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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, with the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{m}),W_{2})$ and the set $\mathcal{P}(\mathbb{R}^{m})$ of Borel probability measures; push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, and $\bar{B}(0_{\mathbb{R}^{m}},R)$ is the closed ball of Euclidean Space and Lebesgue Measure: Standing Notation §space. For a positive $R\in\mathbb{R}$ let $P_{R}:\mathbb{R}^{m}\to\mathbb{R}^{m}$ be the radial retraction onto $\bar{B}(0_{\mathbb{R}^{m}},R)$:

where $\lVert x\rVert^{-1}$ exists because $0<R<\lVert x\rVert$ in the second case.

1. (Retraction)¶ Let $R\in\mathbb{R}$ be positive. Then $P_{R}$ is Borel; for every $x\in\mathbb{R}^{m}$ one has $\lVert P_{R}(x)\rVert\le R$ and $\lVert x-P_{R}(x)\rVert\le\lVert x\rVert$, and $P_{R}(x)=x$ if $\lVert x\rVert\le R$. For every $\mu\in\mathcal{P}(\mathbb{R}^{m})$, $(P_{R})_{\#}\mu\bigl(\bar{B}(0_{\mathbb{R}^{m}},R)\bigr)=1$.

2. (Approximation)¶ Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$. Then $(P_{R})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ for every positive $R\in\mathbb{R}$, and for every positive $\varepsilon\in\mathbb{R}$ there is a positive $R_{0}\in\mathbb{R}$ with $W_{2}\bigl((P_{R})_{\#}\mu,\mu\bigr)<\varepsilon$ for every $R\in\mathbb{R}$ with $R_{0}\le R$.