Basic Properties of the Lift: Law Invariance, the Correspondence on a Rich Space, and Transfer of Boundedness, Lipschitz Constants and Continuity

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 its distance $d_{L^{2}}$, and $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ the Wasserstein space. Let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ with lift $U:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$. Then the following hold.

1. (Law invariance)(i) $U$ is law-invariant.

2. (The correspondence on a rich space)(ii) If $(\Omega,\mathcal{F},P)$ is rich, then for every law-invariant $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ there is exactly one function $v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ whose lift is $\Phi$.

3. (Bounds)(iii) Let $M\in\mathbb{R}$ be nonnegative. If $u$ is bounded with bound $M$, then $U$ is bounded with bound $M$.

4. (Bounds, converse)(iv) Let $M\in\mathbb{R}$ be nonnegative and let $(\Omega,\mathcal{F},P)$ be rich. If $U$ is bounded with bound $M$, then $u$ is bounded with bound $M$.

5. (Lipschitz constants)(v) Let $\kappa\in\mathbb{R}$ be nonnegative. If $u$ is Lipschitz with constant $\kappa$, then $U$ is Lipschitz with constant $\kappa$.

6. (Lipschitz constants, converse)(vi) Let $\kappa\in\mathbb{R}$ be nonnegative and let $(\Omega,\mathcal{F},P)$ be rich. If $U$ is Lipschitz with constant $\kappa$, then $u$ is Lipschitz with constant $\kappa$.

7. (Uniform continuity)(vii) If $u$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $U$ is uniformly continuous on $L^{2}(\Omega;\mathbb{R}^{d})$.

8. (Uniform continuity, converse)(viii) Let $(\Omega,\mathcal{F},P)$ be rich. If $U$ is uniformly continuous on $L^{2}(\Omega;\mathbb{R}^{d})$, then $u$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

9. (Transfer of continuity)(ix) If $u$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$, then $U$ is continuous on $L^{2}(\Omega;\mathbb{R}^{d})$.