Proof of Integration by Parts for the Gibbs Measure of a Gibbs-Sobolev Potential, and Closability of the Noise Gradient in Its Square-Integrable Space

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

Write $w=\exp(-V/\beta)$, $u=\exp(-qV/\beta)$ and $q'=q/(q-1)$; $q$ and $q'$ are conjugate exponents by Conjugate Exponents and Young's Inequality §conjugate. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, $\varphi$ and $\partial_{k}\varphi$ are Borel and bounded 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. 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 $2,2$ applied to it and the constant $1$; we refer to this as (H2).

Step 1 (the weight). The functions $-V/\beta$ and $-qV/\beta$ are Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. The exponential function is smooth on $\mathbb{R}^{1}$ by claim 3 of Basic Properties of the Exponential Function, hence continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §distance, hence Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space; so $w$ and $u$ are Borel by claim 4 of that lemma, and positive by claim 2 of Basic Properties of the Exponential Function. Let $x\in X$. If $V(x)\ge0$ then $-V(x)/\beta\le0$ and $w(x)\le\exp(0)=1$; if $V(x)<0$ then $-V(x)/\beta<-qV(x)/\beta$ because $q>1$, so $w(x)<u(x)$; here $\exp$ is increasing with $\exp(0)=1$ by claims 1 and 4 of Basic Properties of the Exponential Function. Hence $0<w\le1+u$ on $X$. The constant $1$ is integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space and $u$ by Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §exponential, so $1+u$ is integrable by Linearity and Monotonicity of the Lebesgue Integral §integrable, and $\int_{X}w\,d\gamma_{c}\le\int_{X}(1+u)\,d\gamma_{c}<\infty$ by the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative. Thus $w$ is Borel and integrable, the Gibbs measure $\gamma^{V}_{\beta}$ of The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §gibbs is defined and is a Borel probability measure, and its normaliser $Z=Z_{V,\beta}$ is positive by The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §normaliser.

Step 2 (integrals against the Gibbs measure). By The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §gibbs and the homogeneity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\gamma^{V}_{\beta}(A)=\int_{X}\mathbf{1}_{A}(w/Z)\,d\gamma_{c}$ for $A\in\mathcal{B}(X)$, so $\gamma^{V}_{\beta}$ is the measure with the Borel density $w/Z\ge0$ with respect to $\gamma_{c}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables. That claim and linearity give (D): for Borel $f:X\to[0,\infty]$, $\int_{X}f\,d\gamma^{V}_{\beta}=Z^{-1}\int_{X}fw\,d\gamma_{c}$; a Borel $f:X\to\mathbb{R}$ is $\gamma^{V}_{\beta}$-integrable if and only if $fw$ is $\gamma_{c}$-integrable, and then $\int_{X}f\,d\gamma^{V}_{\beta}=Z^{-1}\int_{X}fw\,d\gamma_{c}$. If $N\in\mathcal{B}(X)$ is $\gamma_{c}$-null, then $\gamma^{V}_{\beta}(N)=Z^{-1}\int_{X}\mathbf{1}_{N}w\,d\gamma_{c}=0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral; so, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, integrability and integrals with respect to $\gamma^{V}_{\beta}$ of a member of a class modulo $\gamma_{c}$ do not depend on the member chosen.

Step 3 (claims 1 and 2). First, $w$ is $q$-integrable with respect to $\gamma_{c}$: for $x\in X$, $w(x)^{q}=\exp(q\log w(x))=\exp(-qV(x)/\beta)=u(x)$ by Real Power of a Nonnegative Real Number §power and The Natural Logarithm, and $u$ is integrable. Next, $S=\sum_{k=1}^{\infty}c_{k}$ is a convergent series of positive terms by Variance Sequences and Their Truncations §variances, so for every $k$, $c_{k}$ is at most the $k$-th partial sum $\sum_{i=1}^{k}c_{i}$ (the other terms being nonnegative), which is at most $S$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; thus $c_{k}\le S$ for all $k$, and $0<c_{1}\le S$. Put $\alpha=1/(4S)$; then $2\alpha c_{k}\le\frac12<1$, and Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §exponential, with $\theta=\frac12$, shows that $E(x)=\exp(\alpha|x|^{2})$ is Borel and $\gamma_{c}$-integrable. Fix a natural number $m>q'$ and let $t\ge0$ be real. If $t\le1$ then $t^{q'}\le1^{q'}=\exp(q'\log1)=1$, and if $t>1$ then $t^{m}=t^{q'}t^{m-q'}\ge t^{q'}$, by Properties of Real Powers of Nonnegative Real Numbers §monotone, Properties of Real Powers of Nonnegative Real Numbers §exponents and Properties of Real Powers of Nonnegative Real Numbers §agreement; so $t^{q'}\le1+t^{m}$. Moreover $\exp(\alpha t)=\exp(\alpha t/m)^{m}$ by claim 1 of Basic Properties of the Exponential Function, and $\exp(\alpha t/m)\ge1+\alpha t/m\ge\alpha t/m\ge0$ by claim 4 there, so $t^{m}\le(m/\alpha)^{m}\exp(\alpha t)$. With $t=|x|^{2}$, which is Borel by the preamble of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment, we get $(|x|^{2})^{q'}\le1+(m/\alpha)^{m}E(x)$; the left side is Borel by Power-Integrable Functions and the p-Seminorm §measurable-power, the right side is $\gamma_{c}$-integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space and Linearity and Monotonicity of the Lebesgue Integral §integrable, and so, by the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\int_{X}(|x|^{2})^{q'}\,\gamma_{c}(dx)<\infty$ and $x\mapsto|x|^{2}$ is $q'$-integrable with respect to $\gamma_{c}$. By Hoelder's Inequality, for Two and for Finitely Many Factors §holder, $x\mapsto w(x)|x|^{2}$ is $\gamma_{c}$-integrable, so by (D) $M_{2}(\gamma^{V}_{\beta})=Z^{-1}\int_{X}|x|^{2}w(x)\,\gamma_{c}(dx)<\infty$, and $\gamma^{V}_{\beta}\in\mathcal{P}_{2}(X)$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. Claim 1 is proved.

For claim 2 fix a map $g:X\to X^{c}$ in the class $\nabla_{c}V$, one and the same for all $k$, and for every $k$ the Borel representative $\partial_{k}V(x)=c_{k}^{-1}\langle g(x),e_{k}\rangle$ of Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights §partial; by Step 2 nothing depends on this choice. Now fix $k$. The coordinate $x\mapsto x_{k}$ is Borel by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates, and $x_{k}^{2}=\langle x,e_{k}\rangle^{2}\le|x|^{2}$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space, so $\int_{X}x_{k}^{2}\,\gamma^{V}_{\beta}(dx)\le M_{2}(\gamma^{V}_{\beta})<\infty$. By Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §derivatives, $|\partial_{k}V|^{2q'}$ is $\gamma_{c}$-integrable, and $((\partial_{k}V)^{2})^{q'}=(|\partial_{k}V|^{2})^{q'}=|\partial_{k}V|^{2q'}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement and Properties of Real Powers of Nonnegative Real Numbers §exponents; so $(\partial_{k}V)^{2}$ is $q'$-integrable, $w(\partial_{k}V)^{2}$ is $\gamma_{c}$-integrable by Hoelder's Inequality, for Two and for Finitely Many Factors §holder, and $\partial_{k}V$ is square-integrable with respect to $\gamma^{V}_{\beta}$ by (D). As $k$ was arbitrary, claim 2 is proved. By (H2) and (D), for every $k$, $x\mapsto x_{k}$ and $\partial_{k}V$ are $\gamma^{V}_{\beta}$-integrable, and $x\mapsto x_{k}w(x)$ and $w\,\partial_{k}V$ are $\gamma_{c}$-integrable.

Step 4 (calculus on cylindrical functions). (C) Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ have a representation $(n,\psi)$, and let $H:\mathbb{R}\to\mathbb{R}$ be bounded and of class $C^{1}$ on $\mathbb{R}^{1}$ with $\partial_{1}H$ bounded. 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 $C^{1}$ on $\mathbb{R}^{n}$ with $\partial_{i}(H\circ\psi)=(\partial_{1}H\circ\psi)\,\partial_{i}\psi$, which is bounded; so $H\circ\psi\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, $H\circ\varphi=(H\circ\psi)\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$, and 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 (both sides vanishing for $k>n$) $\partial_{k}(H\circ\varphi)=(\partial_{1}H\circ\varphi)\,\partial_{k}\varphi$ for all $k$. (P) Let $\varphi,\chi\in\mathcal{F}C^{1}_{b}(X)$ have representations $(n,\psi)$ and $(m,\vartheta)$, say $n\le m$ (the other case is symmetric), and let $\pi:\mathbb{R}^{m}\to\mathbb{R}^{n}$ keep the first $n$ coordinates. Its coordinate functions $\pi_{l}$ are smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous at every point of $\mathbb{R}^{m}$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; $\partial_{i}\pi_{l}$ exists everywhere and is the constant $1$ if $i=l$ and $0$ otherwise, its difference quotients in Partial Derivative on a Euclidean Open Set being constant; and these constant functions are smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous at every point by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. So $\pi$ is of class $C^{1}$ by clause 1 of C^k Maps on a Euclidean Open Set. By claims 2 and 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, $\psi\circ\pi$ is of class $C^{1}$ with $\partial_{i}(\psi\circ\pi)=(\partial_{i}\psi)\circ\pi$ for $i\le n$ and $\partial_{i}(\psi\circ\pi)=0$ for $i>n$, so $\psi\circ\pi\in C^{1}_{b}(\mathbb{R}^{m})$; since $p_{n}=\pi\circ p_{m}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $(m,\psi\circ\pi)$ is a representation of $\varphi$. By claims 3 and 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $(\psi\circ\pi)\vartheta$ is of class $C^{1}$ with the bounded partial derivatives $\partial_{i}(\psi\circ\pi)\,\vartheta+(\psi\circ\pi)\,\partial_{i}\vartheta$, so $\varphi\chi=((\psi\circ\pi)\vartheta)\circ p_{m}\in\mathcal{F}C^{1}_{b}(X)$, 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 the representations $(m,(\psi\circ\pi)\vartheta)$, $(m,\psi\circ\pi)$ and $(m,\vartheta)$, gives $\partial_{k}(\varphi\chi)=\partial_{k}\varphi\,\chi+\varphi\,\partial_{k}\chi$ for every $k\in\mathbb{N}$.

Step 5 (a bounded approximation of the weight). For a positive real $R$ and $s\in\mathbb{R}$ let $y(s)=\exp(-s/\beta)>0$ and $G_{R}(s)=Ry(s)/(R+y(s))=R-R^{2}(R+y(s))^{-1}$. Every point of $\mathbb{R}$ is an interior point of the interval $\mathbb{R}$ by claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line. The map $s\mapsto-s/\beta$ has derivative $-1/\beta$ at every point (the identity has all difference quotients equal to $1$ in Derivative at an Interior Point, and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives), so $y'=-y/\beta$ by Chain Rule for One-Dimensional Derivatives and claim 3 of Basic Properties of the Exponential Function; the constant $R$ has derivative $0$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, so $(R+y)'=y'$ by claim 2 there, and claim 1 of Reciprocal Rule for One-Dimensional Derivatives with claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives gives

$$G_{R}'(s)=-\frac{R^{2}\,y(s)}{\beta\,(R+y(s))^{2}}\qquad(s\in\mathbb{R}).$$

The map $s\mapsto-s/\beta$ is continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §lipschitz, and so is the exponential function (Step 1), so their composite $y$ is continuous on $\mathbb{R}$ by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. Constant functions are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants, so $R+y$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set; it does not vanish, so $(R+y)^{-1}$ is continuous by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space; and then $G_{R}=R+(-R^{2})(R+y)^{-1}$ and $G_{R}'=(-R^{2}/\beta)\,y\,(R+y)^{-1}(R+y)^{-1}$ are continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, hence Euclidean continuous by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §equivalent; since $\partial_{1}G_{R}=G_{R}'$ by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $G_{R}$ is of class $C^{1}$ on $\mathbb{R}^{1}$ by clause 1 of C^k Maps on a Euclidean Open Set. Now fix $s\in\mathbb{R}$ and put $t=y(s)>0$. Then

$$G_{R}(s)=t\,\frac{R}{R+t}=R\,\frac{t}{R+t},\qquad G_{R}'(s)=-\frac{t}{\beta}\cdot\frac{R^{2}}{(R+t)^{2}}=-\frac{R}{\beta}\cdot\frac{Rt}{(R+t)^{2}}.$$

Since $R>0$ and $t>0$ we have $0<R<R+t$ and $0<t<R+t$, so $0<\frac{R}{R+t}<1$ and $0<\frac{t}{R+t}<1$, whence $0<G_{R}(s)<t$ and $0<G_{R}(s)<R$. From $R^{2}\le(R+t)^{2}$ the first form of $G_{R}'(s)$ gives $|G_{R}'(s)|\le t/\beta$, and from $Rt\le R^{2}+2Rt+t^{2}=(R+t)^{2}$ the second gives $|G_{R}'(s)|\le R/\beta$. As $s$ was arbitrary, $0<G_{R}<\min(R,y)$ and $|G_{R}'|\le\min(R,y)/\beta$ on $\mathbb{R}$. Moreover, from the same factorisations,

$$t-G_{R}(s)=t\Bigl(1-\frac{R}{R+t}\Bigr)=\frac{t^{2}}{R+t},\qquad G_{R}'(s)+\frac{t}{\beta}=\frac{t}{\beta}\Bigl(1-\frac{R}{R+t}\Bigr)\Bigl(1+\frac{R}{R+t}\Bigr)=\frac{t^{2}}{\beta(R+t)}\Bigl(1+\frac{R}{R+t}\Bigr),$$

and since $R+t\ge R>0$ and $0<\frac{R}{R+t}<1$,

$$0\le t-G_{R}(s)\le\frac{t^{2}}{R},\qquad 0\le G_{R}'(s)+\frac{t}{\beta}\le\frac{2t^{2}}{\beta R}.$$

Given a positive real $\varepsilon$, the Archimedean property gives $R_{0}\in\mathbb{N}$ with $R_{0}\varepsilon>t^{2}+2t^{2}/\beta$, and then $|t-G_{R}(s)|<\varepsilon$ and $|G_{R}'(s)+t/\beta|<\varepsilon$ for every natural $R\ge R_{0}$. So as $R\to\infty$ through $\mathbb{N}$, $G_{R}(s)\to y(s)$ and $G_{R}'(s)\to-y(s)/\beta$, for every $s\in\mathbb{R}$. Note that $y(V(x))=w(x)$.

Step 6 (approximating $V$). By Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments §sobolev the class of $V$ lies in $\mathbb{D}^{1,2}_{c}$, so by The Gaussian Sobolev Space of the Noise Gradient §space and The Gaussian Sobolev Space of the Noise Gradient §gradient, read with $c$ in place of $a$ as in Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights, there are $F_{j}\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert F_{j}-V\rVert_{2}\to0$ and $\lVert\nabla_{c}F_{j}-\nabla_{c}V\rVert_{\gamma_{c}}\to0$. Fix $k$. For $x\in X$ the vector $\eta=\nabla_{c}F_{j}(x)-g(x)$ lies in $X^{c}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and its $k$-th coordinate is $\eta_{k}=c_{k}\partial_{k}F_{j}(x)-\langle g(x),e_{k}\rangle$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient read with $c$ (for $k$ beyond a representation, $\partial_{k}F_{j}(x)$ and that coordinate of $\nabla_{c}F_{j}(x)$ vanish 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). As $c_{k}^{-1}\eta_{k}^{2}$ is a term of the series of nonnegative terms with sum $|\eta|_{c}^{2}$ (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 gives

$$\bigl(\partial_{k}F_{j}(x)-\partial_{k}V(x)\bigr)^{2}=c_{k}^{-1}\cdot c_{k}^{-1}\eta_{k}^{2}\le c_{k}^{-1}\,|\nabla_{c}F_{j}(x)-g(x)|_{c}^{2},$$

and integrating with Linearity and Monotonicity of the Lebesgue Integral §nonnegative and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations gives $\lVert\partial_{k}F_{j}-\partial_{k}V\rVert_{2}^{2}\le c_{k}^{-1}\lVert\nabla_{c}F_{j}-\nabla_{c}V\rVert_{\gamma_{c}}^{2}\to0$. By The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §subsequence, with $p=2$ and the measure $\gamma_{c}$, there are a subsequence of $(\partial_{k}F_{j})$, a square-integrable $h:X\to\mathbb{R}$ and a null set $N_{1}$ such that along the subsequence $\partial_{k}F_{j}(x)\to\partial_{k}V(x)$ and $|\partial_{k}F_{j}(x)|\le h(x)$ for $x\notin N_{1}$. The corresponding subsequence of $(F_{j})$ still converges to $V$ in $L^{2}(\gamma_{c})$ by A Subsequence of a Convergent Sequence Has the Same Limit, and the same claim gives a further subsequence converging to $V$ off a null set $N_{2}$. We pass to this subsequence, still written $(F_{j})$: off the null set $N=N_{1}\cup N_{2}$ (The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union) we have $F_{j}\to V$, $\partial_{k}F_{j}\to\partial_{k}V$ and $|\partial_{k}F_{j}|\le h$, and $h$ is $\gamma_{c}$-integrable by (H2).

Step 7 (claim 3). Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, take $(F_{j})$, $h$ and $N$ as in Step 6 for this $k$, let $B$ bound $|\varphi|$ and $|\partial_{k}\varphi|$, and let $R\in\mathbb{N}$. By Step 4 (C) with $H=G_{R}$ (Step 5), $G_{R}\circ F_{j}\in\mathcal{F}C^{1}_{b}(X)$ with $\partial_{k}(G_{R}\circ F_{j})=G_{R}'(F_{j})\,\partial_{k}F_{j}$. Hence Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space §ibp, applied with $\varphi$ in place of its $F$ and with the companion function $G_{R}\circ F_{j}$ in place of its $\varphi$, gives

$$\int_{X}\partial_{k}\varphi\,G_{R}(F_{j})\,d\gamma_{c}=\int_{X}\varphi(x)\Bigl(\frac{x_{k}}{c_{k}}G_{R}(F_{j}(x))-G_{R}'(F_{j}(x))\,\partial_{k}F_{j}(x)\Bigr)\gamma_{c}(dx).$$

For $x\notin N$, $G_{R}(F_{j}(x))\to G_{R}(V(x))$ and $G_{R}'(F_{j}(x))\to G_{R}'(V(x))$ by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §sequential, and off $N$ (using $|G_{R}|\le R$ and $|G_{R}'|\le R/\beta$ from Step 5) the left integrand is bounded by $BR$ and the single right-hand integrand by the sum $BR|x_{k}|/c_{k}+BRh/\beta$; these bounds are $\gamma_{c}$-integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates, Step 6 and Linearity and Monotonicity of the Lebesgue Integral §integrable. All functions involved are Borel by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Letting $j\to\infty$ with The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated and Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\int_{X}\partial_{k}\varphi\,G_{R}(V)\,d\gamma_{c}=\int_{X}\varphi(x)\Bigl(\frac{x_{k}}{c_{k}}G_{R}(V(x))-G_{R}'(V(x))\,\partial_{k}V(x)\Bigr)\gamma_{c}(dx).$$

Now let $R\to\infty$ through $\mathbb{N}$: by Step 5, at every $x$, $G_{R}(V(x))\to w(x)$ and $G_{R}'(V(x))\to-w(x)/\beta$, and the left integrand is bounded by $Bw$ and the right one by the sum $B|x_{k}|w/c_{k}+Bw|\partial_{k}V|/\beta$, which are $\gamma_{c}$-integrable by Steps 1 and 3 and Linearity and Monotonicity of the Lebesgue Integral §integrable. By Dominated Convergence Theorem,

$$\int_{X}\partial_{k}\varphi\,w\,d\gamma_{c}=\int_{X}\varphi(x)\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)w(x)\,\gamma_{c}(dx).$$

The product of $x\mapsto\varphi(x)(x_{k}/c_{k}+\partial_{k}V(x)/\beta)$ with $w$ is $\gamma_{c}$-integrable, being bounded by $B(|x_{k}|w/c_{k}+w|\partial_{k}V|/\beta)$, so by (D) this function is $\gamma^{V}_{\beta}$-integrable, as is the bounded Borel function $\partial_{k}\varphi$; dividing by $Z$ and applying (D) to both sides yields claim 3.

Step 8 (claim 4). For $\varphi,\chi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$ put $\rho_{\chi,k}(x)=\chi(x)(x_{k}/c_{k}+\partial_{k}V(x)/\beta)-\partial_{k}\chi(x)$; its class lies in $L^{2}(\gamma^{V}_{\beta})$ by Step 3 and claim 2 (which make $x\mapsto x_{k}$ and $\partial_{k}V$ square-integrable), the boundedness of $\chi$ and $\partial_{k}\chi$ (the product of a bounded Borel function with a square-integrable one is square-integrable, by the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and a bounded Borel function is square-integrable, its square being bounded and Borel, hence integrable by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space), and The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §normed. By Step 4 (P), $\varphi\chi\in\mathcal{F}C^{1}_{b}(X)$ with $\partial_{k}(\varphi\chi)=\partial_{k}\varphi\,\chi+\varphi\,\partial_{k}\chi$; so claim 3 for $\varphi\chi$ and Linearity and Monotonicity of the Lebesgue Integral §integrable (all terms being integrable by Step 3 and boundedness) give the identity (8.1):

$$\int_{X}\partial_{k}\varphi\,\chi\,d\gamma^{V}_{\beta}=\int_{X}\varphi\,\rho_{\chi,k}\,d\gamma^{V}_{\beta}.$$

Now let $\varphi_{j}$ and $G$ be as in claim 4, and for $v\in L^{2}(\gamma^{V}_{\beta};X^{a})$ let $v_{k}\in L^{2}(\gamma^{V}_{\beta})$ be its coordinate along $f_{k}$, as in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates. For $x\in X$, $\langle\nabla_{a}\varphi_{j}(x),f_{k}\rangle_{a}=a_{k}^{-1/2}\,a_{k}\partial_{k}\varphi_{j}(x)=a_{k}^{1/2}\partial_{k}\varphi_{j}(x)$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis and The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient (with Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial for $k$ beyond a representation). Since $\gamma^{V}_{\beta}$ is a Borel probability measure (Step 1), $\nabla_{a}\varphi_{j}$ is square-integrable with respect to $\gamma^{V}_{\beta}$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, so its class lies in $L^{2}(\gamma^{V}_{\beta};X^{a})$ and $\nabla_{a}\varphi_{j}-G$ is a vector of that space. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates,

$$\lVert a_{k}^{1/2}\partial_{k}\varphi_{j}-G_{k}\rVert_{L^{2}(\gamma^{V}_{\beta})}^{2}\le\sum_{i=1}^{\infty}\lVert(\nabla_{a}\varphi_{j}-G)_{i}\rVert_{L^{2}(\gamma^{V}_{\beta})}^{2}=\lVert\nabla_{a}\varphi_{j}-G\rVert_{\gamma^{V}_{\beta}}^{2}\to0 .$$

Let $\chi\in\mathcal{F}C^{1}_{b}(X)$; its class lies in $L^{2}(\gamma^{V}_{\beta})$ 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. With the inner product $\langle\cdot,\cdot\rangle$ of $L^{2}(\gamma^{V}_{\beta})$ from The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and The Cauchy-Schwarz Inequality in a Real Inner Product Space, $\langle a_{k}^{1/2}\partial_{k}\varphi_{j},\chi\rangle\to\langle G_{k},\chi\rangle$, while by (8.1) and the same inequality $|\langle\partial_{k}\varphi_{j},\chi\rangle|=|\langle\varphi_{j},\rho_{\chi,k}\rangle|\le\lVert\varphi_{j}\rVert_{L^{2}(\gamma^{V}_{\beta})}\lVert\rho_{\chi,k}\rVert_{L^{2}(\gamma^{V}_{\beta})}\to0$, because $\lVert\varphi_{j}\rVert_{L^{2}(\gamma^{V}_{\beta})}^{2}=\int_{X}\varphi_{j}^{2}\,d\gamma^{V}_{\beta}\to0$. Hence $\langle G_{k},\chi\rangle=0$ for every $\chi\in\mathcal{F}C^{1}_{b}(X)$, and $G_{k}$ is the zero vector of $L^{2}(\gamma^{V}_{\beta})$ 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 with $\mu=\gamma^{V}_{\beta}$. As $k$ was arbitrary, $\lVert G\rVert_{\gamma^{V}_{\beta}}^{2}=\sum_{k=1}^{\infty}\lVert G_{k}\rVert_{L^{2}(\gamma^{V}_{\beta})}^{2}=0$ by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, so $G$ is the zero vector of the real Hilbert space $L^{2}(\gamma^{V}_{\beta};X^{a})$ of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert, its norm being the norm of its inner product.