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

Each result cited is universally quantified over the data in its own statement and is applied here to the data named in the statement of the lemma.

Claim 1. Let $C\in\mathbb{R}$ be nonnegative as in Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound, so that $-C(1+M_{2}(\mu))\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$, where $M_{2}$ is the second moment. Fix $\mu_{1}\in\mathcal{D}$, which is possible because $\mathcal{D}$ is nonempty, and put $c=\mathcal{E}(\mu_{1})$. By Coercive Penalty Pairs on the Wasserstein Space §moment, read at this level $c$, there is $R\in\mathbb{R}$ such that $M_{2}(\mu)\le R$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$. Let $e_{0}$ be the least of the two real numbers $-C(1+R)$ and $c$, which exists by claim 9 of Elementary Order Arithmetic in an Ordered Field.

Let $\mu\in\mathcal{D}$. The order of $\mathbb{R}$ is total (an axiom of Ordered Field), so $\mathcal{E}(\mu)\le c$ or $c\le\mathcal{E}(\mu)$. In the second case $e_{0}\le c\le\mathcal{E}(\mu)$ by the transitivity of $\le$ (Total Order on a Set). In the first case $M_{2}(\mu)\le R$, hence $1+M_{2}(\mu)\le1+R$ by the compatibility of the order with addition (an axiom of Ordered Field), hence $C(1+M_{2}(\mu))\le C(1+R)$ by claim 5 of Elementary Arithmetic in an Ordered Field applied with the nonnegative multiplier $C$. Both this inequality and $-C(1+R)\le-C(1+M_{2}(\mu))$ are equivalent, by claim 3 of Elementary Arithmetic in an Ordered Field, to $0\le C(1+R)-C(1+M_{2}(\mu))$, so the latter holds; with Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound and the transitivity of $\le$ (Total Order on a Set), $e_{0}\le-C(1+R)\le-C(1+M_{2}(\mu))\le\mathcal{E}(\mu)$. In both cases $e_{0}\le\mathcal{E}(\mu)$, which is claim 1.

Claim 2. Let $\mu\in\mathcal{D}$ and let $\varepsilon\in\mathbb{R}$ be positive; we verify the defining condition of Lower Semicontinuous Function on a Subset of a Metric Space at $\mu$ relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$ for this $\varepsilon$. Suppose, to the contrary, that for every positive $r\in\mathbb{R}$ there is $\sigma\in\mathcal{D}$ with $\rho(\mu,\sigma)<r$ for which $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\sigma)$ fails, that is, with $\mathcal{E}(\sigma)\le\mathcal{E}(\mu)-\varepsilon$, the two being complementary because the order of $\mathbb{R}$ is total. Put $c=\mathcal{E}(\mu)-\varepsilon$ and

For $n\in\mathbb{N}$ the inverse $\iota(n)^{-1}$ exists and is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so taking $r=\iota(n)^{-1}$ in the contrary assumption shows that the set

is nonempty for every $n\in\mathbb{N}$. By Axiom of Countable Choice there is a sequence $(\sigma_{n})_{n\in\mathbb{N}}$ with $\sigma_{n}\in A_{n}$ for every $n$; each $\sigma_{n}$ lies in $S$ and satisfies $\rho(\mu,\sigma_{n})<\iota(n)^{-1}$.

The sequence $(\sigma_{n})_{n\in\mathbb{N}}$ converges to $\mu$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$. Indeed, let $\varepsilon'\in\mathbb{R}$ be positive. By claim 2 of The Archimedean Property of the Real Numbers there is $N\in\mathbb{N}$ with $1<\iota(N)\varepsilon'$. Let $n\in\mathbb{N}$ satisfy $N\le n$. By the trichotomy of the order of $\mathbb{N}$ (claim 3 of Properties of the Order on the Natural Numbers) either $N=n$, or $N<n$ and then $\iota(N)<\iota(n)$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; in both cases $\iota(N)\le\iota(n)$. Multiplying by the nonnegative $\varepsilon'$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\iota(N)\varepsilon'\le\iota(n)\varepsilon'$, hence $1<\iota(n)\varepsilon'$ by mixed transitivity (claim 2 of Elementary Order Arithmetic in an Ordered Field). Multiplying by the positive $\iota(n)^{-1}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field) and using $\iota(n)^{-1}\iota(n)=1$ gives $\iota(n)^{-1}<\varepsilon'$, so $\rho(\mu,\sigma_{n})<\varepsilon'$ by mixed transitivity.

By Coercive Penalty Pairs on the Wasserstein Space §compact, read at the level $c$, the set $S$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, so there are $\sigma\in S$ and a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ such that $(\sigma_{n_{k}})_{k\in\mathbb{N}}$ converges to $\sigma$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$. By A Subsequence of a Convergent Sequence Has the Same Limit the same subsequence converges to $\mu$, so $\mu=\sigma$ by Uniqueness of Limits in a Metric Space, and therefore $\mu\in S$, that is, $\mathcal{E}(\mu)\le\mathcal{E}(\mu)-\varepsilon$. By claim 3 of Elementary Arithmetic in an Ordered Field this is equivalent to $0\le-\varepsilon$ and hence, by the same claim, to $\varepsilon\le0$; with $0\le\varepsilon$ and the antisymmetry of the order (an axiom of Ordered Field) this gives $\varepsilon=0$, contradicting the positivity of $\varepsilon$.

Hence there is a positive $r\in\mathbb{R}$ such that every $\sigma\in\mathcal{D}$ with $\rho(\mu,\sigma)<r$ satisfies $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\sigma)$. As $\varepsilon$ and $\mu$ were arbitrary, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$, which is claim 2.

Claim 3. Apply Continuity, Semicontinuity and Lipschitz Bounds Pass from the Centred Heat Gauge to the Wasserstein Distance §lower with $A=\mathcal{D}$ and $f=\mathcal{E}$, whose hypothesis is claim 2.

Claim 4. The function $u$ is continuous, hence bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and its two $\delta$-envelopes relative to the penalty pair are defined and given by the displayed identities: this is Basic Properties of the Delta-Envelopes on the Wasserstein Space §exact, whose remaining hypothesis, that $\mathcal{E}$ be lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, is claim 3.