Proof of Radial Retraction onto a Closed Ball: Compactly Supported Approximation in the Wasserstein Distance

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, and the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, are used without further mention; so are the vector-space operations of $\mathbb{R}^{m}$ and the homogeneity $\lVert c\,x\rVert=|c|\,\lVert x\rVert$ of the Euclidean norm (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Integrals of nonnegative Borel functions are taken in $[0,\infty]$ and compared by claim 1 of Linearity and Monotonicity of the Lebesgue Integral.

Claim 1. Fix a positive $R$. Let $N:\mathbb{R}^{m}\to\mathbb{R}$, $N(x)=\max\{\lVert x\rVert,R\}$. By claims 1 and 2 of Elementary Properties of the Maximum of Two Elements, $R\le N(x)$, $\lVert x\rVert\le N(x)$, and $N(x)=R$ if $\lVert x\rVert\le R$ while $N(x)=\lVert x\rVert$ if $R<\lVert x\rVert$. Hence $N(x)>0$, the quotient $r(x)=R\,N(x)^{-1}$ satisfies $0<r(x)\le1$, and

since $r(x)=RR^{-1}=1$ when $\lVert x\rVert\le R$ and $r(x)=R\lVert x\rVert^{-1}$ otherwise.

Borel. The map $x\mapsto\lVert x\rVert$ is continuous: by the triangle inequality and homogeneity (claims 6 and 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), $\lVert x\rVert\le\lVert x-y\rVert+\lVert y\rVert$ and $\lVert y\rVert\le\lVert x-y\rVert+\lVert x\rVert$, so $|\lVert x\rVert-\lVert y\rVert|\le\lVert x-y\rVert$ (claim 6 of Properties of the Absolute Value in an Ordered Field), and A Lipschitz Map is Uniformly Continuous applies. So it is Borel (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), and $N$ is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the constant $R$ being Borel. The function $r$ is Borel by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line: for real $a\le0$, $\{r>a\}=\mathbb{R}^{m}$ because $r>0$; for real $a>0$, $r(x)>a$ holds if and only if $N(x)<Ra^{-1}$, that is $-N(x)>-Ra^{-1}$, so $\{r>a\}=\{-N>-Ra^{-1}\}$, which is Borel because $-N$ is (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line again. Each component $x\mapsto r(x)x_{l}$ of $P_{R}$ is a product of Borel functions, the coordinate functions being Borel (claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), hence Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); so $P_{R}$ is 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.

Bounds. $\lVert P_{R}(x)\rVert=r(x)\lVert x\rVert=R\,N(x)^{-1}\lVert x\rVert\le R$, because $\lVert x\rVert\le N(x)$. Next $x-P_{R}(x)=(1-r(x))\,x$ with $0\le1-r(x)<1$, so $\lVert x-P_{R}(x)\rVert=(1-r(x))\lVert x\rVert\le\lVert x\rVert$. If $\lVert x\rVert\le R$ then $r(x)=1$ and $P_{R}(x)=x$.

The push-forward. By definition $\bar{B}(0_{\mathbb{R}^{m}},R)$ consists of the $y$ with $\lVert y-0_{\mathbb{R}^{m}}\rVert=\lVert y\rVert\le R$ (Euclidean Space and Lebesgue Measure: Standing Notation §space), and it is Borel by The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §borel. Every $P_{R}(x)$ lies in it, so its preimage under $P_{R}$ is $\mathbb{R}^{m}$ and $(P_{R})_{\#}\mu(\bar{B}(0_{\mathbb{R}^{m}},R))=\mu(\mathbb{R}^{m})=1$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward).

Claim 2. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ and $R>0$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward with $S=\mathrm{id}$ and $T=P_{R}$ (so $S_{\#}\mu=\mu$), the coupling $\pi_{R}=(\mathrm{id},P_{R})_{\#}\mu$ belongs to $\Pi(\mu,(P_{R})_{\#}\mu)$ and

using Claim 1 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, $(P_{R})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, and $W_{2}((P_{R})_{\#}\mu,\mu)^{2}=W_{2}(\mu,(P_{R})_{\#}\mu)^{2}\le I(\pi_{R})$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry and The Quadratic Wasserstein Distance on Euclidean Space §distance.

For $n\in\mathbb{N}$, read in $\mathbb{R}$, let $f_{n}(x)=\lVert x\rVert^{2}\,\mathbf{1}_{\{\lVert x\rVert>n\}}(x)$, a Borel function (the set $\{\lVert x\rVert>n\}$ being Borel as $x\mapsto\lVert x\rVert$ is, and $x\mapsto\lVert x\rVert^{2}$ being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) with $0\le f_{n}\le\lVert x\rVert^{2}$, which is $\mu$-integrable. For each $x$, $f_{n}(x)=0$ once $n\ge\lVert x\rVert$ (claim 1 of The Archimedean Property of the Real Numbers), so Dominated Convergence Theorem gives $\int f_{n}\,d\mu\to0$. Let $\varepsilon>0$ and choose $n$ with $\int f_{n}\,d\mu<\varepsilon^{2}$; put $R_{0}=n$, which is positive, since $1\le n$ (claim 4 of Properties of the Order on the Natural Numbers) and the canonical image of $n$ in $\mathbb{R}$ is positive (claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field). Let $R\ge R_{0}$. For every $x$: if $\lVert x\rVert\le R$ then $\lVert x-P_{R}(x)\rVert^{2}=0\le f_{n}(x)$; if $\lVert x\rVert>R$ then $\lVert x\rVert>n$ and $\lVert x-P_{R}(x)\rVert^{2}\le\lVert x\rVert^{2}=f_{n}(x)$. Hence $W_{2}((P_{R})_{\#}\mu,\mu)^{2}\le I(\pi_{R})\le\int f_{n}\,d\mu<\varepsilon^{2}$, and $W_{2}((P_{R})_{\#}\mu,\mu)<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative. $\blacksquare$