Proof of Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space

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

Conventions. Natural numbers are read in $\mathbb{R}$ through the canonical map, as in The Real Numbers: Standing Notation and Background §numbers; the arithmetic and order manipulations of real numbers, natural powers, finite sums and finite products made below without comment (in particular products of finitely many inequalities between nonnegative numbers, $|y_{k}|\le\lVert y\rVert$ for $y\in\mathbb{R}^{n}$, which follows from $y_{k}^{2}\le\sum_{l=1}^{n}y_{l}^{2}=\lVert y\rVert^{2}$, and the inequality $(a+b+c)^{2}\le3(a^{2}+b^{2}+c^{2})$, which follows from $3(a^{2}+b^{2}+c^{2})-(a+b+c)^{2}=(a-b)^{2}+(b-c)^{2}+(c-a)^{2}\ge0$) are those in force by The Real Numbers: Standing Notation and Background §background, as are the order and arithmetic of limits of real sequences. Let $\rho_{c}$ be the diagonal Gaussian density with variances $c$ and $\lambda_{n}$ Lebesgue measure on $\mathcal{B}(\mathbb{R}^{n})$. By the definition of $\gamma_{c}$ it is the probability measure with density $\rho_{c}$ with respect to $\lambda_{n}$, so by claim 3 of Image Measures, Measures with Densities, and Change of Variables: for Borel $h:\mathbb{R}^{n}\to[0,\infty)$, $\int h\,d\gamma_{c}=\int h\rho_{c}\,d\lambda_{n}$ in $[0,\infty]$, and a Borel $h:\mathbb{R}^{n}\to\mathbb{R}$ is $\gamma_{c}$-integrable exactly when $h\rho_{c}$ is $\lambda_{n}$-integrable. Write $\mathcal{P}$ for the set of polynomial functions on $\mathbb{R}^{n}$, $c_{\max}$ for the largest variance as in The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances, and put $\tau=1/(4c_{\max})>0$ and $E(y)=\exp\bigl(\tau\sum_{k=1}^{n}y_{k}^{2}\bigr)$ for $y\in\mathbb{R}^{n}$.

Step 0 (algebra and regularity of polynomial functions). (a) A constant $b$ is $b$ times the monomial of exponent $0$, and the coordinate function $y\mapsto y_{i}$ is the monomial of exponent $e_{i}$, so both lie in $\mathcal{P}$. (b) Concatenating two finite linear combinations shows that $\mathcal{P}$ is closed under sums and scalar multiples. (c) For multi-indices $\alpha,\beta$ of length $n$, the componentwise sum $\alpha+\beta$ is a multi-index of length $n$ and $\prod_{k}y_{k}^{\alpha_{k}}\prod_{k}y_{k}^{\beta_{k}}=\prod_{k}y_{k}^{\alpha_{k}+\beta_{k}}$; hence, by Generalized Distributivity: Expanding a Product of Finite Sums with two factors (after enumerating the finite index sets $I$ and $I'$ and padding the shorter sum with zero terms), the product of $\sum_{l\in I}a_{l}y^{\alpha^{(l)}}$ and $\sum_{l'\in I'}b_{l'}y^{\beta^{(l')}}$ is $\sum_{(l,l')\in I\times I'}a_{l}b_{l'}y^{\alpha^{(l)}+\beta^{(l')}}$, again a finite linear combination of monomials (here and below such a combination is a sum over a finite family, whose exponents may repeat), and $\mathcal{P}$ is closed under products. (d) If $L\in\mathbb{N}$ and $f_{1},\dots,f_{L}\in\mathcal{P}$, then $\sum_{l=1}^{L}f_{l}$ and $\prod_{l=1}^{L}f_{l}$ lie in $\mathcal{P}$: the set of $l\in\mathbb{N}$ such that $l>L$ or both partial sum and partial product up to $l$ lie in $\mathcal{P}$ contains $1$ and is closed under successors by (b) and (c), so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers. Likewise, for $f\in\mathcal{P}$ the set of $k\in\mathbb{N}$ with $f^{k}\in\mathcal{P}$ contains $1$ and is closed under successors by (c), since $f^{k+1}=f^{k}f$, so it is $\mathbb{N}$. Consequently, if $p(x)=c_{0}+\sum_{k=1}^{K}c_{k}x^{k}$ with real $c_{k}$, then for all real $\alpha,\beta$ and $i\in[n]$ the function $y\mapsto p(\alpha y_{i}+\beta)$ lies in $\mathcal{P}$. (e) The set $\mathbb{R}^{n}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous apply with $U=\mathbb{R}^{n}$, here and in Step 2. The same two inductions, with claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (constants and coordinate functions are smooth, and smoothness is preserved by sums, scalar multiples and products) in place of (a), (b), (c), show that every monomial and every $P\in\mathcal{P}$ is smooth on $\mathbb{R}^{n}$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous every $P\in\mathcal{P}$ is therefore continuous on $\mathbb{R}^{n}$, and hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

Step 1 (an integrable majorant). The function $E$ is Borel: if $y^{(k)}\to y$ in $\mathbb{R}^{n}$ then each coordinate converges, since $|y^{(k)}_{i}-y_{i}|\le\lVert y^{(k)}-y\rVert$, so the set of $l\in\mathbb{N}$ such that $l>n$ or $\sum_{i=1}^{l}(y^{(k)}_{i})^{2}\to\sum_{i=1}^{l}y_{i}^{2}$ as $k\to\infty$ contains $1$ by Arithmetic of Limits of Real Sequences §products and is closed under successors by Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §products, hence is $\mathbb{N}$ by Principle of Induction for the Natural Numbers; taking $l=n$ and using Arithmetic of Limits of Real Sequences §scalar, $\tau\sum_{i}(y^{(k)}_{i})^{2}\to\tau\sum_{i}y_{i}^{2}$, and $E(y^{(k)})\to E(y)$ by claim 3(f) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities; thus $E$ is sequentially continuous, hence Borel by claims 3(a) 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. Since $2\tau c_{k}=c_{k}/(2c_{\max})\le\frac12<1$ for every $k$, The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §exponential (with $t_{k}=\tau$) shows that $E\rho_{c}$ is $\lambda_{n}$-integrable, so $E$ is $\gamma_{c}$-integrable. Moreover, for real $\Lambda\ge0$ and $s\in\mathbb{R}$, $(2\tau|s|-\Lambda)^{2}\ge0$ gives $\Lambda|s|\le\tau s^{2}+\Lambda^{2}/(4\tau)$; summing over the coordinates and using claims 1 and 4 of Basic Properties of the Exponential Function (additivity and monotonicity of $\exp$),

$$\exp\Bigl(\Lambda\sum_{i=1}^{n}|y_{i}|\Bigr)\le\exp\bigl(n\Lambda^{2}/(4\tau)\bigr)\,E(y)\qquad(y\in\mathbb{R}^{n}).$$

Claim 1. Let $P\in\mathcal{P}$ and let $r\ge1$ be real. By the definition in the statement, $P(y)=\sum_{l\in I}a_{l}\,y^{\alpha^{(l)}}$ for every $y\in\mathbb{R}^{n}$, for some finite family $(a_{l},\alpha^{(l)})_{l\in I}$ of reals $a_{l}$ and multi-indices $\alpha^{(l)}$ of length $n$, whose exponents may repeat; adding the term $0\cdot y^{0}$ if necessary, $I$ is nonempty. By Step 0, $P$ is continuous and Borel. Choose $D\in\mathbb{N}$ with $\sum_{k}\alpha^{(l)}_{k}\le D$ for all $l\in I$, put $\Theta=\sum_{l\in I}|a_{l}|$ and $W(y)=1+\lVert y\rVert^{2}\ge1$. For each $k$, $0\le(|y_{k}|-1)^{2}$ gives $|y_{k}|\le1+y_{k}^{2}\le W(y)$, so for each $l\in I$, $|y^{\alpha^{(l)}}|=\prod_{k}|y_{k}|^{\alpha^{(l)}_{k}}\le W(y)^{\sum_{k}\alpha^{(l)}_{k}}\le W(y)^{D}$, the last step by claim 1 of Growth Bound for a Polynomial Function on the Real Line (and trivially when the exponent is $0$). Hence, by the triangle inequality for finite sums, $|P(y)|\le\sum_{l\in I}|a_{l}|\,|y^{\alpha^{(l)}}|\le\Theta\,W(y)^{D}\le V(y)$ with $V=(1+\Theta)W^{D}\ge1$. Choose $N_{r}\in\mathbb{N}$ with $r\le N_{r}$ (Archimedean property). By Properties of Real Powers of Nonnegative Real Numbers §monotone and claims 1 and 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities (for a base $t\ge1$ the real power $t^{b}$ is nondecreasing in $b$, and agrees with the natural power for natural exponents), $|P(y)|^{r}\le V(y)^{r}\le V(y)^{N_{r}}=(1+\Theta)^{N_{r}}W(y)^{K}$ with $K=DN_{r}\in\mathbb{N}$. For $u\ge0$ the partial sums $\sum_{i=0}^{M}u^{i}/i!$ of the series defining $\exp(u)$ in The Real Exponential Function are nondecreasing in $M$ and are $\ge u^{K}/K!$ once $M\ge K$, so $u^{K}/K!\le\exp(u)$. With $u=\tau W(y)$ and claim 1 of Basic Properties of the Exponential Function,

$$|P(y)|^{r}\le(1+\Theta)^{N_{r}}\,K!\,\tau^{-K}\exp(\tau)\,E(y)\qquad(y\in\mathbb{R}^{n}).$$

The map $|P|^{r}$ is Borel by Power-Integrable Functions and the p-Seminorm §measurable-power, so monotonicity of the integral (Linearity and Monotonicity of the Lebesgue Integral §nonnegative) and Step 1 give $\int|P|^{r}\,d\gamma_{c}<\infty$. For $r=2$ this says $P\in\mathcal{L}^{2}(\gamma_{c})$ in the sense of Power-Integrable Functions and the p-Seminorm §space, so its class lies in $L^{2}(\gamma_{c})$ by The Lebesgue Space of Power-Integrable Functions §space.

Claim 2. Let $f$ and $\varepsilon$ be as in the statement. We construct a test function $g$, a trigonometric function $T$ and $P\in\mathcal{P}$ with

$$\int(f-g)^{2}\,d\gamma_{c}\le\frac{\varepsilon}{12},\qquad\int(g-T)^{2}\,d\gamma_{c}\le\frac{\varepsilon}{12},\qquad\int(T-P)^{2}\,d\gamma_{c}\le\frac{\varepsilon}{12};$$

all functions involved are Borel, so integrating the pointwise bound $(f-P)^{2}\le3(f-g)^{2}+3(g-T)^{2}+3(T-P)^{2}$ with Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int(f-P)^{2}\,d\gamma_{c}\le\frac{3\varepsilon}{4}<\varepsilon$. We also use $\int\lVert y\rVert^{2}\,\gamma_{c}(dy)=\int\lVert y\rVert^{2}\rho_{c}(y)\,\lambda_{n}(dy)=\sum_{i=1}^{n}c_{i}$, by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §moments, the map $y\mapsto\lVert y\rVert^{2}$ being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs.

Step 2 (a test function). By the preceding display and $\gamma_{c}(\mathbb{R}^{n})=1$, $\gamma_{c}$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{n})$ of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. Let $\xi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ have first component $f$ and all other components $0$; it is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and $\lVert\xi\rVert^{2}=f^{2}$, so its class lies in the space $L^{2}(\gamma_{c};\mathbb{R}^{n})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. Put $\delta_{1}=\min(1,\varepsilon/12)$, so $\delta_{1}^{2}\le\delta_{1}\le\varepsilon/12$. By Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment §dense there is a test field $\eta$ with $\lVert\xi-\eta\rVert_{\gamma_{c}}\le\delta_{1}$, that is, $\int\lVert\xi-\eta\rVert^{2}\,d\gamma_{c}\le\delta_{1}^{2}$. Let $g=\eta_{1}$, a test function. Since $(f-g)^{2}\le\lVert\xi-\eta\rVert^{2}$ pointwise, $\int(f-g)^{2}\,d\gamma_{c}\le\varepsilon/12$. The function $g$ is smooth, hence continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and Borel; by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable there is a real $B\ge0$ with $|g|\le B$; and its support is compact, hence bounded by Compact Subset of $\mathbb{R}^n$ is Bounded: there are $x_{0}\in\mathbb{R}^{n}$ and $R'>0$ with $\lVert y-x_{0}\rVert\le R'$ for every $y$ in the support. Put $R_{0}=\lVert x_{0}\rVert+R'>0$. If $\lVert y\rVert>R_{0}$, then $y$ is not in the support, by the triangle inequality of Euclidean Distance is a Metric on $\mathbb{R}^n$, and since the support contains $\{g\ne0\}$ (Support of a Real-Valued Function on a Topological Space) we get $g(y)=0$.

Step 3 (periodisation). Put

$$R=R_{0}+1+\frac{24(B+1)^{2}\sum_{i=1}^{n}c_{i}}{\varepsilon},$$

so $R>R_{0}$, $R\ge1$ and $R^{2}\ge R\ge24(B+1)^{2}\bigl(\sum_{i}c_{i}\bigr)/\varepsilon$. Let $\mathbf{1}$ be the point of $\mathbb{R}^{n}$ all of whose coordinates are $1$, and $h(x)=g(2Rx-R\mathbf{1})$. By claim 5 of Partial Derivatives, Continuity and $C^k$ Regularity under a Scaling Substitution (with $U=V=\mathbb{R}^{n}$, open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, the point $-R\mathbf{1}$, the factors $2R$ and $1$), $h$ is smooth, hence continuous. Let $\mathcal{W}=\{x\in\mathbb{R}^{n}:|2Rx_{i}-R|\le R_{0}\text{ for every }i\in[n]\}$. For $x\in\mathcal{W}$, $-R_{0}\le2Rx_{i}-R\le R_{0}$ and $R_{0}<R$ give $0<x_{i}<1$, so $\mathcal{W}$ is contained in the open unit cell $\mathring{Q}$ of The Half-Open Unit Cell Tiles Euclidean Space §cell; for $x\notin\mathcal{W}$ some $|2Rx_{i}-R|>R_{0}$, so $\lVert2Rx-R\mathbf{1}\rVert>R_{0}$ and $h(x)=0$. By Elementary Properties of Lattice-Periodic Functions §periodisation, $u=h\circ\varpi$, with the wrapping map of The Half-Open Unit Cell Tiles Euclidean Space §wrap, written $\varpi$ here, belongs to $C_{\mathrm{per}}$ and $u(x)=h(x)$ for $x\in Q$; also $|u|\le B$.

Step 4 (Fejer approximation). Put $\delta=\min(1,\varepsilon/24)$. By Uniform Convergence of the Fejer Means of a Continuous Periodic Function on the Torus §uniform there is $N\in\mathbb{N}$ with $|\sigma_{N}u(x)-u(x)|\le\delta$ for every $x\in\mathbb{R}^{n}$, where by The Fejer Means of a Continuous Periodic Function on the Torus the Fejer mean is

$$\sigma_{N}u(x)=\sum_{a\in[2N+1]^{n}}\kappa_{a}\,e_{k(a)}(x),\qquad\kappa_{a}=W_{N,a}\bigl\langle[\,u|_{Q}\,],E_{k(a)}\bigr\rangle_{L^{2}}\in\mathbb{R},$$

with $e_{k}$ the trigonometric system and $k(a)\in\mathbb{Z}^{n}$ and the weights $W_{N,a}$ as in The Reproducing Identity for the Fejer Kernels of the Torus; as fixed in The Fejer Means of a Continuous Periodic Function on the Torus, $E_{k}=[\,e_{k}|_{Q}\,]$ is the class in $L^{2}(\mathbb{T}^{n})$ of the restriction of $e_{k}$ to the cell $Q$, and $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is the inner product of $L^{2}(\mathbb{T}^{n})$, so each $\kappa_{a}$ is a fixed real number. Let $\theta(y)$ be the point with coordinates $(y_{i}+R)/(2R)$ and $T(y)=\sigma_{N}u(\theta(y))$. Let $C=\{y:-R\le y_{i}<R\text{ for every }i\}$. If $y\in C$, then $\theta(y)\in Q$, so $u(\theta(y))=h(\theta(y))=g(y)$ and $|g(y)-T(y)|\le\delta$. If $y\notin C$, some $|y_{i}|\ge R>R_{0}$, so $\lVert y\rVert\ge R$, $g(y)=0$, $\lVert y\rVert^{2}/R^{2}\ge1$ and $|T(y)|\le|u(\theta(y))|+\delta\le B+1$. In both cases

$$(g(y)-T(y))^{2}\le\delta^{2}+(B+1)^{2}R^{-2}\lVert y\rVert^{2}.$$

Here $T$ is Borel by Step 7 below. Integrating, with $\gamma_{c}(\mathbb{R}^{n})=1$, $\delta^{2}\le\delta\le\varepsilon/24$ and the choice of $R$, gives $\int(g-T)^{2}\,d\gamma_{c}\le\delta^{2}+(B+1)^{2}R^{-2}\sum_{i}c_{i}\le\varepsilon/12$.

Step 5 (explicit form of $T$). Here $\pi$ is the real number in $C_{m}(t)=\cos(2\pi mt)$, $S_{m}(t)=\sin(2\pi mt)$ of The Trigonometric System on the Torus. For $a\in[2N+1]^{n}$ and $i\in[n]$ let $m=k(a)_{i}\in\mathbb{Z}$, and define reals $\alpha_{a,i},\beta_{a,i}$ and $\psi_{a,i}:\mathbb{R}\to\mathbb{R}$ by: $\alpha_{a,i}=\beta_{a,i}=0$ and $\psi_{a,i}=1$ if $m=0$; $\alpha_{a,i}=\pi m/R$, $\beta_{a,i}=\pi m$ and $\psi_{a,i}(s)=\sqrt2\cos(\alpha_{a,i}s+\beta_{a,i})$ if $m>0$; $\alpha_{a,i}=-\pi m/R$, $\beta_{a,i}=-\pi m$ and $\psi_{a,i}(s)=\sqrt2\sin(\alpha_{a,i}s+\beta_{a,i})$ if $m<0$. Since $2\pi|m|\,(s+R)/(2R)=\alpha_{a,i}s+\beta_{a,i}$, The Trigonometric System on the Torus §one-dimensional gives $\phi_{m}((s+R)/(2R))=\psi_{a,i}(s)$, and The Trigonometric System on the Torus §system gives

$$T(y)=\sum_{a\in[2N+1]^{n}}\kappa_{a}\prod_{i=1}^{n}\psi_{a,i}(y_{i})\qquad(y\in\mathbb{R}^{n}).$$

Step 6 (Taylor polynomials of cosine and sine). For $j\in\mathbb{N}$ and real $x$ put $p^{c}_{j}(x)=1+\sum_{k=1}^{j}c_{k}(x)$ and $p^{s}_{j}(x)=x+\sum_{k=1}^{j}\sigma_{k}(x)$, with $c_{k},\sigma_{k}$ as in Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine; these are polynomial functions of one variable. For real $A>0$ the series $S^{c}(A)=\sum_{k=1}^{\infty}A^{2k}/(2k)!$ and $S^{s}(A)=\sum_{k=1}^{\infty}A^{2k+1}/(2k+1)!$ converge by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials §trigonometric; let $t^{c}_{j}(A)$ and $t^{s}_{j}(A)$ be the sum minus the $j$th partial sum. Fix $x$ and put $A=|x|+1$, so $x\in[-A,A]$. (i) By Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine §uniform with radius $A$, $|\cos x-p^{c}_{j}(x)|\le t^{c}_{j}(A)$ and $|\sin x-p^{s}_{j}(x)|\le t^{s}_{j}(A)$. (ii) By Series of Real Numbers §convergent and Arithmetic of Limits of Real Sequences §scalar, $t^{c}_{j}(A)\to0$ and $t^{s}_{j}(A)\to0$ as $j\to\infty$; so $p^{c}_{j}(x)\to\cos x$ and $p^{s}_{j}(x)\to\sin x$ by claim 3(c) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities. (iii) The terms are nonnegative, so the partial sums are nondecreasing and bounded by the sums, whence $0\le t^{c}_{j}(A)\le S^{c}(A)$ and $0\le t^{s}_{j}(A)\le S^{s}(A)$. For $J\in\mathbb{N}$, regrouping $\sum_{i=1}^{2J}A^{i}/i!=\sum_{k=1}^{J}\bigl(A^{2k-1}/(2k-1)!+A^{2k}/(2k)!\bigr)$ and $\sum_{i=1}^{2J+1}A^{i}/i!=A+\sum_{k=1}^{J}\bigl(A^{2k}/(2k)!+A^{2k+1}/(2k+1)!\bigr)$ shows that the $J$th partial sums of $S^{c}(A)$ and $S^{s}(A)$ are at most partial sums of $\sum_{i=1}^{\infty}A^{i}/i!$, which converges with nonnegative terms by Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials §exponential, and hence at most its sum; that sum is $\exp(A)-1$, because by The Real Exponential Function $\exp(A)$ is the limit of $\sum_{i=0}^{M}A^{i}/i!=1+\sum_{i=1}^{M}A^{i}/i!$. Letting $J\to\infty$, $S^{c}(A)\le\exp(A)-1$ and $S^{s}(A)\le\exp(A)-1$. (iv) By The Pythagorean Identity for Sine and Cosine §bounds, (i) and (iii), $|p^{c}_{j}(x)|\le1+t^{c}_{j}(A)\le\exp(A)$ and $|p^{s}_{j}(x)|\le\exp(A)$; also $|\cos x|,|\sin x|\le1\le1+A\le\exp(A)$ by claim 4 of Basic Properties of the Exponential Function.

Step 7 (polynomial approximation of $T$). For $a$, $i$ as in Step 5 and $j\in\mathbb{N}$ let $q_{a,i,j}(s)$ be $1$ if $k(a)_{i}=0$, $\sqrt2\,p^{c}_{j}(\alpha_{a,i}s+\beta_{a,i})$ if $k(a)_{i}>0$, and $\sqrt2\,p^{s}_{j}(\alpha_{a,i}s+\beta_{a,i})$ if $k(a)_{i}<0$, and put

$$P_{j}(y)=\sum_{a\in[2N+1]^{n}}\kappa_{a}\prod_{i=1}^{n}q_{a,i,j}(y_{i}).$$

By Step 0 (d), each $y\mapsto q_{a,i,j}(y_{i})$ lies in $\mathcal{P}$, and so does $P_{j}$ (finite products and sums). By Step 6 (ii) and Arithmetic of Limits of Real Sequences §scalar, $q_{a,i,j}(s)\to\psi_{a,i}(s)$ as $j\to\infty$, for every $s$. For fixed $y$ and $a$, the set of $l\in\mathbb{N}$ such that $l>n$ or $\prod_{i=1}^{l}q_{a,i,j}(y_{i})\to\prod_{i=1}^{l}\psi_{a,i}(y_{i})$ contains $1$ and is closed under successors by Arithmetic of Limits of Real Sequences §products, so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers, and taking $l=n$ gives $\prod_{i=1}^{n}q_{a,i,j}(y_{i})\to\prod_{i=1}^{n}\psi_{a,i}(y_{i})$. Enumerate the nonempty finite set $[2N+1]^{n}$ without repetition as $a_{1},\dots,a_{L_{0}}$. For fixed $y$, the set of $l\in\mathbb{N}$ such that $l>L_{0}$ or $\sum_{h=1}^{l}\kappa_{a_{h}}\prod_{i=1}^{n}q_{a_{h},i,j}(y_{i})\to\sum_{h=1}^{l}\kappa_{a_{h}}\prod_{i=1}^{n}\psi_{a_{h},i}(y_{i})$ as $j\to\infty$ contains $1$ by Arithmetic of Limits of Real Sequences §scalar and is closed under successors by Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar, so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers; taking $l=L_{0}$ gives $P_{j}(y)\to T(y)$. Hence $T$ is Borel by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. If $k(a)_{i}\ne0$, Step 6 (iv) applied at $x=\alpha_{a,i}s+\beta_{a,i}$, together with $|x|+1\le|\alpha_{a,i}||s|+|\beta_{a,i}|+1$ and the monotonicity of $\exp$ in claim 4 of Basic Properties of the Exponential Function, gives the two bounds below. If $k(a)_{i}=0$, then $q_{a,i,j}=\psi_{a,i}=1$ and $\alpha_{a,i}=\beta_{a,i}=0$, and $1\le1+1\le\exp(1)\le\sqrt2\exp(1)$ by claim 4 of Basic Properties of the Exponential Function (with $u=1$), $1\le\sqrt2$ and $\exp(1)>0$; so the bounds hold in this case as well:

$$|q_{a,i,j}(s)|\le\sqrt2\exp\bigl(|\alpha_{a,i}||s|+|\beta_{a,i}|+1\bigr),\qquad|\psi_{a,i}(s)|\le\sqrt2\exp\bigl(|\alpha_{a,i}||s|+|\beta_{a,i}|+1\bigr).$$

Put $\Lambda_{0}=\sum_{a}\sum_{i}|\alpha_{a,i}|$, $\Gamma_{0}=\sum_{a}\sum_{i}(|\beta_{a,i}|+1)$ and $K_{1}=\bigl(\sum_{a}|\kappa_{a}|\bigr)(\sqrt2)^{n}\exp(\Gamma_{0})$. Multiplying the bounds over $i$, using claim 1 of Basic Properties of the Exponential Function and $|\alpha_{a,i}|\le\Lambda_{0}$, and summing over $a$, we get $|P_{j}(y)|\le K_{1}\exp\bigl(\Lambda_{0}\sum_{i}|y_{i}|\bigr)$ and $|T(y)|\le K_{1}\exp\bigl(\Lambda_{0}\sum_{i}|y_{i}|\bigr)$, hence, by Step 1 with $\Lambda=2\Lambda_{0}$,

$$(P_{j}(y)-T(y))^{2}\le4K_{1}^{2}\exp\Bigl(2\Lambda_{0}\sum_{i=1}^{n}|y_{i}|\Bigr)\le4K_{1}^{2}\exp\bigl(n\Lambda_{0}^{2}/\tau\bigr)E(y).$$

The functions $(P_{j}-T)^{2}$ are Borel (Step 0 and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), converge pointwise to $0$, and are dominated by a $\gamma_{c}$-integrable function (Step 1). By Dominated Convergence Theorem, $\int(P_{j}-T)^{2}\,d\gamma_{c}\to0$, so there is $j$ with $\int(T-P_{j})^{2}\,d\gamma_{c}\le\varepsilon/12$. Taking $P=P_{j}\in\mathcal{P}$ completes the three estimates, and claim 2 follows as explained at its start.