Proof of Gaussian Integration by Parts Relative to a Diagonal Gaussian Measure for Functions with a Bounded Hessian

Each result cited is universally quantified over the data in its own statement. Throughout, $c$, $M$ and $f$ are as in the statement. Borel maps are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the inner product of $L^{2}(\mu;\mathbb{R}^{d})$ is that of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, so that $\langle\eta,\theta\rangle_{\mu}=\int_{\mathbb{R}^{d}}\eta\cdot\theta\,d\mu$ for representatives $\eta,\theta$, and this space is a real Hilbert space by the same clause, in particular a real inner product space (Real Hilbert Space §hilbert). Elementary ordered-field arithmetic and order are used without citation (The Real Numbers: Standing Notation and Background §background). Linearity and monotonicity of the integral of integrable functions are those of Linearity and Monotonicity of the Lebesgue Integral §integrable, and additivity, positive homogeneity and monotonicity of the integral of nonnegative Borel functions are those of Linearity and Monotonicity of the Lebesgue Integral §nonnegative. A function of class $C^{2}$ on $\mathbb{R}^{d}$ (open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) is of class $C^{1}$ and its partial derivatives are of class $C^{1}$, by clause 2 of C^k Maps on a Euclidean Open Set; a smooth function is of class $C^{2}$ by Smooth Map on a Euclidean Open Set.

Step 0: three general facts. (I) Domination. Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$, let $g:\mathbb{R}^{d}\to\mathbb{R}$ be Borel and let $h:\mathbb{R}^{d}\to\mathbb{R}$ be integrable with respect to $\mu$ with $|g(x)|\le h(x)$ for every $x$. Then $h=|h|$, so $\int_{\mathbb{R}^{d}}|g|\,d\mu\le\int_{\mathbb{R}^{d}}|h|\,d\mu<\infty$ by monotonicity of the nonnegative integral and Integrable Function and the Lebesgue Integral, and $g$ is integrable by that definition ($|g|$ being Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). Likewise a nonnegative Borel $u:\mathbb{R}^{d}\to\mathbb{R}$ with $\int_{\mathbb{R}^{d}}u\,d\mu<\infty$ is integrable, since $|u|=u$. Consequently, for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$: the function $x\mapsto\lVert x\rVert^{2}$, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, is integrable, its integral being $M_{2}(\mu)<\infty$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space); constant functions are integrable, being Borel (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and bounded (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures); and for every Borel $\eta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu<\infty$ the function $x\mapsto\lVert\eta(x)\rVert^{2}$, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, is integrable.

(II) Two open sets. Let $r$ be a positive real number, $W=\{y\in\mathbb{R}^{d}:\lVert y\rVert>r\}$ and $B=\{y\in\mathbb{R}^{d}:\lVert y\rVert<r\}$. For $y,y'\in\mathbb{R}^{d}$ we have $y=y'+(y-y')$, $y'=y+(y'-y)$ and $\lVert y-y'\rVert=\lVert y'-y\rVert$ (claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space and the vector space structure of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ with $\lambda=-1$), and $d_{E}(y,y')=\lVert y-y'\rVert$ (claim 2 there). If $y\in W$ and $\lVert y-y'\rVert<\lVert y\rVert-r$, then $\lVert y\rVert\le\lVert y'\rVert+\lVert y-y'\rVert$ by the triangle inequality (claim 6 there), so $\lVert y'\rVert>r$; if $y\in B$ and $\lVert y-y'\rVert<r-\lVert y\rVert$, then $\lVert y'\rVert\le\lVert y\rVert+\lVert y'-y\rVert<r$. Thus each point of $W$, respectively $B$, is the centre of an open ball of positive radius contained in $W$, respectively $B$; so $W$ and $B$ are open in the metric sense, hence open by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$.

(III) Locality of partial derivatives. Let $V\subseteq\mathbb{R}^{d}$ be open and let $u,v:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ with $u(y)=v(y)$ for every $y\in V$. Then $u|_{V}=v|_{V}$, so by claim 1 of Restriction of a $C^k$ Map to an Open Subset, for every $x\in V$ and $l\in[d]$, $\partial_{l}u(x)=\partial_{l}(u|_{V})(x)=\partial_{l}(v|_{V})(x)=\partial_{l}v(x)$. If $u(y)=0$ for every $y\in V$, apply this with $v=0\cdot u$, the function with value $0$ everywhere, which is of class $C^{1}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and has $\partial_{l}v=0\cdot\partial_{l}u=0$ by the scalar-multiple rule of claim 1 there: then $\partial_{l}u(x)=0$ for every $x\in V$ and $l\in[d]$.

Step 1: growth bounds for $f$. Put $a=|f(0_{\mathbb{R}^{d}})|$, $b=\lVert Df(0_{\mathbb{R}^{d}})\rVert$ and $A=a+b+dM$; then $a\ge0$, $b\ge0$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $A\ge0$. Let $x\in\mathbb{R}^{d}$. By Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable, $\lVert Df(x)-Df(0_{\mathbb{R}^{d}})\rVert\le dM\lVert x\rVert$ and $|f(x)|\le a+b\lVert x\rVert+\tfrac{dM}{2}\lVert x\rVert^{2}$. Since $Df(x)=Df(0_{\mathbb{R}^{d}})+(Df(x)-Df(0_{\mathbb{R}^{d}}))$ (claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space and the vector space structure of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space), the triangle inequality (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) gives $\lVert Df(x)\rVert\le b+dM\lVert x\rVert$. As $\tfrac{dM}{2}\le dM$ and $\lVert x\rVert\ge0$,

$$\lVert Df(x)\rVert\le A(1+\lVert x\rVert),\qquad |f(x)|\le A\bigl(1+\lVert x\rVert+\lVert x\rVert^{2}\bigr).\tag{1.1}$$

For $i\in[d]$ the $i$th component of $Df(x)$ is $\partial_{i}f(x)$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives and Gradient of a Real-Valued Function on a Euclidean Open Set), so claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ gives

$$|\partial_{i}f(x)|\le A(1+\lVert x\rVert)\qquad(i\in[d]).\tag{1.2}$$

Step 2: clause 1. By Hessian Matrix of a C^2 Function the diagonal entries of $D^{2}f(x)$ are $\partial_{i}\partial_{i}f(x)$, $i\in[d]$, and by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable each entry of $D^{2}f$ is Borel. Hence $\Delta f=\sum_{i=1}^{d}\partial_{i}\partial_{i}f$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian) is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and for every $x$, by claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, the hypothesis on $f$, claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field,

$$|\Delta f(x)|\le\sum_{i=1}^{d}|\partial_{i}\partial_{i}f(x)|\le\sum_{i=1}^{d}M=dM .\tag{2.1}$$

So $\Delta f$ is bounded and Borel, hence integrable with respect to every probability measure on $\mathbb{R}^{d}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures). The map $S_{c}$ is Borel with $\int_{\mathbb{R}^{d}}\lVert S_{c}\rVert^{2}\,d\mu<\infty$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, by the preamble of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information, and the gradient map $Df$ is Borel with $\int_{\mathbb{R}^{d}}\lVert Df\rVert^{2}\,d\mu<\infty$ for every such $\mu$, by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to the measurable space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ with $u=S_{c}$ and $v=Df$, the map $x\mapsto S_{c}(x)\cdot Df(x)$ is Borel, and by the inequality recorded there

$$|S_{c}(x)\cdot Df(x)|\le\tfrac12\lVert S_{c}(x)\rVert^{2}+\tfrac12\lVert Df(x)\rVert^{2}\qquad(x\in\mathbb{R}^{d}).$$

Fix $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The right side is integrable with respect to $\mu$ by Step 0 (I) and linearity, so $S_{c}\cdot Df$ is integrable by Step 0 (I). Therefore $L_{c}f=\Delta f-S_{c}\cdot Df$ is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and integrable by linearity (with coefficients $1$ and $-1$), and

$$\int_{\mathbb{R}^{d}}L_{c}f\,d\mu=\int_{\mathbb{R}^{d}}\Delta f\,d\mu-\int_{\mathbb{R}^{d}}S_{c}\cdot Df\,d\mu .\tag{2.2}$$

This proves clause 1.

Step 3: reduction of clause 2. Let $\mu$ be as in clause 2. By Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential §score, $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, that is, $\mu$ has finite Fisher information, and with its score $\xi_{\mu}$ we have $\zeta^{c}_{\mu}=\xi_{\mu}+S_{c}$ in $T_{\mu}$, hence in $L^{2}(\mu;\mathbb{R}^{d})$, since $T_{\mu}\subseteq L^{2}(\mu;\mathbb{R}^{d})$ (The Tangent Space of the Wasserstein Space at a Probability Measure §tangent). By additivity in the first argument of the inner product (Real Inner Product Space §inner-product) and Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu,

$$\langle\zeta^{c}_{\mu},Df\rangle_{\mu}=\langle\xi_{\mu},Df\rangle_{\mu}+\langle S_{c},Df\rangle_{\mu}=\langle\xi_{\mu},Df\rangle_{\mu}+\int_{\mathbb{R}^{d}}S_{c}\cdot Df\,d\mu .$$

Comparing with (2.2), clause 2 follows once we show

$$\langle\xi_{\mu},Df\rangle_{\mu}=-\int_{\mathbb{R}^{d}}\Delta f\,d\mu ,\tag{3.1}$$

for then $\langle\zeta^{c}_{\mu},Df\rangle_{\mu}=-\bigl(\int\Delta f\,d\mu-\int S_{c}\cdot Df\,d\mu\bigr)=-\int L_{c}f\,d\mu$. The rest of the proof establishes (3.1) for this fixed $\mu$.

Step 4: the cutoff approximations. The choices are made in this order. First, apply Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=d$: fix a function $\chi$ as in its statement (one exists by Existence of a Smooth Plateau Function on Euclidean Space, as recorded there), with the associated functions $\chi_{R}$ for positive real $R$, and then fix nonnegative real numbers $m_{1},m_{2}$, not depending on $R$, as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff. Next put

$$K_{1}=(3m_{1}+1)A,\qquad K_{2}=7m_{2}A+6m_{1}A+M,$$

both nonnegative. Finally, for $n\in\mathbb{N}$, read in $\mathbb{R}$ through the canonical map, we have $0<n$ and $1\le n$ (Properties of the Canonical Map from the Natural Numbers to an Ordered Field §positivity and Properties of the Canonical Map from the Natural Numbers to an Ordered Field §lower-bound); let $\chi_{n}$ be the cutoff $\chi_{R}$ with $R=n$, and put $f_{n}=\chi_{n}f$ (pointwise product), $W_{n}=\{y:\lVert y\rVert>2n\}$ and $B_{n}=\{y:\lVert y\rVert<n\}$, which are open by Step 0 (II). By Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, $\chi_{n}$ is smooth, $0\le\chi_{n}\le1$, $\chi_{n}(x)=1$ whenever $\lVert x\rVert\le n$, $\chi_{n}(x)=0$ whenever $\lVert x\rVert\ge2n$, and $|\partial_{i}\chi_{n}(x)|\le m_{1}n^{-1}$, $|\partial_{j}\partial_{i}\chi_{n}(x)|\le m_{2}n^{-2}$ for all $x$ and $i,j\in[d]$.

(4a) Regularity and derivatives. As $\chi_{n}$ and $f$ are of class $C^{2}$, so is $f_{n}$, by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. By the product rule of claim 1 there, for all $x$ and $i\in[d]$,

$$\partial_{i}f_{n}(x)=\partial_{i}\chi_{n}(x)\,f(x)+\chi_{n}(x)\,\partial_{i}f(x),$$

and, applying the sum and product rules of the same claim to this expression, whose two summands are products of functions of class $C^{1}$, for all $x$ and $i,j\in[d]$,

$$\partial_{j}\partial_{i}f_{n}(x)=\partial_{j}\partial_{i}\chi_{n}(x)\,f(x)+\partial_{i}\chi_{n}(x)\,\partial_{j}f(x)+\partial_{j}\chi_{n}(x)\,\partial_{i}f(x)+\chi_{n}(x)\,\partial_{j}\partial_{i}f(x).$$

(4b) Vanishing on $W_{n}$. For $y\in W_{n}$, $\lVert y\rVert\ge2n$, so $\chi_{n}(y)=0$ and $f_{n}(y)=0$. By Step 0 (III) applied to $f_{n}$, $\partial_{i}f_{n}=0$ on $W_{n}$ for every $i$; applied again to the function $\partial_{i}f_{n}$, of class $C^{1}$, it gives $\partial_{j}\partial_{i}f_{n}=0$ on $W_{n}$ for all $i,j$.

(4c) Bounds. Let $x\in\mathbb{R}^{d}$ with $\lVert x\rVert\le2n$, and $i,j\in[d]$. From $1\le n$ and $0\le\lVert x\rVert\le2n$ we get $n^{-1}\le1$, $n^{-2}\le1$, $n^{-1}\lVert x\rVert\le2$, $n^{-1}\lVert x\rVert^{2}\le2\lVert x\rVert$, $n^{-2}\lVert x\rVert\le2$ and $n^{-2}\lVert x\rVert^{2}\le4$. By (4a), the bounds on $\chi_{n}$ and its derivatives, (1.1), (1.2), the hypothesis on $f$, and the multiplicativity and triangle inequality of the absolute value (The Real Numbers: Standing Notation and Background §background),

$$|\partial_{i}f_{n}(x)|\le m_{1}n^{-1}A\bigl(1+\lVert x\rVert+\lVert x\rVert^{2}\bigr)+A(1+\lVert x\rVert)\le3m_{1}A(1+\lVert x\rVert)+A(1+\lVert x\rVert)=K_{1}(1+\lVert x\rVert),$$ $$|\partial_{j}\partial_{i}f_{n}(x)|\le m_{2}n^{-2}A\bigl(1+\lVert x\rVert+\lVert x\rVert^{2}\bigr)+2m_{1}n^{-1}A(1+\lVert x\rVert)+M\le7m_{2}A+6m_{1}A+M=K_{2}.$$

If instead $\lVert x\rVert>2n$, both derivatives vanish by (4b), and the same bounds hold because their right sides are nonnegative. Hence for every $n\in\mathbb{N}$, every $x\in\mathbb{R}^{d}$ and all $i,j\in[d]$,

$$|\partial_{i}f_{n}(x)|\le K_{1}(1+\lVert x\rVert),\qquad|\partial_{j}\partial_{i}f_{n}(x)|\le K_{2},\tag{4.1}$$

and moreover $|\partial_{i}f_{n}(x)|\le K_{1}(1+2n)$ for every $x$ (by (4.1) when $\lVert x\rVert\le2n$, by (4b) otherwise).

(4d) The score identity for $f_{n}$. Put $M'_{n}=K_{1}(1+2n)+K_{2}$, a nonnegative real number. By (4c), $f_{n}$ is of class $C^{2}$ with $|\partial_{i}f_{n}(x)|\le M'_{n}$ and $|\partial_{j}\partial_{i}f_{n}(x)|\le M'_{n}$ for all $x$ and $i,j\in[d]$. Since $\mu$ has finite Fisher information with score $\xi_{\mu}$ (Step 3), 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 §score-identity and 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 §tangent, applied with the constant $M'_{n}$ and the function $f_{n}$, give: the gradient map $Df_{n}$ is Borel and square-integrable with respect to $\mu$, $\Delta f_{n}$ is bounded and Borel, and

$$\langle\xi_{\mu},Df_{n}\rangle_{\mu}=-\int_{\mathbb{R}^{d}}\Delta f_{n}\,d\mu .\tag{4.2}$$

(4e) Agreement on $B_{n}$. For $y\in B_{n}$, $\lVert y\rVert\le n$, so $\chi_{n}(y)=1$ and $f_{n}(y)=f(y)$. By Step 0 (III) applied to $f_{n}$ and $f$, $\partial_{i}f_{n}=\partial_{i}f$ on $B_{n}$; applied to $\partial_{i}f_{n}$ and $\partial_{i}f$ (both of class $C^{1}$), $\partial_{j}\partial_{i}f_{n}=\partial_{j}\partial_{i}f$ on $B_{n}$. Hence, by Gradient of a Real-Valued Function on a Euclidean Open Set and The Laplacian of a Twice Continuously Differentiable Function §laplacian, $Df_{n}(x)=Df(x)$ and $\Delta f_{n}(x)=\Delta f(x)$ whenever $\lVert x\rVert<n$.

Step 5: passage to the limit. Fix a representative $\xi$ of $\xi_{\mu}\in T_{\mu}\subseteq L^{2}(\mu;\mathbb{R}^{d})$: a Borel map with $\int_{\mathbb{R}^{d}}\lVert\xi\rVert^{2}\,d\mu<\infty$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). For $n\in\mathbb{N}$ put $u_{n}(x)=\xi(x)\cdot Df_{n}(x)$ and $v_{n}(x)=\Delta f_{n}(x)$; also $u(x)=\xi(x)\cdot Df(x)$. The functions $u_{n}$ and $u$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (with $Df_{n}$ Borel by (4d) and $Df$ Borel by Step 2), and $v_{n}$ is Borel by (4d). By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, $\langle\xi_{\mu},Df_{n}\rangle_{\mu}=\int u_{n}\,d\mu$ and $\langle\xi_{\mu},Df\rangle_{\mu}=\int u\,d\mu$.

Pointwise convergence. Let $x\in\mathbb{R}^{d}$. By claim 1 of The Archimedean Property of the Real Numbers there is $N\in\mathbb{N}$ with $\lVert x\rVert<N$, and for every $n\ge N$ in $\mathbb{N}$ we have $N\le n$ in $\mathbb{R}$ (Properties of the Canonical Map from the Natural Numbers to an Ordered Field §monotone; trivial if $n=N$), so $\lVert x\rVert<n$ and, by (4e), $u_{n}(x)=u(x)$ and $v_{n}(x)=\Delta f(x)$. Hence for every $\varepsilon>0$ and every $n\ge N$, $|u_{n}(x)-u(x)|=0<\varepsilon$ and $|v_{n}(x)-\Delta f(x)|=0<\varepsilon$; by Limit of a Sequence of Real Numbers, $u_{n}(x)\to u(x)$ and $v_{n}(x)\to\Delta f(x)$.

Domination. Let $x\in\mathbb{R}^{d}$ and $n\in\mathbb{N}$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, Gradient of a Real-Valued Function on a Euclidean Open Set, (4.1), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (applied to $|\partial_{i}f_{n}(x)|\le K_{1}(1+\lVert x\rVert)$), claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claim 3 of Properties of Finite Sums and $(1+t)^{2}\le2(1+t^{2})$ for real $t$,

$$\lVert Df_{n}(x)\rVert^{2}=\sum_{i=1}^{d}\bigl(\partial_{i}f_{n}(x)\bigr)^{2}\le dK_{1}^{2}(1+\lVert x\rVert)^{2}\le2dK_{1}^{2}\bigl(1+\lVert x\rVert^{2}\bigr).$$

With the inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions,

$$|u_{n}(x)|\le\tfrac12\lVert\xi(x)\rVert^{2}+\tfrac12\lVert Df_{n}(x)\rVert^{2}\le g(x),\qquad g(x)=\tfrac12\lVert\xi(x)\rVert^{2}+dK_{1}^{2}+dK_{1}^{2}\lVert x\rVert^{2},$$

and, as in (2.1), using (4.1), $|v_{n}(x)|\le\sum_{i=1}^{d}|\partial_{i}\partial_{i}f_{n}(x)|\le dK_{2}$. The function $g$ is integrable with respect to $\mu$ by Step 0 (I) and linearity, and so is the constant $dK_{2}$.

Conclusion. By claim 3 of Dominated Convergence Theorem, applied on the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ to $(u_{n})$ with limit $u$ and dominating function $g$, and to $(v_{n})$ with limit $\Delta f$ and dominating function $dK_{2}$,

$$\int_{\mathbb{R}^{d}}u_{n}\,d\mu\to\int_{\mathbb{R}^{d}}u\,d\mu=\langle\xi_{\mu},Df\rangle_{\mu},\qquad\int_{\mathbb{R}^{d}}\Delta f_{n}\,d\mu\to\int_{\mathbb{R}^{d}}\Delta f\,d\mu .$$

By Arithmetic of Limits of Real Sequences §scalar, $-\int\Delta f_{n}\,d\mu\to-\int\Delta f\,d\mu$. By (4.2) the sequences $\bigl(\int u_{n}\,d\mu\bigr)_{n}$ and $\bigl(-\int\Delta f_{n}\,d\mu\bigr)_{n}$ coincide, so by uniqueness of limits (claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences) $\langle\xi_{\mu},Df\rangle_{\mu}=-\int_{\mathbb{R}^{d}}\Delta f\,d\mu$. This is (3.1), and by Step 3 clause 2 follows.