Proof of Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space

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

Throughout, Borel on $X$ and on each $\mathbb{R}^{m}$ ($m\in\mathbb{N}$, with the distance $d_{E}$) is as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, and $\gamma_{c}\in\mathcal{P}(X)$, so $(X,\mathcal{B}(X),\gamma_{c})$ is a probability space in the sense of Probability Space, Event, and Random Variable. We use the following domination principle repeatedly: if $(E,\mathcal{E},\nu)$ is a measure space, $u:E\to\mathbb{R}$ is measurable, $v:E\to\mathbb{R}$ is integrable and $|u(z)|\le v(z)$ for every $z\in E$, then $u$ is integrable. Indeed $|u|$ is measurable by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $v=|v|$, and the monotonicity of the nonnegative integral in Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int_{E}|u|\,d\nu\le\int_{E}|v|\,d\nu<\infty$, so $u$ is integrable by the criterion recorded in Integrable Function and the Lebesgue Integral. Finite linear combinations of integrable functions are integrable, with the integral of the combination equal to the combination of the integrals, by Linearity and Monotonicity of the Lebesgue Integral §integrable and induction on the number of terms; and $|u|$ is integrable whenever $u$ is, by the same criterion.

Step 1. A growth bound for $\psi$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous (with $k=1$ and $\psi$ regarded as a map into $\mathbb{R}^{1}$), $\psi$ is continuous at every point of $\mathbb{R}^{n}$ relative to $\mathbb{R}^{n}$, as a map from $(\mathbb{R}^{n},d_{E})$ into $(\mathbb{R},d_{\mathbb{R}})$. Fix $y=(y_{1},\dots,y_{n})\in\mathbb{R}^{n}$. For $i\in\{0,1,\dots,n\}$ let $z^{(i)}\in\mathbb{R}^{n}$ have coordinates $z^{(i)}_{l}=y_{l}$ for $l\le i$ and $z^{(i)}_{l}=0$ for $l>i$, so $z^{(0)}=0_{\mathbb{R}^{n}}$ and $z^{(n)}=y$. Fix $i\in[n]$ and, for $s\in\mathbb{R}$, let $w(s)\in\mathbb{R}^{n}$ be the point with $i$th coordinate $s$ and $l$th coordinate $z^{(i)}_{l}$ for $l\ne i$; thus $w(0)=z^{(i-1)}$ and $w(y_{i})=z^{(i)}$. Put $g(s)=\psi(w(s))$. For $s,t\in\mathbb{R}$ the point $w(t)-w(s)$ has the single nonzero coordinate $t-s$ (if any), so by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance, $d_{E}(w(s),w(t))=\lVert w(t)-w(s)\rVert=|t-s|$. Hence, given $s_{0}$ and $\varepsilon>0$, a $\delta>0$ witnessing the continuity of $\psi$ at $w(s_{0})$ in the sense of Continuous Map Between Metric Spaces also witnesses the continuity of $g$ at $s_{0}$; so $g$ is continuous on every closed interval. Moreover, for $s,h\in\mathbb{R}$ the point $w(s+h)$ is obtained from $w(s)$ by adding $h$ to its $i$th coordinate, so the difference quotient $(g(s+h)-g(s))/h$ is exactly the quotient in Partial Derivative on a Euclidean Open Set for $\psi$ at $w(s)$ in the $i$th variable; comparing that definition with Derivative at an Interior Point (every point of an interval $[a,b]$ strictly between $a$ and $b$ being interior), the number $\partial_{i}\psi(w(s))$ satisfies the condition defining the derivative of $g$ at every such point $s$, so $g$ is differentiable at $s$ with $g'(s)=\partial_{i}\psi(w(s))$; a number with that property is unique, since two such numbers $L,L'$ satisfy $|L-L'|<2\varepsilon$ for every $\varepsilon>0$ (choosing an admissible $h$ for the smaller of the two $\delta$'s). If $y_{i}=0$ then $\psi(z^{(i)})=\psi(z^{(i-1)})$. If $y_{i}\ne0$, apply Mean Value Theorem on a Closed Real Interval to the restriction of $g$ to the closed interval with endpoints $0$ and $y_{i}$: there is $s$ strictly between $0$ and $y_{i}$ with $\psi(z^{(i)})-\psi(z^{(i-1)})=g(y_{i})-g(0)=\partial_{i}\psi(w(s))\,y_{i}$. Since $|s|\le|y_{i}|$, every coordinate of $w(s)$ has absolute value at most the corresponding coordinate of $y$, so $\lVert w(s)\rVert^{2}\le\lVert y\rVert^{2}$ and $\lVert w(s)\rVert\le\lVert y\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square; with the hypothesis on $\partial_{i}\psi$ and $|y_{i}|\le\lVert y\rVert$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate),

$$|\psi(z^{(i)})-\psi(z^{(i-1)})|\le M\bigl(1+\lVert y\rVert\bigr)\lVert y\rVert,$$

which also holds trivially when $y_{i}=0$. Writing $\psi(y)-\psi(0_{\mathbb{R}^{n}})$ as the telescoping sum of these $n$ differences and using the triangle inequality,

$$|\psi(y)|\le|\psi(0_{\mathbb{R}^{n}})|+nM\bigl(\lVert y\rVert+\lVert y\rVert^{2}\bigr)\le C\bigl(1+\lVert y\rVert^{2}\bigr),\qquad C=|\psi(0_{\mathbb{R}^{n}})|+2nM,$$

where we used $u\le1+u^{2}$ for real $u\ge0$ (if $u\le1$ then $u\le1$, otherwise $u\le u^{2}$). The constant $C\ge0$ does not depend on $y$.

Step 2. Bounds on $X$. Let $x\in X$ and $q\in\mathbb{N}$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $|P_{q}x|=\lVert p_{q}(x)\rVert$ and $|x|^{2}=|P_{q}x|^{2}+|Q_{q}x|^{2}$, so $\lVert p_{q}(x)\rVert^{2}\le|x|^{2}$ and $\lVert p_{q}(x)\rVert\le|x|$. Since $x_{k}$ is the $k$th coordinate of $p_{k}(x)$, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate gives $|x_{k}|\le\lVert p_{k}(x)\rVert\le|x|$ for every $k\in\mathbb{N}$. Put $W(x)=(1+|x|^{2})^{2}$. Using Step 1, the hypothesis on $\partial_{i}\psi$, and $|x|\le1+|x|^{2}$, $1\le1+|x|^{2}$, we get for all $x\in X$, $i\in[n]$ and $k\in\mathbb{N}$:

$$|\varphi(x)|\le C\,W(x),\qquad|\varphi_{i}(x)|\le M(1+|x|)\le2M\,W(x),\qquad|x_{k}\varphi(x)|\le C|x|(1+|x|^{2})\le C\,W(x),\qquad\varphi(x)^{2}\le C^{2}W(x).$$

Step 3. An integrable majorant. Let $s=\sum_{k=1}^{\infty}c_{k}$, a real number since $c$ is a variance sequence, so that the series converges and every $c_{k}$ is positive. For every $k$, $c_{k}\le\sum_{j=1}^{k}c_{j}$, since the other terms are positive, and $\sum_{j=1}^{k}c_{j}\le s$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; hence $c_{k}\le s$ for every $k$, and in particular $s\ge c_{1}>0$. Put $\alpha=1/(2+4s)$ and $\theta=\tfrac12$. Then $0\le\alpha$, $\theta<1$ and $2\alpha c_{k}\le2s/(2+4s)<\tfrac12=\theta$ for every $k$, so by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §exponential the function $V(x)=\exp(\alpha|x|^{2})$ is Borel and integrable with respect to $\gamma_{c}$. Let $\beta=\alpha/2$, so $0<\beta\le1$. For real $t\ge0$, claims 1 and 4 of Basic Properties of the Exponential Function give $\exp(\alpha t)=\exp(\beta t)\exp(\beta t)\ge(1+\beta t)^{2}\ge\beta^{2}(1+t)^{2}$, the last step because $1+\beta t\ge\beta+\beta t\ge0$. With $t=|x|^{2}$ this gives $W(x)\le\beta^{-2}V(x)$ for every $x\in X$. Consequently every Borel function $u:X\to\mathbb{R}$ with $|u|\le K\,W$ for a real $K\ge0$ is integrable with respect to $\gamma_{c}$, by the domination principle with the integrable majorant $K\beta^{-2}V$.

Step 4. Claim 1 (Borel measurability and integrability). By claim 3 of Borel Measurability and Bounded Integration on a Metric Space and claim 2 of the same lemma (which identifies $\mathcal{B}(\mathbb{R})$ with the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$), together with Step 1, $\psi$ is Borel; each $\partial_{i}\psi$ is continuous at every point of $\mathbb{R}^{n}$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces by clause 1 of C^k Maps on a Euclidean Open Set, hence continuous relative to $\mathbb{R}^{n}$ into $(\mathbb{R},d_{\mathbb{R}})$ by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, hence Borel by the same claims 2 and 3. The map $p_{n}$ and every coordinate function $x\mapsto x_{k}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so $\varphi=\psi\circ p_{n}$ and $\varphi_{i}=(\partial_{i}\psi)\circ p_{n}$ are Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $x\mapsto x_{k}\varphi(x)$ and $\varphi^{2}$ are Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By Steps 2 and 3, $\varphi$, $\varphi_{i}$ ($i\in[n]$), $x\mapsto x_{k}\varphi(x)$ ($k\in\mathbb{N}$) and $\varphi^{2}$ are integrable with respect to $\gamma_{c}$. In particular $\varphi$ is a square-integrable random variable on $(X,\mathcal{B}(X),\gamma_{c})$, the expectation being the integral against $\gamma_{c}$ by Expectation, Variance, and Moments. Let $h\in H_{c}$. By The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel $\ell_{h}$ is Borel and by The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §integrable $(\ell_{h})^{2}$ is integrable, so $\ell_{h}$ is square-integrable; by Square-Integrable Random Variables and the Mean-Square Inner Product the product $\ell_{h}\varphi$ of two square-integrable random variables is a random variable (hence Borel) and is integrable. This proves claim 1.

Step 5. Lifting $\psi$ to $\mathbb{R}^{m}$. Fix $m\in\mathbb{N}$ with $m\ge n$, let $\pi:\mathbb{R}^{m}\to\mathbb{R}^{n}$, $\pi(y)=(y_{1},\dots,y_{n})$, and $f=\psi\circ\pi:\mathbb{R}^{m}\to\mathbb{R}$ (for $m=n$, $\pi$ is the identity and $f=\psi$). For $y,y'\in\mathbb{R}^{m}$, $\sum_{l=1}^{n}(y_{l}-y'_{l})^{2}\le\sum_{l=1}^{m}(y_{l}-y'_{l})^{2}$, the omitted terms being nonnegative. We claim that $f$ is of class $C^{1}$ on $\mathbb{R}^{m}$ (open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) with

$$\partial_{j}f(y)=\partial_{j}\psi(\pi(y))\quad(j\le n),\qquad\partial_{j}f(y)=0\quad(n<j\le m),\qquad y\in\mathbb{R}^{m}.$$

Let $a\in\mathbb{R}^{m}$, $j\in[m]$ and $t\in\mathbb{R}$, and let $a'$ be $a$ with $t$ added to its $j$th coordinate. If $j\le n$, then $\pi(a')$ is $\pi(a)$ with $t$ added to its $j$th coordinate, so the difference quotient $(f(a')-f(a))/t$ in Partial Derivative on a Euclidean Open Set for $f$ at $a$ coincides with the one for $\psi$ at $\pi(a)$; hence the partial derivative of $f$ in the $j$th variable exists at $a$ with value $\partial_{j}\psi(\pi(a))$. If $j>n$, then $\pi(a')=\pi(a)$, every such quotient is $0$, and the partial derivative exists at $a$ with value $0$. These values are the partial derivatives by Uniqueness of the Partial Derivative on a Euclidean Open Set. Next, if $u:\mathbb{R}^{n}\to\mathbb{R}$ is continuous at $\pi(a)$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces, then so is $u\circ\pi$ at $a$: a $\delta$ that works for $u$ and $\varepsilon$ works for $u\circ\pi$, by the inequality between sums of squares above. Applied to $u=\psi$ (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and $u=\partial_{j}\psi$ (clause 1 of C^k Maps on a Euclidean Open Set), and noting that the constant function $0$ is trivially continuous, this shows that $f$ and all $\partial_{j}f$ are continuous at every point; so $f$ is of class $C^{1}$ on $\mathbb{R}^{m}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set. Since $\pi\circ p_{m}=p_{n}$, we have for $x\in X$:

$$f\circ p_{m}=\varphi,\qquad(\partial_{j}f)\circ p_{m}=\varphi_{j}\ (j\le n),\qquad(\partial_{j}f)\circ p_{m}=0\ (n<j\le m),\qquad\pi_{j}\circ p_{m}=\bigl(x\mapsto x_{j}\bigr),$$

where $\pi_{j}(y)=y_{j}$. The functions $f$ and $\partial_{j}f$ are continuous on $(\mathbb{R}^{m},d_{E})$ (claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions); and $\pi_{j}$ is smooth on $\mathbb{R}^{m}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous relative to $\mathbb{R}^{m}$ at every point, 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. So $f$, $\partial_{j}f$ and $\pi_{j}$ are Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space and claim 2 of the same lemma (which identifies $\mathcal{B}(\mathbb{R})$ with the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$), and so is $\pi_{j}f$ by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Step 6. Transfer to $\mathbb{R}^{m}$. Let $\rho=\rho_{c^{(m)}}$ be the diagonal Gaussian density with the variance vector $c^{(m)}=(c_{1},\dots,c_{m})$ of Variance Sequences and Their Truncations §truncations, and $\lambda_{m}$ the Lebesgue measure on $\mathcal{B}(\mathbb{R}^{m})$; $\rho$ is smooth, positive and Borel by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity. By Diagonal Gaussian Measures on a Hilbert Space §measure, $(p_{m})_{\#}\gamma_{c}=\gamma_{c^{(m)}}$, the image measure of $\gamma_{c}$ under the Borel map $p_{m}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward); and by Diagonal Gaussian Measures on Euclidean Space §measure, $\gamma_{c^{(m)}}$ is the measure with density $\rho$ with respect to $\lambda_{m}$. Let $u:\mathbb{R}^{m}\to\mathbb{R}$ be Borel with $u\circ p_{m}$ integrable with respect to $\gamma_{c}$. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, $u$ is integrable with respect to $\gamma_{c^{(m)}}$ and $\int_{X}u\circ p_{m}\,d\gamma_{c}=\int_{\mathbb{R}^{m}}u\,d\gamma_{c^{(m)}}$; by claim 3 of the same lemma, $u\rho$ is integrable with respect to $\lambda_{m}$ and

$$\int_{X}u\circ p_{m}\,d\gamma_{c}=\int_{\mathbb{R}^{m}}u\,\rho\,d\lambda_{m}.$$

By Step 4 and Step 5 this applies to $u=f$, $u=\pi_{j}f$ and $u=\partial_{j}f$ for every $j\in[m]$ (for $j>n$, $(\partial_{j}f)\circ p_{m}$ is the zero function, integrable with integral $0$).

Step 7. Integration by parts on $\mathbb{R}^{m}$. Fix $j\in[m]$. We show

$$\int_{\mathbb{R}^{m}}(\partial_{j}f)\,\rho\,d\lambda_{m}=\frac{1}{c_{j}}\int_{\mathbb{R}^{m}}\pi_{j}f\,\rho\,d\lambda_{m}.$$

By The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §gradient, Gradient of a Real-Valued Function on a Euclidean Open Set and The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling, $\partial_{j}\rho(y)=-\rho(y)\,y_{j}/c_{j}$ for every $y$. Let $\chi_{R}$ ($R>0$) be the cutoffs of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=m$, and $M_{1}\ge0$ the constant of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff. For $N\in\mathbb{N}$ put $g_{N}=\chi_{N}\rho$. By that clause and claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $g_{N}$ is smooth, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set; it vanishes whenever $\lVert y\rVert>2N$, so it is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set; and by the product rule of claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set,

$$\partial_{j}g_{N}(y)=\partial_{j}\chi_{N}(y)\,\rho(y)-\chi_{N}(y)\,\rho(y)\,\frac{y_{j}}{c_{j}}.$$

By Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ §parts (with $q=m$, $i=j$, the $C^{1}$ function $f$ of Step 5 and $g=g_{N}$), the functions $f\,\partial_{j}g_{N}$ and $(\partial_{j}f)\,g_{N}$ are integrable with respect to $\lambda_{m}$ and $\int f\,\partial_{j}g_{N}\,d\lambda_{m}=-\int(\partial_{j}f)\,g_{N}\,d\lambda_{m}$. Hence $F_{N}=f\,\partial_{j}g_{N}+(\partial_{j}f)\,g_{N}$ is integrable with $\int_{\mathbb{R}^{m}}F_{N}\,d\lambda_{m}=0$, and

$$F_{N}(y)=f(y)\,\partial_{j}\chi_{N}(y)\,\rho(y)+\chi_{N}(y)\Bigl(\partial_{j}f(y)-\frac{y_{j}f(y)}{c_{j}}\Bigr)\rho(y).$$

Let $F=(\partial_{j}f)\rho-c_{j}^{-1}\pi_{j}f\rho$ and $G=M_{1}|f\rho|+|(\partial_{j}f)\rho|+c_{j}^{-1}|\pi_{j}f\rho|$; both are integrable with respect to $\lambda_{m}$ by Step 6 and the remarks at the start. Since $\rho>0$, $0\le\chi_{N}\le1$ and $|\partial_{j}\chi_{N}|\le M_{1}N^{-1}\le M_{1}$, we have $|F_{N}(y)|\le G(y)$ for all $y$ and $N$. Fix $y$; for every $N\ge\lVert y\rVert$, $\chi_{N}(y)=1$ by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff, so $|F_{N}(y)-F(y)|=|f(y)\,\partial_{j}\chi_{N}(y)\,\rho(y)|\le M_{1}|f(y)|\rho(y)N^{-1}$, which tends to $0$; thus $F_{N}(y)\to F(y)$ for every $y$. By claim 3 of Dominated Convergence Theorem, $0=\int F_{N}\,d\lambda_{m}\to\int F\,d\lambda_{m}$, so $\int F\,d\lambda_{m}=0$, and linearity (Linearity and Monotonicity of the Lebesgue Integral §integrable) gives the displayed identity.

Step 8. Claim 2. Let $i\in[n]$ and take $m=n$, $j=i$ in Steps 5 to 7. By Step 6 (applied to $u=\partial_{i}f$ and $u=\pi_{i}f$) and Step 7,

$$\int_{X}\varphi_{i}\,d\gamma_{c}=\int_{\mathbb{R}^{n}}(\partial_{i}f)\,\rho\,d\lambda_{n}=\frac{1}{c_{i}}\int_{\mathbb{R}^{n}}\pi_{i}f\,\rho\,d\lambda_{n}=\frac{1}{c_{i}}\int_{X}x_{i}\,\varphi(x)\,\gamma_{c}(dx),$$

which is the first identity of claim 2. Now let $k\in\mathbb{N}$ with $k>n$ and take $m=k$, $j=k$. Then $\partial_{k}f=0$ by Step 5, so the left side of Step 7 is $0$, and by Step 6

$$\int_{X}x_{k}\,\varphi(x)\,\gamma_{c}(dx)=\int_{\mathbb{R}^{k}}\pi_{k}f\,\rho\,d\lambda_{k}=c_{k}\int_{\mathbb{R}^{k}}(\partial_{k}f)\,\rho\,d\lambda_{k}=0,$$

which is the second identity of claim 2. This proves claim 2.

Step 9. Claim 3. Let $h\in H_{c}$ and $S=\sum_{i=1}^{n}h_{i}\int_{X}\varphi_{i}\,d\gamma_{c}$. Let $m\in\mathbb{N}$ with $m\ge n$. By The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §partial-sums, $\ell_{h,m}\varphi=\sum_{k=1}^{m}(h_{k}/c_{k})\,x_{k}\varphi$, a linear combination of functions integrable by claim 1, so by linearity and claim 2,

$$\int_{X}\ell_{h,m}\,\varphi\,d\gamma_{c}=\sum_{k=1}^{n}h_{k}\cdot\frac{1}{c_{k}}\int_{X}x_{k}\varphi\,d\gamma_{c}+\sum_{k=n+1}^{m}\frac{h_{k}}{c_{k}}\cdot0=\sum_{k=1}^{n}h_{k}\int_{X}\varphi_{k}\,d\gamma_{c}=S.$$

By The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §integrable and The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel, $\ell_{h}$ and $\ell_{h,m}$ are square-integrable random variables on $(X,\mathcal{B}(X),\gamma_{c})$, hence so is $\ell_{h}-\ell_{h,m}$ by Square-Integrable Random Variables and the Mean-Square Inner Product; and $\varphi$ is square-integrable by Step 4. Let $D=\int_{X}\ell_{h}\varphi\,d\gamma_{c}-S$, a real number not depending on $m$. By linearity, $D=\int_{X}(\ell_{h}-\ell_{h,m})\varphi\,d\gamma_{c}$, so claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, squared, together with The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §mean-square, gives

$$D^{2}\le\int_{X}(\ell_{h}-\ell_{h,m})^{2}\,d\gamma_{c}\int_{X}\varphi^{2}\,d\gamma_{c}=\Bigl(\sum_{k=m+1}^{\infty}\frac{h_{k}^{2}}{c_{k}}\Bigr)\int_{X}\varphi^{2}\,d\gamma_{c}$$

for every $m\ge n$. By the same clause the right side tends to $0$ as $m\to\infty$; since the nonnegative number $D^{2}$ is bounded by every term of a sequence tending to $0$, $D^{2}=0$, so $D=0$, that is $\int_{X}\ell_{h}\varphi\,d\gamma_{c}=S$. This proves claim 3.