Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances

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-closed noise penalty pair on $\mathcal{P}^{a}_{\rho}$, and for $c\in\mathbb{R}$ write $\mathcal{D}_{c}=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$. Lower semicontinuity on $\mathcal{D}$ is taken relative to $\mathcal{D}$ 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; the restriction of $W_{a}$ to $\mathcal{D}_{c}\times\mathcal{D}_{c}$ is again written $W_{a}$, and completeness is that of that definition. The reference measure $\rho$ belongs to $\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference, so that $W_{a}(\mu,\rho)$ is a nonnegative real number for every $\mu\in\mathcal{P}^{a}_{\rho}$. Then the following hold.

1. (The penalty is bounded below)¶ There is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

2. (The penalty is lower semicontinuous)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

3. (Complete sublevel sets)¶ For every $c\in\mathbb{R}$, $(\mathcal{D}_{c},W_{a})$ is a metric space, and it is complete.

4. (Bounded distances on sublevel sets)¶ Let $c,B\in\mathbb{R}$ satisfy $W_{a}(\mu,\rho)\le B$ for every $\mu\in\mathcal{D}_{c}$, as provided by Noise-Closed Noise Penalty Pairs §bounded. Then $W_{a}(\mu,\nu)\le2B$ for all $\mu,\nu\in\mathcal{D}_{c}$.