Proof of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded $C^1$ Functions

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

Conventions. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures, $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$ is a probability space whose expectation is the integral against $\nu$. For $2$-integrable $f,g$ the inner product of $L^{2}(\nu)$ is $\langle f,g\rangle_{L^{2}(\nu)}=\int_{\mathbb{R}^{d}}fg\,d\nu$ and $\lVert f\rVert_{L^{2}(\nu)}^{2}=\int_{\mathbb{R}^{d}}f^{2}\,d\nu$ by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product; so the pairing $\langle f,g\rangle_{2}=\mathbb{E}[fg]=\int_{\mathbb{R}^{d}}fg\,d\nu$ of square-integrable random variables on this probability space, which appears in Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, equals $\langle f,g\rangle_{L^{2}(\nu)}$. A Borel $f$ is $2$-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space exactly when $\mathbb{E}[f^{2}]=\int f^{2}\,d\nu<\infty$, since pointwise $|f|^{2}$, the power of $|f|$ with real exponent $2$, equals the natural power $|f|\,|f|=f^{2}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement; in particular this applies to the components of a representative of a class in $L^{2}(\nu;\mathbb{R}^{d})$, which are square-integrable random variables by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates. The inner product $\langle\cdot,\cdot\rangle_{\nu}$ of $L^{2}(\nu;\mathbb{R}^{d})$ is that of The Space of Square-Integrable Random Vectors §inner-product, bilinear and symmetric there. We use the following fact (C): a function $f:\mathbb{R}^{d}\to\mathbb{R}$ that is continuous at every point of $\mathbb{R}^{d}$ is Borel. Indeed, if $\lVert y^{j}-y\rVert\to0$ and $\varepsilon>0$, take $\delta>0$ from continuity at $y$; for all large $j$ one has $\sum_{l=1}^{d}(y^{j}_{l}-y_{l})^{2}<\delta^{2}$, hence $|f(y^{j})-f(y)|<\varepsilon$; so $f$ is sequentially continuous, hence measurable with respect to $\mathcal{B}_{d}$ by claim 3(a) of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\mathcal{B}_{d}=\mathcal{B}(\mathbb{R}^{d})$ by claim 5 there.

Claim 1. Let $\psi\in C^{1}_{b}(\mathbb{R}^{d})$ and $i\in[d]$. By Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded (with $q=d$), $\psi$ is of class $C^{1}$ and $\psi,\partial_{1}\psi,\dots,\partial_{d}\psi$ are bounded; taking the largest of these $d+1$ bounds, fix a real $M\ge0$ with $|\psi(y)|\le M$ and $|\partial_{l}\psi(y)|\le M$ for all $y\in\mathbb{R}^{d}$ and $l\in[d]$. By clause 1 of C^k Maps on a Euclidean Open Set, read through its scalar convention, $\psi$ and $\partial_{i}\psi$ are continuous at every point of $\mathbb{R}^{d}$, hence Borel by (C). If $f$ is Borel with $|f|\le M$, then $\int|f|\,d\nu\le M\,\nu(\mathbb{R}^{d})=M$ and, as $|f|^{2}\le M^{2}$ pointwise by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §agreement, $\int|f|^{2}\,d\nu\le M^{2}$, by monotonicity and homogeneity of the integral of nonnegative measurable functions (Linearity and Monotonicity of the Lebesgue Integral §nonnegative); so $\psi$ and $\partial_{i}\psi$ are integrable and $2$-integrable (Power-Integrable Functions and the p-Seminorm §space). The coordinate $y\mapsto y_{i}$ is Borel by claims 1 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $|y_{i}|\le\lVert y\rVert\le\tfrac12(1+\lVert y\rVert^{2})$; since $\int\tfrac12(1+\lVert y\rVert^{2})\,\nu(dy)=\tfrac12(1+M_{2}(\nu))<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, the function $y\mapsto y_{i}$ is integrable. The function $y\mapsto y_{i}\psi(y)$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and satisfies $|y_{i}\psi(y)|\le M|y_{i}|$, so it is integrable by the same monotonicity. By Linearity and Monotonicity of the Lebesgue Integral §integrable, $y\mapsto y_{i}\psi(y)/c_{i}-\partial_{i}\psi(y)$ is integrable and its integral is $c_{i}^{-1}\int y_{i}\psi(y)\,\nu(dy)-\int\partial_{i}\psi\,d\nu$.

Claim 2. We first record two facts. (F1) The constant function $0$ on $\mathbb{R}^{d}$ is smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{1}$; all its partial derivatives vanish, its difference quotients being $0$ (Partial Derivative on a Euclidean Open Set); and it is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. (F2) Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $i\in[d]$. For every natural number $k$, $\psi$ is of class $C^{k+1}$ (Smooth Map on a Euclidean Open Set), so $\partial_{i}\psi$ is of class $C^{k}$ by clause 2 of C^k Maps on a Euclidean Open Set; thus $\partial_{i}\psi$ is smooth, and it is compactly supported by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, so $\partial_{i}\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space). Moreover $\psi\in C^{1}_{b}(\mathbb{R}^{d})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=d$): it is of class $C^{1}$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient, bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable (it is continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), and its partial derivatives are bounded by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient. Applying this to $\partial_{i}\psi$ gives $\partial_{i}\psi\in C^{1}_{b}(\mathbb{R}^{d})$.

($\Rightarrow$) Suppose $\nu$ has finite Fisher information relative to $\gamma_{c}$. 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, $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\zeta^{c}_{\nu}=\xi_{\nu}+S_{c}$ in $T_{\nu}\subseteq L^{2}(\nu;\mathbb{R}^{d})$, where $S_{c}$ is the class of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information. Let $Z=(Z_{1},\dots,Z_{d})$ be a representative of $\zeta^{c}_{\nu}$ and fix $i\in[d]$; then $g_{i}=(\zeta^{c}_{\nu})_{i}$ is the class of $Z_{i}$, an element of $L^{2}(\nu)$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates. We show that $g_{i}$ satisfies the identity of claim 2.

Step 1: $\psi\in C^{1}_{b}(\mathbb{R}^{d})$ compactly supported. Let $\eta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ have $i$-th component $\eta_{i}=\psi$ and $\eta_{l}=0$ for $l\neq i$. By (F1) every component is of class $C^{1}$ and compactly supported, and $\operatorname{div}\eta=\partial_{i}\psi$. By The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field §parts, $\eta$ is Borel and bounded, its class lies in $L^{2}(\nu;\mathbb{R}^{d})$, and $\langle\xi_{\nu},\eta\rangle_{\nu}=-\int\partial_{i}\psi\,d\nu$. Pointwise $S_{c}(y)\cdot\eta(y)=y_{i}\psi(y)/c_{i}$, so $\langle S_{c},\eta\rangle_{\nu}=\int y_{i}\psi(y)/c_{i}\,\nu(dy)$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). By Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations applied to $Z$ and $\eta$,

$$\langle\zeta^{c}_{\nu},\eta\rangle_{\nu}=\mathbb{E}[Z\cdot\eta]=\sum_{l=1}^{d}\langle Z_{l},\eta_{l}\rangle_{2}=\langle Z_{i},\psi\rangle_{2}=\langle g_{i},\psi\rangle_{L^{2}(\nu)},$$

the terms with $l\ne i$ vanishing as $\eta_{l}=0$. By additivity of $\langle\cdot,\cdot\rangle_{\nu}$ in its first argument and claim 1,

$$\langle g_{i},\psi\rangle_{L^{2}(\nu)}=\langle\xi_{\nu},\eta\rangle_{\nu}+\langle S_{c},\eta\rangle_{\nu}=\int_{\mathbb{R}^{d}}\Bigl(\frac{y_{i}}{c_{i}}\psi(y)-\partial_{i}\psi(y)\Bigr)\nu(dy).$$

Step 2: general $\psi\in C^{1}_{b}(\mathbb{R}^{d})$. Let $M$ be as in claim 1. Let $\chi$, $\chi_{R}$ and $M_{1}$ be as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=d$, and for $m\in\mathbb{N}$ put $\psi_{m}=\psi\chi_{m}$. Since $\chi_{m}$ is smooth, hence of class $C^{1}$, $\psi_{m}$ 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 by claim 1 there $\partial_{l}\psi_{m}=\chi_{m}\,\partial_{l}\psi+\psi\,\partial_{l}\chi_{m}$ for $l\in[d]$. Hence $|\psi_{m}|\le M$ and $|\partial_{l}\psi_{m}|\le M+MM_{1}m^{-1}\le M(1+M_{1})$; and $\psi_{m}(y)=0$ for $\lVert y\rVert>2m$, so $\psi_{m}$ is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set. Thus $\psi_{m}\in C^{1}_{b}(\mathbb{R}^{d})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded; as it is compactly supported, Step 1 gives

$$\int_{\mathbb{R}^{d}}Z_{i}\psi_{m}\,d\nu=\int_{\mathbb{R}^{d}}F_{m}\,d\nu,\qquad F_{m}(y)=\frac{y_{i}}{c_{i}}\psi_{m}(y)-\partial_{i}\psi_{m}(y).$$

Fix $y$. For $m\ge\lVert y\rVert$ one has $\chi_{m}(y)=1$, so $\psi_{m}(y)=\psi(y)$ and $|\partial_{i}\psi_{m}(y)-\partial_{i}\psi(y)|=|\psi(y)\partial_{i}\chi_{m}(y)|\le MM_{1}m^{-1}$; hence $Z_{i}\psi_{m}\to Z_{i}\psi$ and $F_{m}\to F$ pointwise, where $F(y)=y_{i}\psi(y)/c_{i}-\partial_{i}\psi(y)$. The functions $Z_{i}\psi_{m}$ are measurable (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) with $|Z_{i}\psi_{m}|\le M|Z_{i}|$, and $Z_{i}$ is integrable: it is Borel with $\int_{\mathbb{R}^{d}}Z_{i}^{2}\,d\nu<\infty$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates, and since $|Z_{i}|\le\tfrac12(1+Z_{i}^{2})$ pointwise and $\nu(\mathbb{R}^{d})=1$, one has $\int|Z_{i}|\,d\nu\le\tfrac12\bigl(1+\int Z_{i}^{2}\,d\nu\bigr)<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative. The functions $F_{m}$ are integrable by claim 1 (applied to $\psi_{m}$), hence measurable, and $|F_{m}(y)|\le Mc_{i}^{-1}|y_{i}|+M(1+M_{1})$, an integrable function of $y$. Indeed, the constant function $1$ on $\mathbb{R}^{d}$ lies in $C^{1}_{b}(\mathbb{R}^{d})$: it is smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{1}$, it is bounded, and its partial derivatives vanish, its difference quotients being $0$ (Partial Derivative on a Euclidean Open Set). Claim 1 applied to it shows that $y\mapsto y_{i}/c_{i}$ is integrable, hence so is $y\mapsto y_{i}$ by Linearity and Monotonicity of the Lebesgue Integral §integrable; so $\int|y_{i}|\,\nu(dy)<\infty$ (Integrable Function and the Lebesgue Integral), and $y\mapsto|y_{i}|$, which is measurable with this same finite integral of its absolute value, is integrable too. The constant $M(1+M_{1})$ is integrable as $\nu(\mathbb{R}^{d})=1$, so the bound is integrable by Linearity and Monotonicity of the Lebesgue Integral §integrable. By Dominated Convergence Theorem (claim 3) applied to both sides, letting $m\to\infty$ gives $\int Z_{i}\psi\,d\nu=\int F\,d\nu$, that is, $\langle g_{i},\psi\rangle_{L^{2}(\nu)}=\int F\,d\nu$.

($\Leftarrow$) Suppose $g_{1},\dots,g_{d}\in L^{2}(\nu)$ satisfy the identity, and let $G_{i}$ be a representative of $g_{i}$. The map $G=(G_{1},\dots,G_{d})$ is Borel by claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\mathbb{E}[\lVert G\rVert^{2}]<\infty$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates, so its class lies in $L^{2}(\nu;\mathbb{R}^{d})$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. By (F2), $\partial_{i}\psi\in C^{1}_{b}(\mathbb{R}^{d})$ for each $i$, so the identity and claim 1 (applied to $\partial_{i}\psi$) give

$$\langle g_{i},\partial_{i}\psi\rangle_{L^{2}(\nu)}=\int_{\mathbb{R}^{d}}\frac{y_{i}}{c_{i}}\partial_{i}\psi(y)\,\nu(dy)-\int_{\mathbb{R}^{d}}\partial_{i}\partial_{i}\psi\,d\nu .$$

The components of $\nabla\psi$ are the $\partial_{i}\psi$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient) and those of $S_{c}$ are $y\mapsto y_{i}/c_{i}$, so Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations gives $\langle S_{c},\nabla\psi\rangle_{\nu}=\sum_{i=1}^{d}\int\frac{y_{i}}{c_{i}}\partial_{i}\psi(y)\,\nu(dy)$ and $\langle G,\nabla\psi\rangle_{\nu}=\sum_{i=1}^{d}\langle g_{i},\partial_{i}\psi\rangle_{L^{2}(\nu)}$; and $\sum_{i=1}^{d}\int\partial_{i}\partial_{i}\psi\,d\nu=\int\Delta\psi\,d\nu$ by The Laplacian of a Twice Continuously Differentiable Function §laplacian and Linearity and Monotonicity of the Lebesgue Integral §integrable. Summing over $i$ and using Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional,

$$\langle G,\nabla\psi\rangle_{\nu}=\langle S_{c},\nabla\psi\rangle_{\nu}-\int_{\mathbb{R}^{d}}\Delta\psi\,d\nu=-\ell^{c}_{\nu}(\psi).$$

The inequality of Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations gives $|\ell^{c}_{\nu}(\psi)|\le\lVert G\rVert_{\nu}\lVert\nabla\psi\rVert_{\nu}$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, so $\nu$ has finite Fisher information relative to $\gamma_{c}$ with $C=\lVert G\rVert_{\nu}$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §finite).

Claim 3. Let $\nu$ have finite Fisher information relative to $\gamma_{c}$ and $i\in[d]$. By the proof of ($\Rightarrow$) in claim 2, $(\zeta^{c}_{\nu})_{i}$ satisfies the identity. Let $g,g'\in L^{2}(\nu)$ both satisfy it and put $h=g-g'$; by linearity of the inner product, $\langle h,\psi\rangle_{L^{2}(\nu)}=0$ for every $\psi\in C^{1}_{b}(\mathbb{R}^{d})$, in particular, by (F2), for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Let $H$ be a representative of $h$ and $\Xi=(H,0,\dots,0):\mathbb{R}^{d}\to\mathbb{R}^{d}$, Borel by claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets with $\mathbb{E}[\lVert \Xi\rVert^{2}]=\int H^{2}\,d\nu<\infty$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates. Let $\varepsilon>0$. By Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment §dense (recall $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$) there is a test field $\eta$ with $\lVert \Xi-\eta\rVert_{\nu}\le\varepsilon$, and by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates applied to $\Xi-\eta$,

$$\lVert h-\eta_{1}\rVert_{L^{2}(\nu)}^{2}=\int_{\mathbb{R}^{d}}(H-\eta_{1})^{2}\,d\nu\le\int_{\mathbb{R}^{d}}(H-\eta_{1})^{2}\,d\nu+\sum_{l=2}^{d}\int_{\mathbb{R}^{d}}\eta_{l}^{2}\,d\nu=\lVert \Xi-\eta\rVert_{\nu}^{2}\le\varepsilon^{2}.$$

Since $\eta_{1}\in C_{c}^{\infty}(\mathbb{R}^{d})$, $\langle h,\eta_{1}\rangle_{L^{2}(\nu)}=0$, so by The Cauchy-Schwarz Inequality in a Real Inner Product Space

$$\lVert h\rVert_{L^{2}(\nu)}^{2}=\langle h,h-\eta_{1}\rangle_{L^{2}(\nu)}\le\lVert h\rVert_{L^{2}(\nu)}\,\varepsilon .$$

This holds for every $\varepsilon>0$. If $\lVert h\rVert_{L^{2}(\nu)}>0$, dividing by it would give $\lVert h\rVert_{L^{2}(\nu)}\le\varepsilon$ for every $\varepsilon>0$, which fails for $\varepsilon=\tfrac12\lVert h\rVert_{L^{2}(\nu)}$. Thus $\lVert h\rVert_{L^{2}(\nu)}=0$, so $h$ is the zero vector and $g=g'$. Hence $(\zeta^{c}_{\nu})_{i}$ is the only element of $L^{2}(\nu)$ satisfying the identity. Finally, with $Z$ a representative of $\zeta^{c}_{\nu}$, Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §information, The Space of Square-Integrable Random Vectors §inner-product and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates give

$$\mathcal{I}(\nu\,|\,\gamma_{c})=\lVert\zeta^{c}_{\nu}\rVert_{\nu}^{2}=\mathbb{E}[\lVert Z\rVert^{2}]=\sum_{i=1}^{d}\int_{\mathbb{R}^{d}}Z_{i}^{2}\,d\nu=\sum_{i=1}^{d}\bigl\lVert(\zeta^{c}_{\nu})_{i}\bigr\rVert_{L^{2}(\nu)}^{2}.$$