Noise-Closed Noise Penalty Pairs

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, so that $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Sequences and their convergence are taken in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space.

(Noise-closed penalty pair)¶ The noise penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is noise-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 $\mu\in\mathcal{P}^{a}_{\rho}$, one has $\mu\in\mathcal{D}$ and $\mathcal{E}(\mu)\le c$.

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