Proof of Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields

Each result cited is universally quantified over the data in its own statement. Throughout, $\lambda_{d}$ is Lebesgue measure on $\mathbb{R}^{d}$ (Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (v)), $e_{i}$ is the $i$th standard basis vector of $\mathbb{R}^{d}$, the dot product $a\cdot x=\sum_{i=1}^{d}a_{i}x_{i}$ is that of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, so that $a\cdot e_{i}=a_{i}$ by claim 7 of Properties of Finite Sums, the set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space Is Open in Itself, and $\mathbb{N}$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background (i). A function of class $C^{1}$ on $\mathbb{R}^{d}$ and its partial derivatives are continuous on $\mathbb{R}^{d}$ by clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ and claim 3 of Euclidean Space Is Open in Itself, hence Borel by claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; a smooth function is of class $C^{k}$ for every $k$ (Smooth Functions on an Open Subset of ℝⁿ). Two elementary facts about partial derivatives are used repeatedly and are proved here from Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$.

(F1) Uniqueness. If $L$ and $L'$ both have the property of that definition at $a$ for $f$ and $i$, then for every $\varepsilon>0$, applying the definition with $\varepsilon/2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) to both values and taking an $h\ne0$ smaller than both radii, the same quotient lies within $\varepsilon/2$ of $L$ and of $L'$, so $|L-L'|<\varepsilon$ by claim 5 of Properties of the Absolute Value in an Ordered Field; hence $|L-L'|\le\varepsilon$ for every $\varepsilon>0$ and $L=L'$ by Comparison of Real Numbers with Arbitrary Positive Slack (iii).

(F2) Locality. If $g:\mathbb{R}^{d}\to\mathbb{R}$ is constant on the set $\{x':\lVert x'-x\rVert<r\}$ for some $r>0$, then $\partial_{i}g(x)$ exists and equals $0$: for $0<|h|<r$ one has $\lVert(x+he_{i})-x\rVert=|h|<r$ by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, so the difference quotient in the definition is $0$, and $L=0$ satisfies the definition with $\delta=r$ for every $\varepsilon$.

Proof of claim 1.

Growth. Let $x\in\mathbb{R}^{d}$. The segment from $0_{\mathbb{R}^{d}}$ to $x$ lies in $\mathbb{R}^{d}$, so part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, applied with $W=\mathbb{R}^{d}$, the points $0_{\mathbb{R}^{d}}$ and $x$, and $M_{1}=M$, gives $|f(x)-f(0_{\mathbb{R}^{d}})|\le\sqrt{d}\,M\lVert x\rVert$, the distance between the two points being $\lVert x\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Hence $|f(x)|\le|f(0_{\mathbb{R}^{d}})|+|f(x)-f(0_{\mathbb{R}^{d}})|\le|f(0_{\mathbb{R}^{d}})|+\sqrt{d}\,M\lVert x\rVert$ by claim 5 of Properties of the Absolute Value in an Ordered Field.

Square integrability. Each $\partial_{i}f$ is Borel, so $Df$ is Borel by the componentwise criterion of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). For every $x$, claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square gives $\lVert Df(x)\rVert^{2}=\sum_{i=1}^{d}(\partial_{i}f(x))^{2}$, and $(\partial_{i}f(x))^{2}=|\partial_{i}f(x)|^{2}\le M^{2}$ by claim 4 of Properties of the Absolute Value in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $\lVert Df(x)\rVert^{2}\le dM^{2}$ by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers. The function $\lVert Df\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), and the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion together with Simple Functions, Nonnegative Simple Functions, and the Simple Integral (the integral of the constant $dM^{2}$ against the probability measure $\mu$ is $dM^{2}$) gives $\int\lVert Df\rVert^{2}\,d\mu\le dM^{2}<\infty$. Thus the class of $Df$ lies in $L^{2}(\mu;\mathbb{R}^{d})$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii)).

Mollification. By Existence of Mollifier Kernels of Every Radius there is a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{d}$; for $\varepsilon>0$ let $\rho_{\varepsilon}(y)=(\varepsilon^{-1})^{d}\rho(\varepsilon^{-1}y)$, a mollifier kernel of radius $\varepsilon$ by Rescaling a Mollifier Kernel: it is smooth, nonnegative, vanishes at every $y$ with $\lVert y\rVert>\varepsilon$, and $\int\rho_{\varepsilon}\,d\lambda_{d}=1$. Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be continuous. Since $\bar B(x,\varepsilon)\subseteq\mathbb{R}^{d}$ for every $x$, the convolution $g*\rho_{\varepsilon}$ is defined on all of $\mathbb{R}^{d}$ (the set $\Omega^{\varepsilon}$ of that definition with $\Omega=\mathbb{R}^{d}$), $(g*\rho_{\varepsilon})(x)=\int g(x-y)\rho_{\varepsilon}(y)\,\lambda_{d}(dy)$, and it is smooth on $\mathbb{R}^{d}$ by claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$. If $|g|\le C$ on $\mathbb{R}^{d}$, then $|g(x-y)\rho_{\varepsilon}(y)|\le C\rho_{\varepsilon}(y)$ for all $y$, so $|(g*\rho_{\varepsilon})(x)|\le\int C\rho_{\varepsilon}\,d\lambda_{d}=C$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (the integrand is integrable by claim 1 of The Convolution Integrand is Continuous, Compactly Supported and Integrable). Likewise, since $\rho_{\varepsilon}(y)=0$ for $\lVert y\rVert>\varepsilon$, the growth bound gives $|f(x-y)\rho_{\varepsilon}(y)|\le\bigl(|f(0_{\mathbb{R}^{d}})|+\sqrt{d}\,M(\lVert x\rVert+\varepsilon)\bigr)\rho_{\varepsilon}(y)$ for all $y$ (claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square: $\lVert x-y\rVert\le\lVert x\rVert+\lVert y\rVert$), whence

$$|(f*\rho_{\varepsilon})(x)|\le|f(0_{\mathbb{R}^{d}})|+\sqrt{d}\,M(\lVert x\rVert+\varepsilon)\qquad(x\in\mathbb{R}^{d}).\tag{G}$$

The derivative of a mollification. Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ with $|\partial_{i}g|\le C$ on $\mathbb{R}^{d}$ for all $i$. We show

$$\partial_{i}(g*\rho_{\varepsilon})=(\partial_{i}g)*\rho_{\varepsilon}\qquad\text{on }\mathbb{R}^{d},\ \text{for every }i\in[d].\tag{D}$$

Fix $x$ and $i$. By claim 2 of Differentiating a Convolution through the Kernel, $\partial_{i}(g*\rho_{\varepsilon})(x)=\int g(x-y)\,\partial_{i}\rho_{\varepsilon}(y)\,\lambda_{d}(dy)$, the integrand being integrable by claim 1 of The Convolution Integrand is Continuous, Compactly Supported and Integrable applied to the kernel $\partial_{i}\rho_{\varepsilon}$ (claim 1 of Differentiating a Convolution through the Kernel). By claim 3 of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ with $a=x$ (applied to the integrable function $y\mapsto g(x-y)\partial_{i}\rho_{\varepsilon}(y)$, evaluated at $x-y$), this equals $\int g(y)\,\partial_{i}\rho_{\varepsilon}(x-y)\,\lambda_{d}(dy)$. Let $k(y)=\rho_{\varepsilon}(x-y)$. For $y\in\mathbb{R}^{d}$ and $j\in[d]$, the function $t\mapsto k(y+te_{j})=\rho_{\varepsilon}\bigl((x-y)+t(-e_{j})\bigr)$ is differentiable at $0$ with derivative $\sum_{l}\partial_{l}\rho_{\varepsilon}(x-y)(-e_{j})_{l}=-\partial_{j}\rho_{\varepsilon}(x-y)$ (claim 7 of Properties of Finite Sums) by Chain Rule Along an Affine Path, since $\rho_{\varepsilon}$ is differentiable at $x-y$ by Continuous Partial Derivatives Imply Differentiability; its difference quotients at $0$ are exactly those of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $k$ at $y$, so $\partial_{j}k(y)=-\partial_{j}\rho_{\varepsilon}(x-y)$ by (F1). These partials are continuous (a composition of continuous maps, claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, the map $y\mapsto x-y$ being continuous as $\lVert(x-y)-(x-y')\rVert=\lVert y'-y\rVert$), as is $k$, so $k$ is of class $C^{1}$ on $\mathbb{R}^{d}$. Moreover $k(y)=0$ whenever $\lVert y\rVert>\lVert x\rVert+\varepsilon$, since then $\lVert x-y\rVert\ge\lVert y\rVert-\lVert x\rVert>\varepsilon$ (claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square: $\lVert y\rVert\le\lVert y-x\rVert+\lVert x\rVert$ and $\lVert y-x\rVert=\lVert x-y\rVert$); so $k$ is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ (ii), applied to $g$ and $k$, gives $\int g\,\partial_{i}k\,d\lambda_{d}=-\int(\partial_{i}g)\,k\,d\lambda_{d}$, that is, $\int g(y)\,\partial_{i}\rho_{\varepsilon}(x-y)\,\lambda_{d}(dy)=\int\partial_{i}g(y)\,\rho_{\varepsilon}(x-y)\,\lambda_{d}(dy)$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion with the factor $-1$). By claim 3 of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ once more, the right side is $\int\partial_{i}g(x-y)\,\rho_{\varepsilon}(y)\,\lambda_{d}(dy)=((\partial_{i}g)*\rho_{\varepsilon})(x)$, the function $\partial_{i}g$ being continuous. This proves (D). Consequently, by the bound above, $|\partial_{i}(g*\rho_{\varepsilon})|\le C$ on $\mathbb{R}^{d}$.

Uniform convergence on balls. Let $g$ be continuous on $\mathbb{R}^{d}$, let $n\in\mathbb{N}$ and let $K_{n}=\bar B(0_{\mathbb{R}^{d}},n)$, which is compact by claim 2 of A Closed Euclidean Ball is Convex and Compact. Claim 2 of Mollification Converges Uniformly on Compact Subsets, applied with $\Omega=\mathbb{R}^{d}$, the kernel $\rho$ of radius $1$ and the compact set $K_{n}$, gives for every $\eta>0$ an $\varepsilon_{0}>0$ such that $|(g*\rho_{\varepsilon})(x)-g(x)|<\eta$ for all $x\in K_{n}$ and $0<\varepsilon<\varepsilon_{0}$.

Cutoff. Fix $\chi$ as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball for the dimension $d$, and for $n\in\mathbb{N}$ let $\chi_{n}$ be the function $\chi_{R}$ of that lemma with $R=n$; by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball (i) it is smooth and compactly supported, $0\le\chi_{n}\le1$, $\chi_{n}(x)=1$ for $\lVert x\rVert\le n$, $\chi_{n}(x)=0$ for $\lVert x\rVert\ge2n$, and $|\partial_{i}\chi_{n}|\le M_{1}n^{-1}$, $|\partial_{j}\partial_{i}\chi_{n}|\le M_{2}n^{-2}$ on $\mathbb{R}^{d}$, with $M_{1},M_{2}$ independent of $n$. By (F2): if $\lVert x\rVert<n$ then $\chi_{n}$ is constant on $\{x':\lVert x'-x\rVert<n-\lVert x\rVert\}$ (claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), so $\partial_{i}\chi_{n}(x)=0$, and then, $\partial_{i}\chi_{n}$ being constant on the same set, $\partial_{j}\partial_{i}\chi_{n}(x)=0$; if $\lVert x\rVert>2n$ then $\chi_{n}$ is constant on $\{x':\lVert x'-x\rVert<\lVert x\rVert-2n\}$ (claim 7 of Properties of the Absolute Value in an Ordered Field applied through claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square: $\lVert x'\rVert\ge\lVert x\rVert-\lVert x-x'\rVert>2n$), so $\chi_{n}(x)=\partial_{i}\chi_{n}(x)=\partial_{j}\partial_{i}\chi_{n}(x)=0$.

For $n\in\mathbb{N}$ and $0<\varepsilon\le n$ put

$$\psi_{n,\varepsilon}=\chi_{n}\cdot(f*\rho_{\varepsilon}).$$

It is smooth on $\mathbb{R}^{d}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, and it vanishes at every $x$ with $\lVert x\rVert>2n$, as $\chi_{n}$ does, so it is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. Hence $\psi_{n,\varepsilon}\in C_{c}^{\infty}(\mathbb{R}^{d})$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (i)), and $\nabla\psi_{n,\varepsilon}\in G_{\mu}$ (The Tangent Space of the Wasserstein Space at a Probability Measure (i)). By claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions and (D) applied to $g=f$,

$$\partial_{i}\psi_{n,\varepsilon}=\partial_{i}\chi_{n}\cdot(f*\rho_{\varepsilon})+\chi_{n}\cdot\bigl((\partial_{i}f)*\rho_{\varepsilon}\bigr).\tag{P}$$

The gradient estimate. Let $n\in\mathbb{N}$, $\eta>0$, and let $\varepsilon_{0}$ be provided by the uniform convergence paragraph for the $d$ continuous functions $\partial_{1}f,\dots,\partial_{d}f$ on $K_{n}$ simultaneously (the least of the $d$ radii, obtained by applying claim 9 of Elementary Order Arithmetic in an Ordered Field repeatedly). Let $0<\varepsilon<\varepsilon_{0}$ with $\varepsilon\le n$. For $\lVert x\rVert<n$, (P) and the cutoff paragraph give $\partial_{i}\psi_{n,\varepsilon}(x)=((\partial_{i}f)*\rho_{\varepsilon})(x)$, so $|\partial_{i}\psi_{n,\varepsilon}(x)-\partial_{i}f(x)|<\eta$ and, by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, $\lVert\nabla\psi_{n,\varepsilon}(x)-Df(x)\rVert^{2}\le d\eta^{2}$. For $\lVert x\rVert>2n$, $\partial_{i}\psi_{n,\varepsilon}(x)=0$ by (P). For $n\le\lVert x\rVert\le2n$, (P), (G), the cutoff bounds, the bound $|(\partial_{i}f)*\rho_{\varepsilon}|\le M$ and $\varepsilon\le n$, $1\le n$ give

$$|\partial_{i}\psi_{n,\varepsilon}(x)|\le M_{1}n^{-1}\bigl(|f(0_{\mathbb{R}^{d}})|+3n\sqrt{d}\,M\bigr)+M\le M_{1}|f(0_{\mathbb{R}^{d}})|+3\sqrt{d}\,MM_{1}+M=:C_{1},$$

using claims 4 and 5 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field. Hence for $\lVert x\rVert\ge n$, $\lVert\nabla\psi_{n,\varepsilon}(x)\rVert\le\sqrt{d}\,C_{1}$ and $\lVert Df(x)\rVert\le\sqrt{d}\,M$ (as above), so $\lVert\nabla\psi_{n,\varepsilon}(x)-Df(x)\rVert^{2}\le C_{2}^{2}$ with $C_{2}=\sqrt{d}(C_{1}+M)$, by claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Writing $A_{n}=\{x:\lVert x\rVert\ge n\}$, a Borel set by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) and The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, the pointwise bound $\lVert\nabla\psi_{n,\varepsilon}-Df\rVert^{2}\le d\eta^{2}+C_{2}^{2}\mathbf{1}_{A_{n}}$ holds on $\mathbb{R}^{d}$, and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion with Simple Functions, Nonnegative Simple Functions, and the Simple Integral gives

$$\lVert\nabla\psi_{n,\varepsilon}-Df\rVert_{\mu}^{2}\le d\eta^{2}+C_{2}^{2}\,\mu(A_{n})\le d\eta^{2}+C_{2}^{2}\,n^{-2}M_{2}(\mu),\tag{E1}$$

where $\mu(A_{n})=\mu(\{\lVert x\rVert^{2}\ge n^{2}\})\le n^{-2}\int\lVert x\rVert^{2}\mu(dx)=n^{-2}M_{2}(\mu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, The Lebesgue Integral and Null Sets (vi) and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i).

Conclusion of claim 1. Let $k\in\mathbb{N}$. By The Archimedean Property of the Real Numbers choose $n_{k}\in\mathbb{N}$ with $C_{2}^{2}M_{2}(\mu)n_{k}^{-2}<(2k^{2})^{-1}$, then $\eta_{k}>0$ with $d\eta_{k}^{2}<(2k^{2})^{-1}$ (claim 8 of Elementary Order Arithmetic in an Ordered Field and Existence and Uniqueness of the Nonnegative Square Root), then $\varepsilon_{k}$ as in the gradient estimate for $(n_{k},\eta_{k})$, and put $\psi_{k}=\psi_{n_{k},\varepsilon_{k}}$. Then $\lVert\nabla\psi_{k}-Df\rVert_{\mu}^{2}<k^{-2}$, i.e. $d_{\mu}(\nabla\psi_{k},Df)<k^{-1}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). Hence $\nabla\psi_{k}\to Df$ in the metric space $(L^{2}(\mu;\mathbb{R}^{d}),d_{\mu})$ (Convergent Sequence in a Metric Space, with The Archimedean Property of the Real Numbers), and since every $\nabla\psi_{k}\in G_{\mu}$, the class $Df$ lies in the closure $\overline{G_{\mu}}=T_{\mu}$ by Sequential Characterization of the Closure in a Metric Space and The Tangent Space of the Wasserstein Space at a Probability Measure (ii).

Proof of claim 2. Let $a\in\mathbb{R}^{d}$ and $f_{a}(x)=a\cdot x=\sum_{i=1}^{d}a_{i}x_{i}$. For $x\in\mathbb{R}^{d}$, $i\in[d]$ and $h\ne0$, the difference quotient of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ is $(f_{a}(x+he_{i})-f_{a}(x))/h=(a\cdot(he_{i}))/h=a_{i}h/h=a_{i}$ (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$; claims 2 and 3 of Properties of Finite Sums for the additivity and homogeneity of the dot product in its second argument, and $a\cdot e_{i}=a_{i}$), so $\partial_{i}f_{a}(x)=a_{i}$ by that definition and (F1); the partials are constant, hence continuous, and $f_{a}$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space, so $f_{a}$ is of class $C^{1}$ on $\mathbb{R}^{d}$; and $|\partial_{i}f_{a}|=|a_{i}|\le\lVert a\rVert$ by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. By claim 1 with $M=\lVert a\rVert$, the class of $Df_{a}$, the constant map with value $a$, lies in $T_{\mu}$; that class is the class written $a$.

Proof of claim 3. Now $f$ is of class $C^{2}$ with $|\partial_{j}\partial_{i}f|\le M$. Each $\partial_{j}\partial_{i}f$ is continuous (clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ), so $\Delta f=\sum_{i}\partial_{i}\partial_{i}f$ (The Laplacian of a Twice Continuously Differentiable Function (i)) is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space, hence Borel, and $|\Delta f|\le dM$ by claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers; being bounded and Borel it is integrable with respect to $\mu$ (Integrable Functions and Their Lebesgue Integral and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion).

Retain the sequence $\psi_{k}=\psi_{n_{k},\varepsilon_{k}}$ of the proof of claim 1, choosing $\varepsilon_{k}$ now also smaller than the radius provided by the uniform convergence paragraph for the $d$ continuous functions $\partial_{i}\partial_{i}f$ on $K_{n_{k}}$ with the same $\eta_{k}$; (E1) still holds. By Finite Fisher Information, the Score and the Fisher Information of a Probability Measure (ii), $\langle\xi_{\mu},\nabla\psi_{k}\rangle_{\mu}=-\int\Delta\psi_{k}\,d\mu$ for every $k$. The left side converges to $\langle\xi_{\mu},Df\rangle_{\mu}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (v), since $\nabla\psi_{k}\to Df$ in $L^{2}(\mu;\mathbb{R}^{d})$. It remains to show $\int\Delta\psi_{k}\,d\mu\to\int\Delta f\,d\mu$; then $\langle\xi_{\mu},Df\rangle_{\mu}=-\int\Delta f\,d\mu$ by claim 3 of Arithmetic of Limits of Real Sequences (negation of limits) and claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences (uniqueness of limits).

Fix $n,\varepsilon$ as in the construction and write $\psi=\psi_{n,\varepsilon}$, $F=f*\rho_{\varepsilon}$. By (D) applied to $g=f$ and then to $g=\partial_{i}f$ (of class $C^{1}$ with partials bounded by $M$), $\partial_{i}F=(\partial_{i}f)*\rho_{\varepsilon}$ and $\partial_{j}\partial_{i}F=(\partial_{j}\partial_{i}f)*\rho_{\varepsilon}$, so $|\partial_{i}F|\le M$ and $|\partial_{j}\partial_{i}F|\le M$. Differentiating (P) with claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions,

$$\partial_{i}\partial_{i}\psi=\partial_{i}\partial_{i}\chi_{n}\cdot F+2\,\partial_{i}\chi_{n}\cdot\partial_{i}F+\chi_{n}\cdot\partial_{i}\partial_{i}F,$$

and summing over $i$ (Properties of Finite Sums, claim 2), $\Delta\psi=\Delta\chi_{n}\cdot F+2\sum_{i}\partial_{i}\chi_{n}\,\partial_{i}F+\chi_{n}\,\Delta F$. For $\lVert x\rVert<n$ the cutoff paragraph gives $\Delta\psi(x)=\Delta F(x)=\sum_{i}((\partial_{i}\partial_{i}f)*\rho_{\varepsilon})(x)$, so $|\Delta\psi(x)-\Delta f(x)|\le\sum_{i}|((\partial_{i}\partial_{i}f)*\rho_{\varepsilon})(x)-\partial_{i}\partial_{i}f(x)|<d\eta$ (claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers) when $\varepsilon$ is below the radius chosen for these $d$ functions. For $\lVert x\rVert>2n$, $\Delta\psi(x)=0$, so $|\Delta\psi(x)-\Delta f(x)|\le dM$. For $n\le\lVert x\rVert\le2n$, by (G), the cutoff bounds and $1\le n$, $\varepsilon\le n$,

$$|\Delta\psi(x)|\le dM_{2}n^{-2}\bigl(|f(0_{\mathbb{R}^{d}})|+3n\sqrt{d}\,M\bigr)+2dM_{1}n^{-1}M+dM\le dM_{2}\bigl(|f(0_{\mathbb{R}^{d}})|+3\sqrt{d}\,M\bigr)+2dM_{1}M+dM=:C_{3}',$$

so $|\Delta\psi(x)-\Delta f(x)|\le C_{3}'+dM=:C_{3}$, and $C_{3}\ge dM$. Hence $|\Delta\psi-\Delta f|\le d\eta+C_{3}\mathbf{1}_{A_{n}}$ on $\mathbb{R}^{d}$, and by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (both functions are bounded Borel, hence integrable), claim 1 of that theorem, Simple Functions, Nonnegative Simple Functions, and the Simple Integral and the Markov bound above,

$$\Bigl|\int\Delta\psi\,d\mu-\int\Delta f\,d\mu\Bigr|\le\int|\Delta\psi-\Delta f|\,d\mu\le d\eta+C_{3}\,n^{-2}M_{2}(\mu).\tag{E2}$$

With the choices made for $\psi_{k}$ (and, if necessary, $n_{k}$ chosen also so large that $C_{3}M_{2}(\mu)n_{k}^{-2}<(2k)^{-1}$ and $\eta_{k}$ also with $d\eta_{k}<(2k)^{-1}$, which does not affect (E1)), (E2) gives $|\int\Delta\psi_{k}\,d\mu-\int\Delta f\,d\mu|<k^{-1}$ for every $k$, so $\int\Delta\psi_{k}\,d\mu\to\int\Delta f\,d\mu$ by Limit of a Sequence of Real Numbers and The Archimedean Property of the Real Numbers. This proves claim 3.

Proof of claim 4. Let $a\in\mathbb{R}^{d}$ and $f_{a}(x)=a\cdot x$ as in the proof of claim 2. Its partials $\partial_{i}f_{a}=a_{i}$ are constant, so for all $i,j$ the difference quotients of $\partial_{i}f_{a}$ vanish and $\partial_{j}\partial_{i}f_{a}=0$ by Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ and (F1); these are continuous, so $f_{a}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ) with $|\partial_{i}f_{a}|\le\lVert a\rVert$ and $|\partial_{j}\partial_{i}f_{a}|=0\le\lVert a\rVert$, and $\Delta f_{a}=0$. Claim 3 with $M=\lVert a\rVert$ gives $\langle\xi_{\mu},a\rangle_{\mu}=\langle\xi_{\mu},Df_{a}\rangle_{\mu}=-\int0\,d\mu=0$ (Simple Functions, Nonnegative Simple Functions, and the Simple Integral; and $-0=0$).

Proof of claim 5. Let $a\in\mathbb{R}^{d}$ and $\eta\in T_{\mu}$, and write $\mu_{a}=(\tau_{a})_{\#}\mu$, a probability measure on $\mathbb{R}^{d}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), $\tau_{a}$ being Borel by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii). By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iv) with $S=\mathrm{id}$, $T=\tau_{a}$, the plan $(\mathrm{id},\tau_{a})_{\#}\mu$ is a coupling of $\mu$ and $\mu_{a}$ with cost $\int\lVert x-(x+a)\rVert^{2}\mu(dx)=\lVert a\rVert^{2}<\infty$ (claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and Simple Functions, Nonnegative Simple Functions, and the Simple Integral), so $\mu_{a}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii).

Transport of classes. For a Borel $\zeta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ write $\zeta^{a}=\zeta\circ\tau_{-a}$, the map $x\mapsto\zeta(x-a)$; it is Borel as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)), and $\zeta^{a}\circ\tau_{a}=\zeta$ since $\tau_{-a}(\tau_{a}(x))=x$. By claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (change of variables for the image measure $\mu_{a}$ of $\mu$ under $\tau_{a}$), for every Borel $\zeta$,

$$\int\lVert\zeta^{a}\rVert^{2}\,d\mu_{a}=\int\lVert\zeta^{a}\circ\tau_{a}\rVert^{2}\,d\mu=\int\lVert\zeta\rVert^{2}\,d\mu.\tag{T}$$

If $\zeta,\zeta'$ are Borel with $\mu(\{\zeta=\zeta'\})=1$, then $\{\zeta^{a}=\zeta'^{a}\}=\tau_{a}(\{\zeta=\zeta'\})$ has $\tau_{a}$-preimage $\{\zeta=\zeta'\}$, so $\mu_{a}(\{\zeta^{a}=\zeta'^{a}\})=\mu(\{\zeta=\zeta'\})=1$ by claim 1 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (the set $\{\zeta^{a}=\zeta'^{a}\}=\{x:\lVert\zeta^{a}(x)-\zeta'^{a}(x)\rVert=0\}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv)). Hence, for a representative $\eta$ of the given class, $\eta^{a}$ is Borel with $\int\lVert\eta^{a}\rVert^{2}d\mu_{a}=\int\lVert\eta\rVert^{2}d\mu<\infty$, its class in $L^{2}(\mu_{a};\mathbb{R}^{d})$ does not depend on the representative, and $\lVert\eta^{a}\rVert_{\mu_{a}}=\lVert\eta\rVert_{\mu}$ by (T) and Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). This proves all assertions of claim 5 except tangency.

Translates of test functions. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $\psi^{a}=\psi\circ\tau_{-a}$. For every function $g$ on $\mathbb{R}^{d}$, every $x$ and $i$, the difference quotients of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $g^{a}=g\circ\tau_{-a}$ at $x$ are those of $g$ at $x-a$, since $g^{a}(x+he_{i})=g(x-a+he_{i})$; so $\partial_{i}g^{a}$ exists at $x$ if and only if $\partial_{i}g$ exists at $x-a$, and then $\partial_{i}(g^{a})=(\partial_{i}g)^{a}$ by (F1). Also $g^{a}$ is continuous whenever $g$ is (claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, $\tau_{-a}$ being continuous as $\lVert\tau_{-a}(x)-\tau_{-a}(x')\rVert=\lVert x-x'\rVert$). By induction on $k$ (Principle of Induction for the Natural Numbers, on the set of $k$ such that for every $g$ of class $C^{k}$ the function $g^{a}$ is of class $C^{k}$ with $\partial_{i}(g^{a})=(\partial_{i}g)^{a}$), using clauses 1 and 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ, $\psi^{a}$ is of class $C^{k}$ for every $k$, i.e. smooth. By claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set there is $R>0$ with $\psi(y)=0$ for $\lVert y\rVert>R$; if $\lVert x\rVert>R+\lVert a\rVert$ then $\lVert x-a\rVert\ge\lVert x\rVert-\lVert a\rVert>R$ (claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), so $\psi^{a}(x)=\psi(x-a)=0$, and $\psi^{a}$ is compactly supported by the same claim. Thus $\psi^{a}\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $\nabla(\psi^{a})=(\nabla\psi)^{a}$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians (ii)), so $(\nabla\psi)^{a}\in G_{\mu_{a}}$.

Tangency. Since $\eta\in T_{\mu}=\overline{G_{\mu}}$, Sequential Characterization of the Closure in a Metric Space gives $\psi_{k}\in C_{c}^{\infty}(\mathbb{R}^{d})$ with $\lVert\nabla\psi_{k}-\eta\rVert_{\mu}\to0$. By (T) applied to $\zeta=\nabla\psi_{k}-\eta$, and since $(\nabla\psi_{k}-\eta)^{a}=(\nabla\psi_{k})^{a}-\eta^{a}$ pointwise, $\lVert(\nabla\psi_{k})^{a}-\eta^{a}\rVert_{\mu_{a}}=\lVert\nabla\psi_{k}-\eta\rVert_{\mu}\to0$. As $(\nabla\psi_{k})^{a}=\nabla(\psi_{k}^{a})\in G_{\mu_{a}}$, the class $\eta^{a}$ lies in $\overline{G_{\mu_{a}}}=T_{\mu_{a}}$ by Sequential Characterization of the Closure in a Metric Space and The Tangent Space of the Wasserstein Space at a Probability Measure (ii).