Basic Properties of a Coercive Penalty Pair: a Lower Bound for the Penalty, Semicontinuity, and Exact Delta-Envelopes

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\rho$ be the centred heat gauge. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, which is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, so that $\mathcal{D}$ is nonempty; it is regarded as a subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures and as a subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$ of The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality §metric, lower semicontinuity on $\mathcal{D}$ being read in the metric space named in each claim. That a function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is continuous, and the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of a function $u$ relative to the penalty pair, are as fixed there. Then the following hold.

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

2. (Lower semicontinuity for the gauge)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$.

3. (Lower semicontinuity for the Wasserstein distance)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$.

4. (Exact $\delta$-envelopes)¶ Let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be continuous and let $\delta\in\mathbb{R}$ be positive. Then both $\delta$-envelopes of $u$ relative to the penalty pair are defined and