Proof of Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound

Each result cited is universally quantified over the data in its own statement. Elementary real order and arithmetic are carried by The Real Numbers: Standing Notation and Background §background and are not cited again; this covers in particular manipulations of inequalities, squares and finite sums, the limit laws for sums and real multiples of convergent sequences, the Archimedean property, the fact that for a nonnegative real $r$ and a positive real $\delta$ one has $r<\delta$ if and only if $r^{2}<\delta^{2}$, and the two inequalities

$$pq\le2\beta p^{2}+\frac{q^{2}}{8\beta},\qquad 2pq\le\frac{p^{2}}{8\beta}+8\beta q^{2}\qquad(p,q\in\mathbb{R}),$$

which hold because $2\beta\bigl(p-q/(4\beta)\bigr)^{2}\ge0$ and $(8\beta)^{-1}(p-8\beta q)^{2}\ge0$, $\beta$ being positive.

Notation. Write $w=w_{V,\beta}$, $Z=Z_{V,\beta}$, $\gamma^{V}=\gamma^{V}_{\beta}$, and fix once and for all a constant $b$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below, so that $w$ is Borel with $0<w(x)\le\exp(b/\beta)$ for every $x\in X$ by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight, and $Z$ is a positive real number by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser. Put $h=Z^{-1}w:X\to\mathbb{R}$; it is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $h(x)>0$ for every $x$. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, $V$ and every $\partial_{k}V$ are continuous on $X$, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, $-b\le V(x)$ for every $x\in X$, and $|\nabla_{a}V(x)|_{a}^{2}=\sum_{k=1}^{d}a_{k}\,\partial_{k}V(x)^{2}$, so that $|\nabla_{a}V|_{a}^{2}$ is Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. $\phi:[0,\infty)\to\mathbb{R}$ is the function $\phi(0)=0$, $\phi(s)=s\log s$ ($s>0$) of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, the function used in Relative Entropy of Probability Measures §relative-entropy; test functions in $\mathcal{F}C^{1}_{b}(X)$ are written $\varphi$. Integrals and integrability are those of Measure Spaces and the Lebesgue Integral: Standing Notation §integral: a measurable real function is integrable exactly when the integral of its absolute value is finite, and for a nonnegative integrable function the two integrals agree by Integrable Function and the Lebesgue Integral. Linearity and monotonicity of integrals, for nonnegative measurable and for integrable functions, are Linearity and Monotonicity of the Lebesgue Integral §nonnegative and Linearity and Monotonicity of the Lebesgue Integral §integrable, cited below as linearity and monotonicity; constants are integrable against the Borel probability measures used here by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space.

Step 1 (Claim 1, change of measure). The function $h$ is measurable with values in $[0,\infty)$, so claim 3 of Image Measures, Measures with Densities, and Change of Variables provides the measure $\nu_{h}$ with density $h$ with respect to $\gamma_{c}$. For $A\in\mathcal{B}(X)$, linearity (with the constant $Z^{-1}$) and The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §gibbs give

$$\nu_{h}(A)=\int_{X}\mathbf{1}_{A}\,h\,d\gamma_{c}=\frac{1}{Z}\int_{X}\mathbf{1}_{A}\,w\,d\gamma_{c}=\gamma^{V}(A),$$

so $\gamma^{V}=\nu_{h}$. Hence, by claim 3 of Image Measures, Measures with Densities, and Change of Variables and linearity, for every Borel $f:X\to[0,\infty)$

$$\int_{X}f\,d\gamma^{V}=\int_{X}f\,h\,d\gamma_{c}=\frac{1}{Z}\int_{X}f\,w\,d\gamma_{c}; \tag{1.1}$$

and a Borel $f:X\to\mathbb{R}$ is integrable with respect to $\gamma^{V}$ if and only if $fh$ is integrable with respect to $\gamma_{c}$, which by linearity (the two functions $fh$ and $fw=Z\,fh$ being real multiples of one another) holds if and only if $fw$ is integrable with respect to $\gamma_{c}$; in that case (1.1) holds in $\mathbb{R}$, again by claim 3 of Image Measures, Measures with Densities, and Change of Variables and linearity. This is the displayed formula of Claim 1 and the integrability criterion.

Null sets. Let $A\in\mathcal{B}(X)$. If $\gamma_{c}(A)=0$, the nonnegative Borel function $\mathbf{1}_{A}w$ vanishes off the null set $A$, so $\int_{X}\mathbf{1}_{A}w\,d\gamma_{c}=0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral and $\gamma^{V}(A)=0$. Conversely, if $\gamma^{V}(A)=0$ then $\int_{X}\mathbf{1}_{A}w\,d\gamma_{c}=0$, so $\mathbf{1}_{A}w=0$ $\gamma_{c}$-almost everywhere by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing; since $w>0$ everywhere, the set where this fails is $A$ itself, which is therefore $\gamma_{c}$-null in the sense of A Property Holding Almost Everywhere, that is, contained in some $B\in\mathcal{B}(X)$ with $\gamma_{c}(B)=0$ by Null Set of a Measure; so $\gamma_{c}(A)\le\gamma_{c}(B)=0$ by monotonicity of measures. Thus $\gamma^{V}$ and $\gamma_{c}$ have the same null sets.

Step 2 (Two identities). (a) For $s\in[0,\infty)$ and positive $t\in\mathbb{R}$,

$$\phi(st)=t\,\phi(s)+st\log t .$$

If $s=0$ both sides are $0$, since $\phi(0)=0$. If $s>0$ then $st>0$ and $\log(st)=\log s+\log t$ by The Natural Logarithm, so $\phi(st)=st\log s+st\log t=t\,\phi(s)+st\log t$.

(b) For every $x\in X$, $\log h(x)=-V(x)/\beta-\log Z$. Indeed $h(x)=\exp(-V(x)/\beta)\,Z^{-1}$, and by The Natural Logarithm $\log\exp(-V(x)/\beta)=-V(x)/\beta$ and $\log(Z^{-1})=-\log Z$, the latter because $\log Z+\log(Z^{-1})=\log1=\log\exp(0)=0$, using $\exp(0)=1$ from claim 1 of Basic Properties of the Exponential Function.

Consequently, if $g:X\to[0,\infty)$ and $f=gh$, then for every $x\in X$

$$\phi(f(x))=h(x)\,\phi(g(x))-\frac{1}{\beta}\,f(x)V(x)-f(x)\log Z . \tag{2.1}$$

Step 3 (Claim 2, finite entropy relative to $\gamma^{V}$ implies the right-hand conditions, and the formula). Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma^{V}$: by Relative Entropy of Probability Measures §relative-entropy there is a density $g$ of $\mu$ with respect to $\gamma^{V}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities (a measurable $g:X\to[0,\infty)$ with $\mu(A)=\int_{X}\mathbf{1}_{A}g\,d\gamma^{V}$ for $A\in\mathcal{B}(X)$) such that $\phi\circ g$ is integrable with respect to $\gamma^{V}$, and $H(\mu\,|\,\gamma^{V})=\int_{X}\phi\circ g\,d\gamma^{V}$. Put $f=gh$, a Borel function with values in $[0,\infty)$ by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By (1.1) applied to $\mathbf{1}_{A}g$,

$$\int_{X}\mathbf{1}_{A}f\,d\gamma_{c}=\int_{X}\mathbf{1}_{A}g\,d\gamma^{V}=\mu(A)\qquad(A\in\mathcal{B}(X)),$$

so $f$ is a density of $\mu$ with respect to $\gamma_{c}$; it is integrable with respect to $\gamma_{c}$ with integral $\mu(X)=1$ by The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §uniqueness, and $\mu$ is the measure $\nu_{f}$ with density $f$ with respect to $\gamma_{c}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables. The function $\phi\circ g$ is Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous and integrable with respect to $\gamma^{V}$, so by Claim 1 (Step 1) $(\phi\circ g)\,w$, and hence $h\,(\phi\circ g)=Z^{-1}(\phi\circ g)\,w$, is integrable with respect to $\gamma_{c}$, with $\int_{X}h\,(\phi\circ g)\,d\gamma_{c}=\int_{X}\phi\circ g\,d\gamma^{V}$.

Integrability of $\phi\circ f$. The function $\phi\circ f$ is Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, and $-\exp(-1)\le\phi\circ f$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §lower. Since $f\ge0$, $\beta>0$ and $-V\le b$, (2.1) gives

$$\phi\circ f\le F,\qquad F=h\,(\phi\circ g)+\Bigl(\frac{b}{\beta}-\log Z\Bigr)f,$$

and $F$ is integrable with respect to $\gamma_{c}$ by linearity. Hence $|\phi\circ f|\le|F|+\exp(-1)$, whose integral is finite, so $\phi\circ f$ is integrable with respect to $\gamma_{c}$ by monotonicity. Thus $\mu$ has finite relative entropy with respect to $\gamma_{c}$, with $H(\mu\,|\,\gamma_{c})=\int_{X}\phi\circ f\,d\gamma_{c}$.

Integrability of $V$. By (2.1), $\beta^{-1}fV=h\,(\phi\circ g)-\phi\circ f-(\log Z)\,f$, which is integrable with respect to $\gamma_{c}$ by linearity; so $fV$ is integrable with respect to $\gamma_{c}$, and since $\mu=\nu_{f}$, claim 3 of Image Measures, Measures with Densities, and Change of Variables shows that $V$ is integrable with respect to $\mu$ with $\int_{X}V\,d\mu=\int_{X}fV\,d\gamma_{c}$.

Formula. Integrating (2.1) with linearity,

$$H(\mu\,|\,\gamma^{V})=\int_{X}h\,(\phi\circ g)\,d\gamma_{c}=\int_{X}\phi\circ f\,d\gamma_{c}+\frac{1}{\beta}\int_{X}fV\,d\gamma_{c}+\log Z\int_{X}f\,d\gamma_{c}=H(\mu\,|\,\gamma_{c})+\frac{1}{\beta}\int_{X}V\,d\mu+\log Z .$$

Step 4 (Claim 2, the converse). Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma_{c}$, with a density $f$ with respect to $\gamma_{c}$ for which $\phi\circ f$ is integrable with respect to $\gamma_{c}$, and let $V$ be integrable with respect to $\mu$. As in Step 3, $f$ is integrable with respect to $\gamma_{c}$ and $\mu=\nu_{f}$, so $fV$ is integrable with respect to $\gamma_{c}$ by claim 3 of Image Measures, Measures with Densities, and Change of Variables. The function $x\mapsto\exp(V(x)/\beta)$ is continuous, hence Borel, by the argument of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight with $-V$ replaced by $V$; put $g=Z\,f\,\exp(V/\beta)$, a Borel function with values in $[0,\infty)$ by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $\exp(V/\beta)\exp(-V/\beta)=\exp(0)=1$ by claim 1 of Basic Properties of the Exponential Function, $gh=f$. By (1.1), $\int_{X}\mathbf{1}_{A}g\,d\gamma^{V}=\int_{X}\mathbf{1}_{A}f\,d\gamma_{c}=\mu(A)$ for $A\in\mathcal{B}(X)$, so $g$ is a density of $\mu$ with respect to $\gamma^{V}$. By (2.1), $h\,(\phi\circ g)=\phi\circ f+\beta^{-1}fV+(\log Z)\,f$, which is integrable with respect to $\gamma_{c}$ by linearity; hence so is $(\phi\circ g)\,w=Z\,h\,(\phi\circ g)$, and $\phi\circ g$ (Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous) is integrable with respect to $\gamma^{V}$ by Claim 1. So $\mu$ has finite relative entropy with respect to $\gamma^{V}$, and the formula of Claim 2 holds by Step 3. This proves Claim 2.

Step 5 (Claim 3). Let $\mu$ have finite relative entropy with respect to $\gamma^{V}$. By Step 3, $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $V$ is integrable with respect to $\mu$. The first gives $\mu\in\mathcal{P}_{2}(X)$ by Relative Entropy with Respect to a Diagonal Gaussian Measure on a Hilbert Space: the Moment Bound, the Cutoff Projections, and Bounded, Tight, Weakly Closed, Wasserstein-Closed and Weakly Sequentially Compact Sublevel Sets §moment; the second gives that $|\nabla_{a}V|_{a}$ and every $\partial_{k}V$ are integrable with respect to $\mu$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable. In particular $\mu\in\mathcal{P}_{2}(X)$ and $\partial_{k}V$ is integrable with respect to $\mu$ for every $k\in[d]$, which are the standing requirements of The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential.

Step 6 (Square-integrable slope). (a) Let $\mu\in\mathcal{P}(X)$ satisfy $G=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. Let $k\in[d]$. Since every term of $\sum_{j=1}^{d}a_{j}\,\partial_{j}V(x)^{2}=|\nabla_{a}V(x)|_{a}^{2}$ is nonnegative, $\partial_{k}V(x)^{2}\le a_{k}^{-1}|\nabla_{a}V(x)|_{a}^{2}$ for every $x$, so $\int_{X}(\partial_{k}V)^{2}\,d\mu\le a_{k}^{-1}G<\infty$ by monotonicity and linearity. The power $|\partial_{k}V|^{2}$ of Power-Integrable Functions and the p-Seminorm §measurable-power is the natural square $(\partial_{k}V)^{2}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement, so the Borel function $\partial_{k}V$ is $2$-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space, its class lies in $L^{2}(\mu)$, and by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and Real Inner Product Space §norm

$$\lVert\partial_{k}V\rVert_{L^{2}(\mu)}^{2}=\int_{X}(\partial_{k}V)^{2}\,d\mu .$$

For $k>d$, $\partial_{k}V=0$, whose class is the zero vector, of norm $0$. Hence the partial sums of the series $\sum_{k=1}^{\infty}a_{k}\lVert\partial_{k}V\rVert_{L^{2}(\mu)}^{2}$ are, from the $d$-th on, all equal to $\sum_{k=1}^{d}a_{k}\int_{X}(\partial_{k}V)^{2}\,d\mu=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu=G$, by linearity; so by Series of Real Numbers §convergent the series converges, with sum $G$.

(b) In every real inner product space $E$, for $u,v\in E$ and $t\in\mathbb{R}$, $|u+tv|^{2}\le|u+tv|^{2}+|u-tv|^{2}=2|u|^{2}+2t^{2}|v|^{2}$ by Elementary Identities in a Real Inner Product Space §parallelogram and Elementary Identities in a Real Inner Product Space §homogeneity.

Step 7 (Claim 4, the implication from the Gaussian side). Let $\mu\in\mathcal{P}_{2}(X)$ with $V$ integrable with respect to $\mu$ (so $\partial_{k}V$ is integrable for $k\in[d]$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable), let $\mu$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ in the sense of The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score, with finite Fisher information relative to $\gamma_{c}$ with weights $a$ in the sense of Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information, and let $G=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. By Step 6, $\partial_{k}V\in L^{2}(\mu)$ for every $k$; put $\zeta^{V}_{k}=\zeta_{k}+\beta^{-1}\partial_{k}V\in L^{2}(\mu)$. Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$. The functions $x_{k}\varphi/c_{k}-\partial_{k}\varphi$ and $\partial_{k}V\varphi$ are integrable with respect to $\mu$, as recorded in the preamble of The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential; so by linearity of the inner product (Real Inner Product Space §inner-product), The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score and linearity of the integral,

$$\langle\zeta^{V}_{k},\varphi\rangle_{L^{2}(\mu)}=\langle\zeta_{k},\varphi\rangle_{L^{2}(\mu)}+\frac{1}{\beta}\int_{X}\partial_{k}V\,\varphi\,d\mu=\int_{X}\Bigl(\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)\varphi(x)-\partial_{k}\varphi(x)\Bigr)\,\mu(dx).$$

So $(\zeta^{V}_{k})_{k\in\mathbb{N}}$ is a relative score of $\mu$ with respect to $\gamma^{V}$ by The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score, and by the uniqueness recorded there it is the relative score; this proves the formula for its components. By Step 6(b) with $t=\beta^{-1}$, $0\le a_{k}\lVert\zeta^{V}_{k}\rVert^{2}\le2a_{k}\lVert\zeta_{k}\rVert^{2}+2\beta^{-2}a_{k}\lVert\partial_{k}V\rVert^{2}$ (norms in $L^{2}(\mu)$); the series of the right-hand sides converges, its partial sums being $2$ times those of the convergent series $\sum_{k}a_{k}\lVert\zeta_{k}\rVert^{2}$ plus $2\beta^{-2}$ times those of the series of Step 6(a). By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, $\sum_{k}a_{k}\lVert\zeta^{V}_{k}\rVert^{2}$ converges, that is, $\mu$ has finite Fisher information relative to $\gamma^{V}$ with weights $a$ in the sense of The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information.

Step 8 (Constants for the converse). The following choices are made in this order, before any measure $\mu$ is considered. By Existence of a Smooth Plateau Function on Euclidean Space (with $q=d$, $x_{0}=0_{\mathbb{R}^{d}}$, $r=1$, $s=2$) fix a smooth $\chi:\mathbb{R}^{d}\to\mathbb{R}$ as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=d$, with the functions $\chi_{R}(u)=\chi(R^{-1}u)$, and fix the constant $M_{1}\ge0$ of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff; thus for every $n\in\mathbb{N}$ (a positive real number), $\chi_{n}$ is smooth and compactly supported, $0\le\chi_{n}\le1$, $\chi_{n}(u)=1$ when $\lVert u\rVert\le n$, $\chi_{n}(u)=0$ when $\lVert u\rVert\ge2n$, and $|\partial_{i}\chi_{n}(u)|\le M_{1}n^{-1}\le M_{1}$ for all $u\in\mathbb{R}^{d}$ and $i\in[d]$. Put $\varepsilon=1/(8\beta)$, let $C_{\varepsilon}$ be a constant given for this $\varepsilon$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature, and put $C^{+}=\max(C_{\varepsilon},0)$; since $1+\sum_{k=1}^{d}u_{k}^{2}>0$, the inequality of that clause also holds with $C^{+}$ in place of $C_{\varepsilon}$. Put

$$\Lambda=\sum_{k=1}^{d}\frac{a_{k}}{c_{k}^{2}},\qquad\alpha=\sum_{k=1}^{d}a_{k},\qquad C_{1}=4+4\beta^{2}\Lambda+2\beta C^{+}+16\beta^{2}M_{1}^{2}\alpha,\qquad C_{*}=2+3C_{1},$$

with $c_{k}>0$ by Variance Sequences and Their Truncations §variances. These depend only on $v$ (hence on $V$), $\beta$, $a$, $c$ and the choice of $\chi$, which is made once from $d$ alone; in particular $C_{*}$ does not depend on $\mu$.

Step 9 (Test functions). Fix $n\in\mathbb{N}$ and $k\in[d]$, and write $\psi_{k,n}=\partial_{k}v\cdot\chi_{n}\cdot\chi_{n}:\mathbb{R}^{d}\to\mathbb{R}$ (pointwise product). The profile $v$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry), so $\partial_{k}v$ is of class $C^{1}$ on $\mathbb{R}^{d}$ by clause 2 of C^k Maps on a Euclidean Open Set; $\chi_{n}$ is smooth, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set. By clause 1 of C^k Maps on a Euclidean Open Set (read through its clause 3), $\partial_{k}v$, $\chi_{n}$ and their partial derivatives $\partial_{i}\partial_{k}v$ and $\partial_{i}\chi_{n}$ ($i\in[d]$) exist at every point and are continuous at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces. For a real-valued function on $\mathbb{R}^{d}$ that notion of continuity at a point coincides with continuity at that point relative to $\mathbb{R}^{d}$ from $(\mathbb{R}^{d},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$: by Euclidean Distance on $\mathbb{R}^n$, $d_{E}(u,u')$ is the nonnegative square root of $\sum_{i=1}^{d}(u_{i}-u'_{i})^{2}$, and for nonnegative $r$ and positive $\delta$, $r<\delta$ exactly when $r^{2}<\delta^{2}$, while $|t|^{2}=t^{2}$; so the two $\varepsilon$-$\delta$ conditions are the same.

Partial derivatives. Fix $u\in\mathbb{R}^{d}$ and $i\in[d]$, and take $\rho$ and the interval $I$ of claims 1 and 2 of Slice Function and the Partial Derivative for $U=\mathbb{R}^{d}$. The slice function of $\psi_{k,n}$ at $u$ in the $i$-th variable is the pointwise product of the slice functions of $\partial_{k}v$, $\chi_{n}$ and $\chi_{n}$, which by claim 2 of that lemma are differentiable at $u_{i}$ with derivatives $\partial_{i}\partial_{k}v(u)$, $\partial_{i}\chi_{n}(u)$ and $\partial_{i}\chi_{n}(u)$. Applying the product rule, claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, twice, the slice of $\psi_{k,n}$ is differentiable at $u_{i}$, and by claim 2 of Slice Function and the Partial Derivative again

$$\partial_{i}\psi_{k,n}(u)=\partial_{i}\partial_{k}v(u)\,\chi_{n}(u)^{2}+2\,\partial_{k}v(u)\,\chi_{n}(u)\,\partial_{i}\chi_{n}(u). \tag{9.1}$$

By Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, $\psi_{k,n}$ and the right-hand side of (9.1) are continuous on $\mathbb{R}^{d}$, hence continuous at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces by the coincidence above; so $\psi_{k,n}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ by clause 1 of C^k Maps on a Euclidean Open Set. Since $\chi_{n}(u)=0$ when $\lVert u\rVert\ge2n$, $\psi_{k,n}$ and every $\partial_{i}\psi_{k,n}$ vanish when $\lVert u\rVert>2n$, so they are compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set and bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable. Hence $\psi_{k,n}\in C^{1}_{b}(\mathbb{R}^{d})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, and $\varphi_{k,n}=\psi_{k,n}\circ p_{d}\in\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with $\partial_{k}\varphi_{k,n}=(\partial_{k}\psi_{k,n})\circ p_{d}$ 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.

Write $\kappa_{n}=\chi_{n}\circ p_{d}:X\to\mathbb{R}$. It is Borel: $\chi_{n}$ is continuous relative to $\mathbb{R}^{d}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous (smooth case), hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, $p_{d}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and the composite is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Moreover $0\le\kappa_{n}\le1$, and $\kappa_{n}(x)=1$ whenever $\lVert p_{d}(x)\rVert\le n$. Since $\partial_{k}V=(\partial_{k}v)\circ p_{d}$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, (9.1) with $i=k$ gives, for every $x\in X$,

$$\varphi_{k,n}(x)=\partial_{k}V(x)\,\kappa_{n}(x)^{2},\qquad\partial_{k}\varphi_{k,n}(x)=\partial_{k}\partial_{k}v(p_{d}(x))\,\kappa_{n}(x)^{2}+2\,\partial_{k}V(x)\,\kappa_{n}(x)\,\partial_{k}\chi_{n}(p_{d}(x)). \tag{9.2}$$

Step 10 (Claim 4, the converse: an integral identity). Let $\mu\in\mathcal{P}_{2}(X)$ with $V$ integrable with respect to $\mu$, so that $\partial_{k}V$ is integrable for $k\in[d]$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable, and let $\mu$ have a relative score $(\zeta^{V}_{k})_{k\in\mathbb{N}}$ with respect to $\gamma^{V}$ and finite Fisher information relative to $\gamma^{V}$ with weights $a$; write $I^{V}=\mathcal{I}_{a}(\mu\,|\,\gamma^{V})$ and norms and inner products in $L^{2}(\mu)$ as $\lVert\cdot\rVert$ and $\langle\cdot,\cdot\rangle$. Let $s(x)=\sum_{k=1}^{d}x_{k}^{2}$ for $x\in X$; $s$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $p_{d}(x)=(x_{1},\dots,x_{d})$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, Euclidean Norm on $\mathbb{R}^n$ gives $s(x)=\lVert p_{d}(x)\rVert^{2}$, and Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity gives $\lVert p_{d}(x)\rVert^{2}=|P_{d}x|^{2}\le|P_{d}x|^{2}+|Q_{d}x|^{2}=|x|^{2}$. As $\int_{X}|x|^{2}\,\mu(dx)<\infty$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, monotonicity shows that $s$ is integrable with respect to $\mu$; put $m=\int_{X}s\,d\mu=\int_{X}\sum_{k=1}^{d}x_{k}^{2}\,\mu(dx)\ge0$. Fix $n\in\mathbb{N}$.

(a) For $k\in[d]$, $\varphi_{k,n}$ is 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, say $|\varphi_{k,n}|\le B$; so $\partial_{k}V\,\varphi_{k,n}=(\partial_{k}V)^{2}\kappa_{n}^{2}$ is Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) with $|\partial_{k}V\,\varphi_{k,n}|\le B|\partial_{k}V|$, hence integrable with respect to $\mu$ by monotonicity. Therefore $|\nabla_{a}V|_{a}^{2}\kappa_{n}^{2}=\sum_{k=1}^{d}a_{k}\,\partial_{k}V\,\varphi_{k,n}$ is integrable, and

$$A_{n}=\int_{X}|\nabla_{a}V|_{a}^{2}\,\kappa_{n}^{2}\,d\mu=\sum_{k=1}^{d}a_{k}\int_{X}(\partial_{k}V)^{2}\kappa_{n}^{2}\,d\mu$$

is a nonnegative real number.

(b) For $k\in[d]$, the functions $x\mapsto x_{k}\varphi_{k,n}(x)$, $\partial_{k}V\,\varphi_{k,n}$ and $\partial_{k}\varphi_{k,n}$ are integrable with respect to $\mu$ 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, (a) and 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. So The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score with $\varphi=\varphi_{k,n}$ and linearity give

$$\langle\zeta^{V}_{k},\varphi_{k,n}\rangle=\frac{1}{c_{k}}\int_{X}x_{k}\varphi_{k,n}(x)\,\mu(dx)+\frac{1}{\beta}\int_{X}\partial_{k}V\,\varphi_{k,n}\,d\mu-\int_{X}\partial_{k}\varphi_{k,n}\,d\mu .$$

Multiplying by $a_{k}$, summing over $k\in[d]$ and using (a) and linearity,

$$\frac{1}{\beta}A_{n}=S_{1}+S_{2}+S_{3}, \tag{10.1}$$ $$S_{1}=\sum_{k=1}^{d}a_{k}\langle\zeta^{V}_{k},\varphi_{k,n}\rangle,\qquad S_{2}=-\int_{X}\sum_{k=1}^{d}\frac{a_{k}}{c_{k}}x_{k}\varphi_{k,n}(x)\,\mu(dx),\qquad S_{3}=\int_{X}\sum_{k=1}^{d}a_{k}\,\partial_{k}\varphi_{k,n}\,d\mu .$$

Step 11 (The estimate). With the data of Step 10:

$S_{1}$. For $k\in[d]$, $\langle\zeta^{V}_{k},\varphi_{k,n}\rangle\le\lVert\zeta^{V}_{k}\rVert\,\lVert\varphi_{k,n}\rVert$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space in the inner product space $L^{2}(\mu)$, and by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, (9.2), $\kappa_{n}^{4}\le\kappa_{n}^{2}$ and monotonicity, $\lVert\varphi_{k,n}\rVert^{2}=\int_{X}(\partial_{k}V)^{2}\kappa_{n}^{4}\,d\mu\le\int_{X}(\partial_{k}V)^{2}\kappa_{n}^{2}\,d\mu$. With $pq\le2\beta p^{2}+q^{2}/(8\beta)$,

$$a_{k}\langle\zeta^{V}_{k},\varphi_{k,n}\rangle\le2\beta a_{k}\lVert\zeta^{V}_{k}\rVert^{2}+\frac{a_{k}}{8\beta}\int_{X}(\partial_{k}V)^{2}\kappa_{n}^{2}\,d\mu .$$

Summing over $k\in[d]$, and using that $\sum_{k=1}^{d}a_{k}\lVert\zeta^{V}_{k}\rVert^{2}$ is a partial sum of the series of nonnegative terms defining $I^{V}$ in The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information, hence at most $I^{V}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, and (a): $S_{1}\le2\beta I^{V}+A_{n}/(8\beta)$.

$S_{2}$. Fix $x\in X$ and $k\in[d]$, and apply $pq\le2\beta p^{2}+q^{2}/(8\beta)$ with $p=|x_{k}|\kappa_{n}(x)/c_{k}$ and $q=|\partial_{k}V(x)|\kappa_{n}(x)$: by (9.2), $a_{k}/c_{k}^{2}\le\Lambda$ and $\kappa_{n}^{2}\le1$,

$$\Bigl|\frac{a_{k}}{c_{k}}x_{k}\varphi_{k,n}(x)\Bigr|=a_{k}pq\le2\beta\frac{a_{k}}{c_{k}^{2}}x_{k}^{2}+\frac{a_{k}}{8\beta}\partial_{k}V(x)^{2}\kappa_{n}(x)^{2}\le2\beta\Lambda x_{k}^{2}+\frac{a_{k}}{8\beta}\partial_{k}V(x)^{2}\kappa_{n}(x)^{2}.$$

Summing over $k$, $-\sum_{k=1}^{d}\frac{a_{k}}{c_{k}}x_{k}\varphi_{k,n}(x)\le2\beta\Lambda s(x)+(8\beta)^{-1}|\nabla_{a}V(x)|_{a}^{2}\kappa_{n}(x)^{2}$; both sides are integrable, so monotonicity gives $S_{2}\le2\beta\Lambda m+A_{n}/(8\beta)$.

$S_{3}$. Fix $x\in X$ and put $u=p_{d}(x)$, so $u_{k}=x_{k}$ and $\partial_{k}v(u)=\partial_{k}V(x)$. By Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §curvature with $\varepsilon=1/(8\beta)$ and $C^{+}$ (Step 8),

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v(u)\le\frac{1}{8\beta}\sum_{k=1}^{d}a_{k}\,\partial_{k}V(x)^{2}+C^{+}\bigl(1+s(x)\bigr);$$

multiplying by $\kappa_{n}(x)^{2}\in[0,1]$ and using $C^{+}(1+s(x))\ge0$, the first part of $\sum_{k}a_{k}\partial_{k}\varphi_{k,n}(x)$ in (9.2) is at most $(8\beta)^{-1}|\nabla_{a}V(x)|_{a}^{2}\kappa_{n}(x)^{2}+C^{+}(1+s(x))$. For the second part, $|\partial_{k}\chi_{n}(u)|\le M_{1}$ (Step 8) and $2pq\le p^{2}/(8\beta)+8\beta q^{2}$ with $p=|\partial_{k}V(x)|\kappa_{n}(x)$, $q=M_{1}$ give

$$2a_{k}\,\partial_{k}V(x)\,\kappa_{n}(x)\,\partial_{k}\chi_{n}(u)\le2a_{k}pq\le\frac{a_{k}}{8\beta}\partial_{k}V(x)^{2}\kappa_{n}(x)^{2}+8\beta M_{1}^{2}a_{k},$$

whose sum over $k\in[d]$ is $(8\beta)^{-1}|\nabla_{a}V(x)|_{a}^{2}\kappa_{n}(x)^{2}+8\beta M_{1}^{2}\alpha$. Hence

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\varphi_{k,n}(x)\le\frac{1}{4\beta}|\nabla_{a}V(x)|_{a}^{2}\kappa_{n}(x)^{2}+C^{+}\bigl(1+s(x)\bigr)+8\beta M_{1}^{2}\alpha ;$$

both sides are integrable, so monotonicity gives $S_{3}\le A_{n}/(4\beta)+C^{+}(1+m)+8\beta M_{1}^{2}\alpha$.

Combining with (10.1), $\beta^{-1}A_{n}\le(2\beta)^{-1}A_{n}+2\beta I^{V}+2\beta\Lambda m+C^{+}(1+m)+8\beta M_{1}^{2}\alpha$. Subtracting the real number $(2\beta)^{-1}A_{n}$ and multiplying by $2\beta$,

$$A_{n}\le4\beta^{2}I^{V}+\bigl(4\beta^{2}\Lambda+2\beta C^{+}\bigr)m+2\beta C^{+}+16\beta^{2}M_{1}^{2}\alpha\le C_{1}\bigl(1+\beta^{2}I^{V}+m\bigr), \tag{11.1}$$

the last step because each of the three coefficients is at most $C_{1}$ and $1$, $\beta^{2}I^{V}$, $m$ are nonnegative.

Step 12 (Removing the cutoff). With the data of Step 10, for $n\in\mathbb{N}$ let $B_{n}=\{x\in X:s(x)\le n^{2}\}$, which lies in $\mathcal{B}(X)$ since its complement $\{s>n^{2}\}$ does, by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable; let $f_{n}=|\nabla_{a}V|_{a}^{2}\,\mathbf{1}_{B_{n}}$, nonnegative and Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $n^{2}\le(n+1)^{2}$, $f_{n}\le f_{n+1}$; and for each $x$ there is $n\in\mathbb{N}$ with $s(x)\le n\le n^{2}$, so $x\in B_{N}$ for all $N\ge n$ and $\sup_{n}f_{n}(x)=|\nabla_{a}V(x)|_{a}^{2}$. For $x\in B_{n}$, $\lVert p_{d}(x)\rVert^{2}=s(x)\le n^{2}$, so $\lVert p_{d}(x)\rVert\le n$ and $\kappa_{n}(x)=1$ (Step 9); hence $f_{n}\le|\nabla_{a}V|_{a}^{2}\kappa_{n}^{2}$ everywhere, and $\int_{X}f_{n}\,d\mu\le A_{n}\le C_{1}(1+\beta^{2}I^{V}+m)$ by monotonicity and (11.1). By Monotone Convergence Theorem,

$$G=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu=\sup_{n}\int_{X}f_{n}\,d\mu\le C_{1}\bigl(1+\beta^{2}I^{V}+m\bigr)<\infty . \tag{12.1}$$

Step 13 (Claim 4, the converse concluded). With the data of Step 10, (12.1) and Step 6(a) give $\partial_{k}V\in L^{2}(\mu)$ for every $k\in\mathbb{N}$ and the convergence of $\sum_{k}a_{k}\lVert\partial_{k}V\rVert^{2}$ with sum $G$. Put $\zeta_{k}=\zeta^{V}_{k}-\beta^{-1}\partial_{k}V\in L^{2}(\mu)$. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$, $\partial_{k}V\varphi$ and $x_{k}\varphi/c_{k}-\partial_{k}\varphi$ are integrable (preamble of The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential), so by linearity of the inner product, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score and linearity of the integral,

$$\langle\zeta_{k},\varphi\rangle=\langle\zeta^{V}_{k},\varphi\rangle-\frac{1}{\beta}\int_{X}\partial_{k}V\,\varphi\,d\mu=\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\varphi(x)-\partial_{k}\varphi(x)\Bigr)\,\mu(dx).$$

So $(\zeta_{k})_{k\in\mathbb{N}}$ is a relative score of $\mu$ with respect to $\gamma_{c}$ by The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score, the relative score by the uniqueness recorded there, and $\zeta^{V}_{k}=\zeta_{k}+\beta^{-1}\partial_{k}V$. By Step 6(b) with $t=-\beta^{-1}$, $0\le a_{k}\lVert\zeta_{k}\rVert^{2}\le2a_{k}\lVert\zeta^{V}_{k}\rVert^{2}+2\beta^{-2}a_{k}\lVert\partial_{k}V\rVert^{2}$, and the series of the right-hand sides converges with sum $2I^{V}+2\beta^{-2}G$ (its partial sums being the corresponding combination of partial sums). By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, $\mu$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$ (Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information) and

$$\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le2I^{V}+2\beta^{-2}G . \tag{13.1}$$

Together with $G<\infty$ from (12.1) and Step 7, this proves the equivalence of Claim 4; in either case both scores exist, so the component formula proved in Step 7 holds, and the class of $\partial_{k}V$ lies in $L^{2}(\mu)$ by Step 6(a).

Step 14 (Claim 5). Let $\mu$ be as in Claim 5; by Claim 4, $\zeta^{V}_{k}=\zeta_{k}+\beta^{-1}\partial_{k}V$ for every $k$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$, so the class of $\nabla_{a}V$ lies in $L^{2}(\mu;X^{a})$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent, and $Z^{a}_{\mu}\in L^{2}(\mu;X^{a})$ has coordinate $a_{k}^{1/2}\zeta_{k}$ along $f_{k}$ by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field. Coordinates of $\nabla_{a}V$: fix $x\in X$ and $k\in\mathbb{N}$, and let $y=\nabla_{a}V(x)=\sum_{j=1}^{d}a_{j}\partial_{j}V(x)e_{j}\in X^{a}$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient). Its coordinate along $f_{k}$ is $\langle y,f_{k}\rangle_{a}=a_{k}^{-1/2}\langle y,e_{k}\rangle$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis. Since $(e_{1},\dots,e_{d})$ is orthonormal, $\langle y,e_{k}\rangle=a_{k}\partial_{k}V(x)$ for $k\le d$ by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §coefficients, and for $k>d$, $\langle y,e_{k}\rangle=\sum_{j=1}^{d}a_{j}\partial_{j}V(x)\langle e_{j},e_{k}\rangle=0=a_{k}\partial_{k}V(x)$ by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and orthonormality of $(e_{j})_{j\in\mathbb{N}}$. As $a_{k}^{-1/2}a_{k}=a_{k}^{1/2}$, the coordinate is $a_{k}^{1/2}\partial_{k}V(x)$ in all cases, so by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the coordinate of the class of $\nabla_{a}V$ along $f_{k}$ is $a_{k}^{1/2}\partial_{k}V\in L^{2}(\mu)$ (Step 6(a)). By the linearity of coordinates in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the coordinate of $\beta Z^{a}_{\mu}+\nabla_{a}V$ along $f_{k}$ is

$$\beta a_{k}^{1/2}\zeta_{k}+a_{k}^{1/2}\partial_{k}V=\beta a_{k}^{1/2}\bigl(\zeta_{k}+\beta^{-1}\partial_{k}V\bigr)=\beta a_{k}^{1/2}\zeta^{V}_{k}.$$

By the norm identity of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates and Elementary Identities in a Real Inner Product Space §homogeneity, $\lVert\beta Z^{a}_{\mu}+\nabla_{a}V\rVert_{\mu}^{2}=\sum_{k=1}^{\infty}\beta^{2}a_{k}\lVert\zeta^{V}_{k}\rVert^{2}$; the partial sums of this series are $\beta^{2}$ times those of the series defining $\mathcal{I}_{a}(\mu\,|\,\gamma^{V})$ in The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information, so its sum is $\beta^{2}\mathcal{I}_{a}(\mu\,|\,\gamma^{V})$. This proves Claim 5.

Step 15 (Claim 6). Let $C_{*}$ be the constant of Step 8, and let $\mu\in\mathcal{P}_{2}(X)$ be as in Claim 6; these are the data of Step 10. With $G$ and $m$ as there, (13.1) gives $\beta^{2}\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le2\beta^{2}I^{V}+2G$, and (12.1) gives $G\le C_{1}(1+\beta^{2}I^{V}+m)$. Hence

$$\beta^{2}\mathcal{I}_{a}(\mu\,|\,\gamma_{c})+G\le2\beta^{2}I^{V}+3G\le2\beta^{2}I^{V}+3C_{1}\bigl(1+\beta^{2}I^{V}+m\bigr)\le(2+3C_{1})\bigl(1+\beta^{2}I^{V}+m\bigr)=C_{*}\Bigl(1+\beta^{2}\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})+\int_{X}\sum_{k=1}^{d}x_{k}^{2}\,\mu(dx)\Bigr),$$

using that $1$, $\beta^{2}I^{V}$ and $m$ are nonnegative. Since $C_{*}$ was fixed in Step 8 from $V$, $\beta$, $a$ and $c$ alone, this proves Claim 6. $\qquad\blacksquare$