Wasserstein-Closed Penalty Pairs

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, so that $\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, and let $M_{2}$ be the second moment of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions. Sequences and their convergence are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions.

(Wasserstein-closed penalty pair)¶ The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is Wasserstein-closed if the following two conditions hold for every $c\in\mathbb{R}$.

1. (Closed sublevel sets)¶ For every sequence $(\mu_{n})_{n\in\mathbb{N}}$ in $\mathcal{D}$ with $\mathcal{E}(\mu_{n})\le c$ for every $n\in\mathbb{N}$ that converges to some $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, one has $\mu\in\mathcal{D}$ and $\mathcal{E}(\mu)\le c$.

2. (Bounded sublevel sets)¶ There is $B\in\mathbb{R}$ with $M_{2}(\mu)\le B$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$.