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

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) the space $L^{2}(\mu;\mathbb{R}^{d})$ is a real Hilbert space with norm $\lVert\cdot\rVert_{\mu}$ given by $\lVert\eta\rVert_{\mu}^{2}=\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu$, and a class is denoted by the same symbol as a representative of it; the triangle inequality $\lVert\eta+\eta'\rVert_{\mu}\le\lVert\eta\rVert_{\mu}+\lVert\eta'\rVert_{\mu}$ is claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity. Let $\iota$ denote the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Since $\mu\in\mathcal{P}(\mathbb{R}^{d})$ it is a Borel measure on $(\mathbb{R}^{d},d_{E})$ with $\mu(\mathbb{R}^{d})=1$, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), so Inner and Outer Regularity of a Finite Borel Measure on a Metric Space, and Lipschitz Approximation of Indicators applies to it. For $A\in\mathcal{B}(\mathbb{R}^{d})$, $\mathbf{1}_{A}$ is its indicator, and $\int_{\mathbb{R}^{d}}\mathbf{1}_{A}\,d\mu=\mu(A)$ by that lemma.

Fix a Borel representative of $\xi$, again written $\xi$, so $\int_{\mathbb{R}^{d}}\lVert\xi\rVert^{2}\,d\mu<\infty$. Its components $\xi_{1},\dots,\xi_{d}$ are Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\lVert v\rVert^{2}=\sum_{i=1}^{d}v_{i}^{2}$ for $v\in\mathbb{R}^{d}$ by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

Step 1 (Truncation). For $K\in\mathbb{N}$ let $A_{K}=\{x\in\mathbb{R}^{d}:\lVert\xi(x)\rVert\le\iota(K)\}$, a member of $\mathcal{B}(\mathbb{R}^{d})$ because $x\mapsto\lVert\xi(x)\rVert^{2}$ 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 $A_{K}$ is the preimage of a ray under it, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Let $\xi^{K}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ agree with $\xi$ on $A_{K}$ and vanish off $A_{K}$; it is Borel, since for $B\in\mathcal{B}(\mathbb{R}^{d})$ its preimage is $(A_{K}\cap\xi^{-1}(B))\cup(C_{K}\cap E_{B})$ with $C_{K}=\mathbb{R}^{d}\setminus A_{K}$ and $E_{B}=\mathbb{R}^{d}$ or $\varnothing$ according as $0_{\mathbb{R}^{d}}\in B$ or not, and it satisfies $\lVert\xi^{K}(x)\rVert\le\iota(K)$ for every $x$.

The functions $f_{K}=\lVert\xi-\xi^{K}\rVert^{2}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), satisfy $0\le f_{K}\le\lVert\xi\rVert^{2}$ (they vanish on $A_{K}$ and equal $\lVert\xi\rVert^{2}$ off it), and converge pointwise to $0$: given $x$, the real number $\lVert\xi(x)\rVert$ satisfies $\lVert\xi(x)\rVert<\iota(K)$ for some $K\in\mathbb{N}$ by claim 1 of The Archimedean Property of the Real Numbers, and then $x\in A_{K'}$ and $f_{K'}(x)=0$ for every $K'\ge K$, by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Since $\lVert\xi\rVert^{2}$ is integrable with respect to $\mu$, claim 7 of The Lebesgue Integral and Null Sets gives that $\bigl(\int_{\mathbb{R}^{d}}f_{K}\,d\mu\bigr)_{K\in\mathbb{N}}$ converges to $0$, that is, $\lVert\xi-\xi^{K}\rVert_{\mu}^{2}\to0$.

Fix the positive real number $\varepsilon$ first. Choose $K\in\mathbb{N}$ with $\lVert\xi-\xi^{K}\rVert_{\mu}\le\tfrac{\varepsilon}{2}$, which is possible by the previous paragraph together with Existence and Uniqueness of the Nonnegative Square Root and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Write $M=\iota(K)$.

Step 2 (Uniform approximation of a bounded component by a simple function). Let $i\in[d]$ and let $\delta$ be a positive real number. The function $\xi^{K}_{i}+M$ is Borel and satisfies $0\le\xi^{K}_{i}(x)+M\le2M$ for every $x$, by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claim 3 of Properties of the Absolute Value in an Ordered Field. By Approximation of Measurable Functions by Dyadic Simple Functions (ii) there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of nonnegative simple functions on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ with $0\le(\xi^{K}_{i}(x)+M)-s_{m}(x)\le(\tfrac{1}{2})^{m}$ for every $x\in\mathbb{R}^{d}$ and every $m\in\mathbb{N}$. Since $\bigl((\tfrac{1}{2})^{m}\bigr)_{m\in\mathbb{N}}$ converges to $0$ by claim 4 of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series, there is $m\in\mathbb{N}$ with $(\tfrac{1}{2})^{m}\le\delta$; put $s=s_{m}$, so that $0\le(\xi^{K}_{i}(x)+M)-s(x)\le\delta$ for every $x$. The function $\sigma_{i}=s-M$ is again simple, having finite range and being measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and

$$|\xi^{K}_{i}(x)-\sigma_{i}(x)|\le\delta\qquad\text{for every }x\in\mathbb{R}^{d}.$$

Let $c_{1},\dots,c_{r}$ be the distinct values of $\sigma_{i}$ and $A_{j}=\sigma_{i}^{-1}(\{c_{j}\})\in\mathcal{B}(\mathbb{R}^{d})$, so that $\sigma_{i}=\sum_{j=1}^{r}c_{j}\mathbf{1}_{A_{j}}$.

Step 3 (Replacing the indicators by Lipschitz functions). Let $\delta'$ be a positive real number. By Inner and Outer Regularity of a Finite Borel Measure on a Metric Space, and Lipschitz Approximation of Indicators (iv), applied to $\mu$ and to each $A_{j}$ with the positive real number $\delta'$, there are maps $h_{j}:\mathbb{R}^{d}\to\mathbb{R}$, each Lipschitz with a constant and with values in $[0,1]$, and sets $N_{j}\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(N_{j})\le\delta'$, such that $h_{j}=\mathbf{1}_{A_{j}}$ off $N_{j}$. Put

$$\zeta_{i}=\sum_{j=1}^{r}c_{j}h_{j},\qquad N^{i}=\bigcup_{j=1}^{r}N_{j}.$$

Write $C_{i}=\sum_{j=1}^{r}|c_{j}|$ and let $L_{j}$ be a Lipschitz constant for $h_{j}$. Then $\zeta_{i}$ is Lipschitz with constant $\sum_{j=1}^{r}|c_{j}|L_{j}$: by Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, claim 4 of Properties of the Absolute Value in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field applied to $|h_{j}(x)-h_{j}(y)|\le L_{j}\,d_{E}(x,y)$ with the nonnegative factor $|c_{j}|$, and claims 2 and 3 of Properties of Finite Sums,

$$|\zeta_{i}(x)-\zeta_{i}(y)|\le\sum_{j=1}^{r}|c_{j}|\,|h_{j}(x)-h_{j}(y)|\le\Bigl(\sum_{j=1}^{r}|c_{j}|L_{j}\Bigr)d_{E}(x,y).$$

Likewise $|\zeta_{i}(x)|\le\sum_{j=1}^{r}|c_{j}|\,h_{j}(x)\le C_{i}$ for every $x$, since $0\le h_{j}\le1$, so $\zeta_{i}$ is bounded by $C_{i}$; and $\zeta_{i}=\sigma_{i}$ on $\mathbb{R}^{d}\setminus N^{i}$. Moreover $\mu(N^{i})\le r\delta'$ by claim 4 of Basic Properties of a Measure, applied to the sequence whose first $r$ terms are $N_{1},\dots,N_{r}$ and whose remaining terms are $\varnothing$, and $|\zeta_{i}-\sigma_{i}|\le2C_{i}$ everywhere.

Step 4 (Assembly). Apply Steps 2 and 3 for each $i\in[d]$, with $\delta$ and $\delta'$ still to be fixed, and let $\zeta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map with components $\zeta_{1},\dots,\zeta_{d}$, which exists by claim 2 of Euclidean Points as Tuples of Real Numbers. It is bounded, by $\bigl(\sum_{i=1}^{d}C_{i}^{2}\bigr)^{1/2}$, and Lipschitz: if $L_{i}$ is a Lipschitz constant for $\zeta_{i}$ then, by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square,

$$\lVert\zeta(x)-\zeta(y)\rVert^{2}=\sum_{i=1}^{d}\bigl(\zeta_{i}(x)-\zeta_{i}(y)\bigr)^{2}\le\Bigl(\sum_{i=1}^{d}L_{i}^{2}\Bigr)d_{E}(x,y)^{2},$$

so $\zeta$ is Lipschitz with constant $\bigl(\sum_{i}L_{i}^{2}\bigr)^{1/2}$. Hence $\zeta$ is bounded Lipschitz in the sense of the statement, and its class lies in $L^{2}(\mu;\mathbb{R}^{d})$.

By claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, the integrands being nonnegative,

$$\lVert\xi^{K}-\zeta\rVert_{\mu}^{2}=\sum_{i=1}^{d}\int_{\mathbb{R}^{d}}(\xi^{K}_{i}-\zeta_{i})^{2}\,d\mu .$$

For each $i$, split the integrand: off $N^{i}$ one has $\zeta_{i}=\sigma_{i}$ and hence $|\xi^{K}_{i}-\zeta_{i}|\le\delta$ by Step 2, while on $N^{i}$ one has $|\xi^{K}_{i}-\zeta_{i}|\le M+C_{i}$. Therefore, by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and The Integral of an Indicator Function is the Measure of the Set,

$$\int_{\mathbb{R}^{d}}(\xi^{K}_{i}-\zeta_{i})^{2}\,d\mu\le\delta^{2}+(M+C_{i})^{2}\,\mu(N^{i})\le\delta^{2}+(M+C_{i})^{2}r\,\delta' .$$

Now fix the order of choice: having fixed $\varepsilon$ and then $K$ (hence $M$), choose $\delta$ positive with $d\,\delta^{2}\le\tfrac{\varepsilon^{2}}{8}$; this determines, through Step 2, the simple functions $\sigma_{i}$, hence the numbers $r$ and $C_{i}$; then choose $\delta'$ positive with $\sum_{i=1}^{d}(M+C_{i})^{2}r\,\delta'\le\tfrac{\varepsilon^{2}}{8}$. Summing over $i$ gives $\lVert\xi^{K}-\zeta\rVert_{\mu}^{2}\le\tfrac{\varepsilon^{2}}{4}$, so $\lVert\xi^{K}-\zeta\rVert_{\mu}\le\tfrac{\varepsilon}{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Finally, by the triangle inequality,

\lVert\xi-\zeta\rVert_{\mu}\le\lVert\xi-\xi^{K}\rVert_{\mu}+\lVert\xi^{K}-\zeta\rVert_{\mu}\le\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon ,$$ which proves claim 1.