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

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, read with a natural number $n$ in place of the dimension written $d$ there, with the Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ and integrals as in Measure Spaces and the Lebesgue Integral: Standing Notation. Let $c$ be a variance vector in $\mathbb{R}^{n}$ and $\gamma_{c}$ the diagonal Gaussian measure on $\mathbb{R}^{n}$ with variances $c$, and let $L^{2}(\gamma_{c})=L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\gamma_{c})$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with norm $\lVert\cdot\rVert_{2}$. For a multi-index of length $n$ $\alpha=(\alpha_{1},\dots,\alpha_{n})$, the monomial of exponent $\alpha$ is the function $y\mapsto\prod_{k=1}^{n}y_{k}^{\alpha_{k}}$ on $\mathbb{R}^{n}$, with $y_{k}^{0}=1$; a polynomial function on $\mathbb{R}^{n}$ is a finite linear combination of monomials.

1. (Integrability)¶ Every polynomial function $P$ on $\mathbb{R}^{n}$ is continuous, hence Borel, and $\int_{\mathbb{R}^{n}}|P|^{r}\,d\gamma_{c}<\infty$ for every real $r\ge1$; in particular the class of $P$ lies in $L^{2}(\gamma_{c})$.

2. (Density)¶ For every Borel $f:\mathbb{R}^{n}\to\mathbb{R}$ with $\int_{\mathbb{R}^{n}}f^{2}\,d\gamma_{c}<\infty$ and every real $\varepsilon>0$ there is a polynomial function $P$ on $\mathbb{R}^{n}$ with

$$\int_{\mathbb{R}^{n}}(f-P)^{2}\,d\gamma_{c}<\varepsilon .$$