The Lift of a Function on the Wasserstein Space to the Space of Square-Integrable Random Vectors

In the setting of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors, $\mathcal{P}_{2}(\mathbb{R}^{d})$ the Wasserstein space and $\Lambda:L^{2}(\Omega;\mathbb{R}^{d})\to\mathcal{P}_{2}(\mathbb{R}^{d})$ the law map, and let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$.

(The lift)(i) The lift of $u$ is the function

$$U=u\circ\Lambda:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R},\qquad U(X)=u(\mathcal{L}(X)).$$