Basic Properties of a Wasserstein-Coercive Penalty Pair

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: 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)¶ 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)¶ 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)¶ The function $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

4. (The $\delta$-envelopes are exact)¶ Let $\delta\in\mathbb{R}$ be positive and let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be such that $u$ is upper semicontinuous and $v$ lower semicontinuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, and such that there are $b,b'\in\mathbb{R}$ with $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then the $\delta$-envelope $u^{-}_{\delta}$ of $u$ and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the penalty pair are defined, and

No continuity of $u$ or of $v$ is used in claim 4: a function already semicontinuous of the right kind is its own envelope.