Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $L^2(\mathbb{R}^d)$ denote the Bochner space of all Borel functions $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$ satisfying $\int \, \lVert f(x)\rVert^2dx<\infty$.

Let $\mathcal{X}\subseteq \mathcal{W}_2(\mathbb{R}^d)$ consist of all measures ${\nu}$ for which there is some $f\in L^2(\lambda,\mathbb{R}^d)$ satisfying: $$ {\nu}=f_{\#}\lambda $$ where $\lambda$ is the uniform measure on $[0,1]^d$.

How is $\mathcal{X}$ related to $\mathcal{W}_2(\mathbb{R}^d)$? Is $\mathcal{X}$ a dense subset of $\mathcal{W}_2(\mathbb{R}^d)$?