Proof of Bounded $C^1$ Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions

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

Claim 1. For each $q\in\mathbb{N}$, $C^{1}_{b}(\mathbb{R}^{q})$ is the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with that $q$; as recorded there, $\mathbb{R}^{q}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and class $C^{1}$ and partial derivatives are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, that is of C^k Maps on a Euclidean Open Set read through its clause 3 and of Partial Derivative on a Euclidean Open Set. We first record three facts. (F) Let $q\in\mathbb{N}$, $f:\mathbb{R}^{q}\to\mathbb{R}$ and $a\in\mathbb{R}^{q}$. Then $f$ is continuous at $a$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces if and only if it is continuous at $a$ relative to $\mathbb{R}^{q}$ as a map from $(\mathbb{R}^{q},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$: by Euclidean Distance on $\mathbb{R}^n$, $d_{E}(x,a)^{2}=\sum_{i=1}^{q}(x_{i}-a_{i})^{2}$, also $(f(x)-f(a))^{2}=|f(x)-f(a)|^{2}$, and for nonnegative reals $s,t$ one has $s<t$ exactly when $s^{2}<t^{2}$. (G) Let $n\le m$ be natural numbers, let $\pi:\mathbb{R}^{m}\to\mathbb{R}^{n}$, $\pi(y_{1},\dots,y_{m})=(y_{1},\dots,y_{n})$, and let $\psi\in C^{1}_{b}(\mathbb{R}^{n})$. Then $\psi\circ\pi\in C^{1}_{b}(\mathbb{R}^{m})$, with $\partial_{i}(\psi\circ\pi)=(\partial_{i}\psi)\circ\pi$ for $i\le n$ and $\partial_{i}(\psi\circ\pi)=0$ for $n<i\le m$. Indeed, for $l\in[n]$ the $l$th coordinate function $\pi_{l}$ of $\pi$ is the $l$th coordinate function $y\mapsto y_{l}$ of $\mathbb{R}^{m}$, which 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 of class $C^{1}$ there by Smooth Map on a Euclidean Open Set; since clause 1 of C^k Maps on a Euclidean Open Set for a map into $\mathbb{R}^{n}$ imposes on each coordinate function exactly the condition which, by clause 3 there, makes that function of class $C^{1}$, the map $\pi$ is of class $C^{1}$ on $\mathbb{R}^{m}$. As $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and $\pi(y)\in\mathbb{R}^{n}$ for all $y$, claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ with $k=1$, $U=\mathbb{R}^{m}$, $V=\mathbb{R}^{n}$, $p=1$, $F=\pi$ and $G=\psi$ shows that $\psi\circ\pi$ is of class $C^{1}$ on $\mathbb{R}^{m}$, and claim 1 there gives, for $y\in\mathbb{R}^{m}$ and $i\in[m]$,

$$\partial_{i}(\psi\circ\pi)(y)=\sum_{l=1}^{n}\partial_{l}\psi\bigl(\pi(y)\bigr)\,\partial_{i}\pi_{l}(y).$$

For every real $h\ne0$ the difference quotient of $\pi_{l}$ at $y$ in the $i$th variable is $\frac{(y_{l}+h)-y_{l}}{h}=1$ if $i=l$ and $\frac{y_{l}-y_{l}}{h}=0$ if $i\ne l$, so in Partial Derivative on a Euclidean Open Set any $\delta$ serves and $\partial_{i}\pi_{l}(y)$ is $1$ if $i=l$ and $0$ otherwise, by Uniqueness of the Partial Derivative on a Euclidean Open Set. Hence $\partial_{i}(\psi\circ\pi)=(\partial_{i}\psi)\circ\pi$ for $i\le n$ and $\partial_{i}(\psi\circ\pi)=0$ for $n<i\le m$, so $\psi\circ\pi$ and its partial derivatives are bounded by the bounds of $\psi$ and of the $\partial_{i}\psi$, or by $0$. (H) If $f,g\in C^{1}_{b}(\mathbb{R}^{m})$ and $c\in\mathbb{R}$, then $f+g$ and $cf$ belong to $C^{1}_{b}(\mathbb{R}^{m})$, with $\partial_{i}(f+g)=\partial_{i}f+\partial_{i}g$ and $\partial_{i}(cf)=c\,\partial_{i}f$. Indeed, $f+g$ and $cf$ are of class $C^{1}$ on $\mathbb{R}^{m}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set with $k=1$ and $U=\mathbb{R}^{m}$, their partial derivatives are as stated by claim 1 there, and bounds are obtained by adding the bounds, respectively multiplying them by $|c|$. Now let $c\in\mathbb{R}$ and let $\psi_{c}:\mathbb{R}^{1}\to\mathbb{R}$ be constant with value $c$. It is smooth on $\mathbb{R}^{1}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{1}$ there by Smooth Map on a Euclidean Open Set; its partial derivative with respect to the first variable is $0$ everywhere, since all difference quotients vanish (any $\delta$ serves in Partial Derivative on a Euclidean Open Set; uniqueness by Uniqueness of the Partial Derivative on a Euclidean Open Set); and $\psi_{c}$, $\partial_{1}\psi_{c}$ are bounded by $|c|$ and $0$. So $\psi_{c}\in C^{1}_{b}(\mathbb{R}^{1})$ and the constant function on $X$ with value $c$ is $\psi_{c}\circ p_{1}\in\mathcal{F}C^{1}_{b}(X)$. Next let $\varphi,\varphi'\in\mathcal{F}C^{1}_{b}(X)$ have representations $(n,\psi)$ and $(n',\psi')$, let $c\in\mathbb{R}$, let $m$ be the larger of $n,n'$, and let $\pi:\mathbb{R}^{m}\to\mathbb{R}^{n}$ and $\pi':\mathbb{R}^{m}\to\mathbb{R}^{n'}$ be the maps of (G). By the definition of the coordinate maps in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $p_{n}=\pi\circ p_{m}$ and $p_{n'}=\pi'\circ p_{m}$, so $\varphi+\varphi'=(\psi\circ\pi+\psi'\circ\pi')\circ p_{m}$ and $c\varphi=(c\psi)\circ p_{n}$, where $\psi\circ\pi+\psi'\circ\pi'\in C^{1}_{b}(\mathbb{R}^{m})$ by (G) and (H), and $c\psi\in C^{1}_{b}(\mathbb{R}^{n})$ by (H). Hence $\varphi+\varphi'$ and $c\varphi$ belong to $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical.

Claim 2. Let $x\in X$, put $a=p_{n}(x)$ and $p=\sum_{i=1}^{n}\partial_{i}\psi(a)\,e_{i}\in X$. By A Real-Valued C^1 Function is Differentiable at Every Point, $\psi$ is differentiable at $a$ with derivative matrix the $1\times n$ matrix with entries $\partial_{i}\psi(a)$. Since the Euclidean norm of a point $s$ of $\mathbb{R}^{1}$ is $\sqrt{s^{2}}=|s|$ (Euclidean Norm on $\mathbb{R}^n$) and the matrix-vector product is given by Matrix-Vector Product, this says: for every $\varepsilon>0$ there is $\delta>0$ such that every $k\in\mathbb{R}^{n}$ with $0<\lVert k\rVert<\delta$ satisfies $|\psi(a+k)-\psi(a)-\sum_{i=1}^{n}\partial_{i}\psi(a)k_{i}|\le\varepsilon\lVert k\rVert$, and this inequality holds trivially for $k=0$. Let $\varepsilon>0$ be given and choose $\delta$ in this way. Let $z\in X$ with $|z|<\delta$ and put $k=p_{n}(z)$. By the additivity in the first argument of the inner product, $(x+z)_{i}=x_{i}+z_{i}$, so $p_{n}(x+z)=a+k$; by Elementary Identities in a Real Inner Product Space §zero, $p_{n}(0_{X})=0$, and $z-0_{X}=z+0_{X}=z$ because $-0_{X}=(-1)0_{X}=0_{X}$ by claims 5 and 4 of Elementary Identities in a Vector Space, so that $|z-0_{X}|=|z|$; hence Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity gives $\lVert k\rVert=\lVert p_{n}(z)-p_{n}(0_{X})\rVert\le|z|<\delta$. By the symmetry, additivity and homogeneity of the inner product, $\langle p,z\rangle=\sum_{i=1}^{n}\partial_{i}\psi(a)\langle z,e_{i}\rangle=\sum_{i=1}^{n}\partial_{i}\psi(a)k_{i}$. Therefore

$$\bigl|\varphi(x+z)-\varphi(x)-\langle p,z\rangle\bigr|=\Bigl|\psi(a+k)-\psi(a)-\sum_{i=1}^{n}\partial_{i}\psi(a)k_{i}\Bigr|\le\varepsilon\lVert k\rVert\le\varepsilon|z|,$$

and $x+z\in X$. Thus $\varphi$ is differentiable at $x$ with gradient $p$ in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable, so $D\varphi(x)=p$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient; as $x$ was arbitrary, $\varphi$ is differentiable on $X$ in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set.

Claim 3. Let $x\in X$. By Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial, which applies since $\varphi$ is differentiable on $X$ by claim 2, $\partial_{k}\varphi(x)=\langle D\varphi(x),e_{k}\rangle$; so by claim 2 and the additivity and homogeneity of the inner product in its first argument, $\partial_{k}\varphi(x)=\sum_{i=1}^{n}\partial_{i}\psi(p_{n}(x))\langle e_{i},e_{k}\rangle$. Since $(e_{k})$ is an orthonormal basis, hence an orthonormal sequence, $\langle e_{i},e_{k}\rangle=0$ for $i\ne k$ and $\langle e_{k},e_{k}\rangle=|e_{k}|^{2}=1$. Hence $\partial_{k}\varphi(x)=\partial_{k}\psi(p_{n}(x))$ if $k\le n$ and $\partial_{k}\varphi(x)=0$ if $k>n$.

Claim 4. Let $(n,\psi)$ be a representation of $\varphi$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $\psi$ is continuous on $\mathbb{R}^{n}$ as a map into $(\mathbb{R},d_{\mathbb{R}})$; for $k\le n$, $\partial_{k}\psi$ is continuous at every point 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 on $\mathbb{R}^{n}$ into $(\mathbb{R},d_{\mathbb{R}})$ by (F) of claim 1. By claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space, $\psi$ and $\partial_{k}\psi$ are measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R})$; $p_{n}$ is 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, for $k\le n$, $\partial_{k}\varphi=(\partial_{k}\psi)\circ p_{n}$ (claim 3) are Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. For $k>n$, $\partial_{k}\varphi$ is the constant function $0$ by claim 3, which is measurable by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and bounded by $0$. If $M\ge0$ is a bound for $\psi$, respectively for $\partial_{k}\psi$ with $k\le n$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$), then $|\varphi(x)|=|\psi(p_{n}(x))|\le M$, respectively $|\partial_{k}\varphi(x)|=|\partial_{k}\psi(p_{n}(x))|\le M$, for every $x\in X$, so these functions are bounded.

Claim 5. Let $(n,\psi)$ be a representation of $\varphi$. For every $k\in\mathbb{N}$ the function $x\mapsto\langle D\varphi(x),e_{k}\rangle=\partial_{k}\varphi(x)$ (by Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial, as $\varphi$ is differentiable on $X$ by claim 2) is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$ by claim 4, so $D\varphi:X\to X$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(X)$, that is Borel, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §measurable applied with $(\Omega,\mathcal{F})=(X,\mathcal{B}(X))$. Let $M\ge0$ be a common bound of $\partial_{1}\psi,\dots,\partial_{n}\psi$ (the largest of their bounds). For $x\in X$ put $y=(\partial_{1}\psi(p_{n}(x)),\dots,\partial_{n}\psi(p_{n}(x)))\in\mathbb{R}^{n}$; by claim 2 and the definition of $p_{n}^{*}$ in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $D\varphi(x)=p_{n}^{*}(y)$, so Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity gives $|D\varphi(x)|=\lVert y\rVert$. As $\lVert y\rVert^{2}=\sum_{i=1}^{n}y_{i}^{2}\le nM^{2}\le(nM)^{2}$ and both $\lVert y\rVert$ and $nM$ are nonnegative, $|D\varphi(x)|\le nM$, and $B=nM$ serves.

Claim 6. Let $f$ be $\varphi$ or $\partial_{k}\varphi$. By claim 4, $f$ is Borel with a bound $M\ge0$, and $\mu(X)=1<\infty$, so $f$ is integrable with respect to $\mu$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. The function $|f|^{2}$ of Power-Integrable Functions and the p-Seminorm §measurable-power is measurable by that clause, nonnegative, and satisfies $|f(x)|^{2}\le M^{2}$ for every $x$ by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §agreement; by claims 6(b) and 6(c) of Borel Measurability and Bounded Integration on a Metric Space its integral as a nonnegative measurable function is a real number. Hence $\int_{X}|f|^{2}\,d\mu<\infty$, that is $f$ is $2$-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space, and its class lies in $L^{2}(\mu)$ by The Lebesgue Space of Power-Integrable Functions §space.

Claim 7. Let $M\ge0$ be a bound of $\varphi$ (claim 4) and $f(x)=x_{k}\varphi(x)$. The coordinate function $x\mapsto x_{k}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and $\varphi$ is Borel by claim 4, so $f$ and $|f|$ are measurable by claims 3 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity with $n=k$ and $x'=0_{X}$, using $p_{k}(0_{X})=0$ and $|x-0_{X}|=|x|$ as in claim 2, $x_{k}^{2}\le\sum_{i=1}^{k}x_{i}^{2}=\lVert p_{k}(x)\rVert^{2}\le|x|^{2}$, so $|x_{k}|\le|x|$. Since $0\le(|x|-1)^{2}$ gives $2|x|\le1+|x|^{2}$, we get $|f(x)|\le M|x|\le\tfrac{M}{2}+\tfrac{M}{2}|x|^{2}$ for every $x\in X$. The function $x\mapsto|x|^{2}$ is Borel and nonnegative by the preamble of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment, and $M_{2}(\mu)<\infty$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space; the constant $\tfrac{M}{2}$ has integral $\tfrac{M}{2}\mu(X)=\tfrac{M}{2}$ by claims 6(a) and 6(c) of Borel Measurability and Bounded Integration on a Metric Space. By Linearity and Monotonicity of the Lebesgue Integral §nonnegative (additivity, homogeneity and monotonicity of the nonnegative integral),

$$\int_{X}|f|\,d\mu\le\frac{M}{2}+\frac{M}{2}M_{2}(\mu)<\infty ,$$

so $f$ is integrable with respect to $\mu$ by the criterion of Integrable Function and the Lebesgue Integral.

Claim 8. Write $L^{2}=L^{2}(X,\mathcal{B}(X),\mu)$, a real Hilbert space by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, and for $\varphi\in\mathcal{F}C^{1}_{b}(X)$ let $[\varphi]\in L^{2}$ be its class (claim 6); by the convention of The Lebesgue Space of Power-Integrable Functions §convention, $\langle g,\varphi\rangle_{L^{2}(\mu)}$ means $\langle g,[\varphi]\rangle_{L^{2}(\mu)}$. Let $\mathcal{L}=\{[\varphi]:\varphi\in\mathcal{F}C^{1}_{b}(X)\}$. The zero vector of $L^{2}$ is $0\,[\varphi]=[0\cdot\varphi]=[0]$ for any $\varphi$, by claim 3 of Elementary Identities in a Vector Space and The Lebesgue Space of Power-Integrable Functions §space, and it lies in $\mathcal{L}$ since the constant $0$ belongs to $\mathcal{F}C^{1}_{b}(X)$ by claim 1; and $[\varphi]+[\varphi']=[\varphi+\varphi']$, $c[\varphi]=[c\varphi]$ lie in $\mathcal{L}$ by The Lebesgue Space of Power-Integrable Functions §space and claim 1. So $\mathcal{L}$ is a linear subspace of $L^{2}$.

Step 1 (the orthogonality statement). Let $g\in L^{2}$ satisfy $\langle g,[\varphi]\rangle=0$ for every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, and let $f$ be a $2$-integrable representative of $g$. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, for every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ (which is $2$-integrable by claim 6) the product $f\varphi$ is integrable and $\int_{X}f\varphi\,d\mu=\langle g,[\varphi]\rangle=0$; with $\varphi$ the constant $1$, $f$ is integrable. Its positive and negative parts $f^{+},f^{-}$ are measurable with values in $[0,\infty)$, with finite integrals, and $f=f^{+}-f^{-}$, by Integrable Function and the Lebesgue Integral. By claim 3 of Image Measures, Measures with Densities, and Change of Variables, $\nu_{+}(A)=\int_{X}\mathbf{1}_{A}f^{+}\,d\mu$ and $\nu_{-}(A)=\int_{X}\mathbf{1}_{A}f^{-}\,d\mu$ ($A\in\mathcal{B}(X)$) define measures on $(X,\mathcal{B}(X))$, that is Borel measures on $(X,d)$, with $\nu_{\pm}(X)=\int_{X}f^{\pm}\,d\mu<\infty$. Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$. Being Borel and bounded (claim 4), $\varphi$ is integrable with respect to $\nu_{+}$ and $\nu_{-}$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space, so by claim 3 of Image Measures, Measures with Densities, and Change of Variables the functions $\varphi f^{\pm}$ are integrable with respect to $\mu$ and $\int_{X}\varphi\,d\nu_{\pm}=\int_{X}\varphi f^{\pm}\,d\mu$. Since $f\varphi=\varphi f^{+}-\varphi f^{-}$ pointwise, Linearity and Monotonicity of the Lebesgue Integral §integrable with $a=1$ and $b=-1$ yields

$$\int_{X}\varphi\,d\nu_{+}-\int_{X}\varphi\,d\nu_{-}=\int_{X}f\varphi\,d\mu=0\qquad(\varphi\in\mathcal{F}C^{1}_{b}(X)),$$

an identity referred to as $(*)$. For $\varphi$ the constant $1$, claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space turns $(*)$ into $\nu_{+}(X)=\nu_{-}(X)$. Fix $n\in\mathbb{N}$. Since $p_{n}$ is Borel (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity), claim 1 of Image Measures, Measures with Densities, and Change of Variables makes $(p_{n})_{\#}\nu_{\pm}$ Borel measures on $(\mathbb{R}^{n},d_{E})$ of total mass $\nu_{\pm}(X)<\infty$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be a test function. It is of class $C^{1}$ in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient; it is continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, hence bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable and Borel by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space; 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. So $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$, and $\psi\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical. By claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space, $\psi$ is integrable with respect to $(p_{n})_{\#}\nu_{\pm}$, and by claim 2 of Image Measures, Measures with Densities, and Change of Variables and $(*)$,

$$\int_{\mathbb{R}^{n}}\psi\,d(p_{n})_{\#}\nu_{+}=\int_{X}\psi\circ p_{n}\,d\nu_{+}=\int_{X}\psi\circ p_{n}\,d\nu_{-}=\int_{\mathbb{R}^{n}}\psi\,d(p_{n})_{\#}\nu_{-}.$$

The test functions of Test Functions Approximate Bounded Continuous Functions Pointwise, Determine a Finite Borel Measure, and Detect a Vanishing Density are those of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test, namely those of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space; applying Test Functions Approximate Bounded Continuous Functions Pointwise, Determine a Finite Borel Measure, and Detect a Vanishing Density §determination with $d=n$ gives $(p_{n})_{\#}\nu_{+}=(p_{n})_{\#}\nu_{-}$. As $n$ was arbitrary and $\nu_{+}(X)=\nu_{-}(X)<\infty$, Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §determination gives $\nu_{+}=\nu_{-}$. Let $A=\{x\in X:f(x)>0\}$ and $A'=\{x\in X:f(x)<0\}$, which belong to $\mathcal{B}(X)$ because $f$ and $-f$ are measurable (Measure Spaces and the Lebesgue Integral: Standing Notation §measurable). Pointwise $\mathbf{1}_{A}f^{+}=f^{+}$, $\mathbf{1}_{A}f^{-}=0$, $\mathbf{1}_{A'}f^{-}=f^{-}$ and $\mathbf{1}_{A'}f^{+}=0$, and the zero function has integral $0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral with the null set $\emptyset$. Hence

$$\int_{X}f^{+}\,d\mu=\nu_{+}(A)=\nu_{-}(A)=0,\qquad\int_{X}f^{-}\,d\mu=\nu_{-}(A')=\nu_{+}(A')=0,$$

so $f^{+}=0$ and $f^{-}=0$ almost everywhere by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, and $f=f^{+}-f^{-}=0$ outside the union of two null sets, which is null by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union. Thus $f\sim0$ in the sense of The Lebesgue Space of Power-Integrable Functions §equivalence, and $g=[f]=[0]$ is the zero vector of $L^{2}$. This proves the second sentence of claim 8.

Step 2 (density). By Step 1 every element of the orthogonal complement $\mathcal{L}^{\perp}$ is the zero vector, and the zero vector lies in $\mathcal{L}^{\perp}$ by Elementary Identities in a Real Inner Product Space §zero; so $\mathcal{L}^{\perp}=\{0_{L^{2}}\}$. Since $\mathcal{L}$ is a linear subspace of the real Hilbert space $L^{2}$, Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §density shows that $\mathcal{L}$ is dense in $L^{2}$ in the sense of Real Hilbert Space §topology.