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

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for adding, multiplying and comparing inequalities between real numbers, the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, and the norm properties of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ (claim 4: $|v_{l}|\le\lVert v\rVert$; claim 5: homogeneity; claim 6: triangle inequality) are used without further mention. Integrals of nonnegative Borel functions are taken in $[0,\infty]$ and compared by claim 1 of Linearity and Monotonicity of the Lebesgue Integral. For real $a,b$, $(a+b)^{2}\le2a^{2}+2b^{2}$, since the difference is $(a-b)^{2}\ge0$.

Let $\xi\in L^{2}(\mu;\mathbb{R}^{d})$ and $\varepsilon>0$.

Step 1 (a bounded Lipschitz field). By The Bounded Lipschitz Vector Fields are Dense in the Square-Integrable Vector Fields Against a Probability Measure §dense there is a bounded Lipschitz map $\zeta:\mathbb{R}^{d}\to\mathbb{R}^{d}$, in the sense of that lemma, with $\lVert\xi-\zeta\rVert_{\mu}\le\varepsilon/3$; let $M\ge0$ satisfy $\lVert\zeta(x)\rVert\le M$ for all $x$. Each component $\zeta_{l}$ satisfies $|\zeta_{l}(x)|\le M$ and $|\zeta_{l}(x)-\zeta_{l}(y)|\le\lVert\zeta(x)-\zeta(y)\rVert$, so it is Lipschitz, hence continuous (A Lipschitz Map is Uniformly Continuous).

Step 2 (the radius). Put $R=1+9M^{2}M_{2}(\mu)\varepsilon^{-2}$, where $M_{2}(\mu)=\int\lVert x\rVert^{2}\,\mu(dx)<\infty$ is the second moment. Then $R\ge1$, and pointwise $\mathbf{1}_{\{\lVert x\rVert>R\}}(x)\le R^{-2}\lVert x\rVert^{2}$, so

the set being Borel as $x\mapsto\lVert x\rVert^{2}$ is Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and $\{\lVert x\rVert>R\}=\{\lVert x\rVert^{2}>R^{2}\}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), a set of the form covered by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line.

Step 3 (mollification). Let $\rho$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$ (Existence of Mollifier Kernels of Every Radius), and for $\delta>0$ let $\rho_{\delta}(y)=(\delta^{-1})^{d}\rho(\delta^{-1}y)$, a mollifier kernel of radius $\delta$ (Rescaling a Mollifier Kernel). With $\Omega=\mathbb{R}^{d}$ in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel, $\Omega^{\delta}=\mathbb{R}^{d}$, so $\zeta_{l}*\rho_{\delta}$ is defined on $\mathbb{R}^{d}$, and it is smooth there by claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$. Let $K=\bar{B}(0_{\mathbb{R}^{d}},2R)$, compact by A Closed Euclidean Ball is Convex and Compact, and $\tau=\varepsilon/(3d)$, the natural number $d\ge1$ read in $\mathbb{R}$. For each $l\in[d]$, claim 2 of Mollification Converges Uniformly on Compact Subsets (with $n=d$, $\Omega=\mathbb{R}^{d}$, $f=\zeta_{l}$, radius $1$, the compact set $K$ and $\eta=\tau$) gives $\varepsilon_{0,l}>0$. Let $\delta$ be half the least of $\varepsilon_{0,1},\dots,\varepsilon_{0,d}$ and $1$; then $0<\delta<\varepsilon_{0,l}$ for every $l$, and

Step 4 (the test field). Let $\chi_{R}$ be the cutoff of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff (dimension $d$): smooth, $0\le\chi_{R}\le1$, $\chi_{R}(x)=1$ for $\lVert x\rVert\le R$ and $\chi_{R}(x)=0$ for $\lVert x\rVert\ge2R$. Put $\eta_{l}=\chi_{R}\,(\zeta_{l}*\rho_{\delta})$. It is smooth by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and vanishes whenever $\lVert x\rVert>2R$, so it is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set; hence $\eta_{l}\in C_{c}^{\infty}(\mathbb{R}^{d})$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space), and $\eta=(\eta_{1},\dots,\eta_{d})$ is a test field.

Step 5 (the estimate). Let $x\in\mathbb{R}^{d}$ and write $\zeta*\rho_{\delta}$ for the map with components $\zeta_{l}*\rho_{\delta}$. We claim

If $\lVert x\rVert>2R$, then $\eta(x)=0$ and $\lVert\eta(x)-\zeta(x)\rVert=\lVert\zeta(x)\rVert\le M$. If $\lVert x\rVert\le2R$, then $x\in K$ and $\sum_{l}\bigl((\zeta_{l}*\rho_{\delta})(x)-\zeta_{l}(x)\bigr)^{2}<d\tau^{2}\le(\varepsilon/3)^{2}$, so $\lVert(\zeta*\rho_{\delta})(x)-\zeta(x)\rVert<\varepsilon/3$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); since $\eta(x)-\zeta(x)=\chi_{R}(x)\bigl((\zeta*\rho_{\delta})(x)-\zeta(x)\bigr)-(1-\chi_{R}(x))\zeta(x)$,

and $1-\chi_{R}(x)=0$ when $\lVert x\rVert\le R$, while $1-\chi_{R}(x)\le1$ always. This proves (P). The map $\eta-\zeta$ is Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and squaring (P),

by Step 2, so $\lVert\eta-\zeta\rVert_{\mu}\le2\varepsilon/3$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). By the triangle inequality in $L^{2}(\mu;\mathbb{R}^{d})$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity), $\lVert\xi-\eta\rVert_{\mu}\le\lVert\xi-\zeta\rVert_{\mu}+\lVert\zeta-\eta\rVert_{\mu}\le\varepsilon$. $\blacksquare$