Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$, with norm $\lVert\cdot\rVert_{\mu}$; the test functions form the set $C_{c}^{\infty}(\mathbb{R}^{d})$ fixed there. Call a map $\eta=(\eta_{1},\dots,\eta_{d}):\mathbb{R}^{d}\to\mathbb{R}^{d}$ a test field if $\eta_{l}\in C_{c}^{\infty}(\mathbb{R}^{d})$ for every $l\in[d]$. Each $\eta_{l}$ is then continuous, hence Borel, and there is a real number $B\ge0$ with $|\eta_{l}(x)|\le B$ for all $x\in\mathbb{R}^{d}$ and $l\in[d]$, by A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable and claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; so $\eta$ is Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), $\lVert\eta(x)\rVert^{2}\le dB^{2}$ for every $x$, and the class of $\eta$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\eta$.

1. (Density)¶ Let $\xi\in L^{2}(\mu;\mathbb{R}^{d})$ and let $\varepsilon$ be a positive real number. Then there is a test field $\eta$ with $\lVert\xi-\eta\rVert_{\mu}\le\varepsilon$.