Basic Properties of a Wasserstein-Coercive Penalty Pair

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ be the Wasserstein space with its distance and $M_{2}$ the second moment. Upper and lower semicontinuity of a real-valued function on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$, relative to that subset, are understood in this metric space. Then the following hold.

1. (Bounded second moments on sublevel sets)(i) For every $c\in\mathbb{R}$ there is $R\in\mathbb{R}$ such that $M_{2}(\mu)\le R$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$.

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

3. (The penalty is lower semicontinuous)(iii) The function $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.