The Bounded Lipschitz Vector Fields are Dense in the Square-Integrable Vector Fields Against a Probability Measure

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu$ belong to the set $\mathcal{P}(\mathbb{R}^{d})$ of probability measures on $\mathbb{R}^{d}$, and let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$, with norm $\lVert\cdot\rVert_{\mu}$.

Call a map $\zeta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ bounded Lipschitz if it is Lipschitz with a constant from $(\mathbb{R}^{d},d_{E})$ to $(\mathbb{R}^{d},d_{E})$ and there is a real number $M\ge0$ with $\lVert\zeta(x)\rVert\le M$ for every $x\in\mathbb{R}^{d}$. Such a $\zeta$ is continuous and hence Borel; the function $x\mapsto\lVert\zeta(x)\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) and bounded by $M^{2}$, so $\int_{\mathbb{R}^{d}}\lVert\zeta\rVert^{2}\,d\mu\le M^{2}<\infty$ by claim 6 of Borel Measurability and Bounded Integration on a Metric Space, and the class of $\zeta$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\zeta$.

1. (Density)(i) Let $\xi\in L^{2}(\mu;\mathbb{R}^{d})$ and let $\varepsilon$ be a positive real number. Then there is a bounded Lipschitz map $\zeta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with

$$\lVert\xi-\zeta\rVert_{\mu}\le\varepsilon .$$