Law-Invariant Function on 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 with the law $\mathcal{L}(X)$ of its elements as fixed there.

(Law-invariant function)¶ A function $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ is law-invariant if $\Phi(X)=\Phi(Y)$ for all $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mathcal{L}(Y)$.