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu)$.

Step 8 (claim 7). Let $\psi$, $n$ and $\tilde\psi$ be as in claim 7. By claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, $\tilde\psi\in C^{1}_{b}(\mathbb{R}^{n})$, so $\psi\in\mathcal{F}C^{1}_{b}(X)$; for $k\le n$, $\chi_{k}=\partial_{k}\psi=(\partial_{k}\tilde\psi)\circ p_{n}$ 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, where $\partial_{k}\tilde\psi$ is of class $C^{1}$ (clause 2 of C^k Maps on a Euclidean Open Set) and bounded with bounded partial derivatives, so $\chi_{k}\in\mathcal{F}C^{1}_{b}(X)$ and $\partial_{k}\chi_{k}=\partial_{k}\partial_{k}\psi$. Let $\rho_{k}(x)=(x_{k}/c_{k}+\partial_{k}V(x)/\beta)\chi_{k}(x)-\partial_{k}\chi_{k}(x)$. It is $\mu$-integrable: $x\mapsto x_{k}\chi_{k}(x)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §coordinate-integrable (as $\mu\in\mathcal{P}_{2}(X)$), $\partial_{k}V\chi_{k}$ since $\partial_{k}V$ is $\mu$-integrable and $\chi_{k}$ bounded Borel, and $\partial_{k}\chi_{k}$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable; hence $f\rho_{k}$ is $\nu$-integrable (Step 6).

For $m\in\mathbb{N}$ let $q_{m}(s)=1+s^{2}/m\ge1$ and $H_{m}(s)=s^{2}q_{m}(s)^{-1}$. Every point of $\mathbb{R}$ is interior (claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line) and the identity has derivative $1$ (its difference quotients are $1$, Derivative at an Interior Point); by Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and claim 1 of Reciprocal Rule for One-Dimensional Derivatives, $q_{m}'(s)=2s/m$ and $H_{m}'(s)=2sq_{m}(s)^{-1}-s^{2}(2s/m)q_{m}(s)^{-2}=2s\,q_{m}(s)^{-2}$. These functions are continuous (Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space), hence Euclidean continuous (Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §equivalent), and $\partial_{1}H_{m}=H_{m}'$ (claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line), so $H_{m}$ is of class $C^{1}$ on $\mathbb{R}^{1}$ (clause 1 of C^k Maps on a Euclidean Open Set). Also $0\le H_{m}(s)\le s^{2}$ and $|H_{m}'(s)|\le2|s|$; $|H_{m}'(s)|\le2m$, since for $|s|>m$ we have $q_{m}(s)^{-2}\le q_{m}(s)^{-1}\le m/s^{2}$; and $H_{m}(s)\to s^{2}$, $H_{m}'(s)\to2s$ as $m\to\infty$, because $q_{m}(s)\to1$.

Fix $k\le n$. By claim 3 the class of $H_{m}\circ u$ lies in $\mathbb{D}$ with gradient the class of $H_{m}'(u)g$, and Step 7 with $v=H_{m}\circ u$, $G=H_{m}'(u)g$ and $\chi=\chi_{k}$ gives

$$\int_{X}H_{m}'(u)\,a_{k}^{-1}\langle g,e_{k}\rangle\,\chi_{k}\,d\nu=\int_{X}H_{m}(u)\,\rho_{k}\,d\nu .$$

As $m\to\infty$ the left integrand tends pointwise to $2u\,a_{k}^{-1}\langle g,e_{k}\rangle\chi_{k}$ and is bounded by $2Bu\,|a_{k}^{-1}\langle g,e_{k}\rangle|$, with $B$ a bound of $|\chi_{k}|$, which is integrable as $u$ and $a_{k}^{-1}\langle g,e_{k}\rangle$ lie in $L^{2}(\nu)$; the right integrand tends to $u^{2}\rho_{k}=f\rho_{k}$ and is bounded by $f|\rho_{k}|$. By Dominated Convergence Theorem and Step 6,

$$\int_{X}2u\,\langle g,e_{k}\rangle\,\partial_{k}\psi\,d\nu=a_{k}\int_{X}f\rho_{k}\,d\nu=a_{k}\int_{X}\rho_{k}\,d\mu .$$

Finally, $\nabla_{a}\psi$ and $\zeta$ lie in $L^{2}(\mu;X^{a})$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient and claim 5), so $\langle\zeta,\nabla_{a}\psi\rangle_{\mu}=\int_{X}\langle\zeta,\nabla_{a}\psi\rangle_{a}\,d\mu$ with a $\mu$-integrable integrand (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations). By The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, the terms $a_{i}^{-1}\langle\zeta(x),e_{i}\rangle\,a_{i}\partial_{i}\psi(x)$ of the series for $\langle\zeta(x),\nabla_{a}\psi(x)\rangle_{a}$ vanish for $i>n$, so it equals $\sum_{k=1}^{n}\langle\zeta(x),e_{k}\rangle\partial_{k}\psi(x)$; and $\langle\zeta,e_{k}\rangle f=2u\langle g,e_{k}\rangle$ pointwise, both sides vanishing where $f=0$ and $2u^{-1}f=2u$ where $f>0$. By Step 6 and Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\langle\zeta,\nabla_{a}\psi\rangle_{\mu}=\int_{X}\langle\zeta,\nabla_{a}\psi\rangle_{a}f\,d\nu=\sum_{k=1}^{n}\int_{X}2u\,\langle g,e_{k}\rangle\,\partial_{k}\psi\,d\nu=\sum_{k=1}^{n}a_{k}\int_{X}\rho_{k}\,d\mu,$$

which is claim 7.