Proof of Basic Properties of a Wasserstein-Coercive Penalty Pair

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 $c\in\mathbb{R}$ and put $K=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$. If $K$ is empty the assertion holds with $R=0$, there being no $\mu$ to test. Suppose then that $K$ is nonempty. By Wasserstein-Coercive Penalty Pairs §coercive, read at the level $c$, the set $K$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, hence compact by A Sequentially Compact Subset of a Metric Space is Compact.

The function $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ with $g(\mu)=\sqrt{M_{2}(\mu)}$, the square root being that of Existence and Uniqueness of the Nonnegative Square Root applied to the nonnegative $M_{2}(\mu)$, is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$: given $\mu$ and a positive $\varepsilon$, every $\sigma$ with $W_{2}(\mu,\sigma)<\varepsilon$ satisfies $d_{\mathbb{R}}(g(\sigma),g(\mu))=|g(\sigma)-g(\mu)|\le W_{2}(\mu,\sigma)<\varepsilon$ by claim 2 of The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance, read with $m=d$, the symmetry of $W_{2}$ (a metric axiom of Metric Space) and mixed transitivity (claim 2 of Elementary Order Arithmetic in an Ordered Field). Its restriction to $K$ is continuous on $K$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map.

By Extreme Value Theorem on a Compact Subset of a Metric Space the restriction of $g$ to the nonempty compact $K$ attains a greatest value, at some $\mu^{*}\in K$; put $R=M_{2}(\mu^{*})$. For $\mu\in K$ one has $g(\mu)\le g(\mu^{*})$, and both are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $g(\mu)^{2}\le g(\mu^{*})^{2}$, that is $M_{2}(\mu)\le R$, the squares of the square roots being the numbers themselves by Existence and Uniqueness of the Nonnegative Square Root. This is claim 1.

Claim 2. 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 claim 1, 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 2.

Claim 3. 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 $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ for this $\varepsilon$. Suppose, to the contrary, that for every positive $r\in\mathbb{R}$ there is $\sigma\in\mathcal{D}$ with $W_{2}(\mu,\sigma)<r$ 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 $W_{2}(\mu,\sigma_{n})<\iota(n)^{-1}$.

The sequence $(\sigma_{n})_{n\in\mathbb{N}}$ converges to $\mu$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. 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 $W_{2}(\mu,\sigma_{n})<\varepsilon'$ by mixed transitivity.

By Wasserstein-Coercive Penalty Pairs §coercive, read at the level $c$, the set $S$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, 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 $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. 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 $W_{2}(\mu,\sigma)<r$ satisfies $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\sigma)$. As $\varepsilon$ and $\mu$ were arbitrary, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, which is claim 3.

Claim 4. Every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfies $u(\sigma)\le b$, so $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, the constant $b$ serving on every ball; hence $u^{-}_{\delta}$ is defined on $\mathcal{D}$ by The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair §minus. Likewise $b'\le v(\sigma)$ for every $\sigma$, so $v$ is bounded below near each point and $v^{+}_{\delta}$ is defined on $\mathcal{D}$ by The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair §plus.

The restriction of $u$ to $\mathcal{D}$ is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map. By claim 3 the function $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$, so $\delta\,\mathcal{E}$ is lower semicontinuous there by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, applied with the nonnegative multiplier $\delta$, and $-\delta\,\mathcal{E}$ is upper semicontinuous by Semicontinuity Under Negation and Characterization of Continuity. Hence $u-\delta\mathcal{E}$, the sum of two upper semicontinuous functions on $\mathcal{D}$, is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. An upper semicontinuous function is its own upper semicontinuous envelope by Properties of the Upper Semicontinuous Envelope §fixed, so

Symmetrically, the restriction of $v$ to $\mathcal{D}$ is lower semicontinuous relative to $\mathcal{D}$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map, $\delta\mathcal{E}$ is lower semicontinuous as just shown, their sum $v+\delta\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, and a lower semicontinuous function is its own lower semicontinuous envelope by Properties of the Lower Semicontinuous Envelope, by Duality §fixed, so $v^{+}_{\delta}(\mu)=v(\mu)+\delta\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. No continuity of $u$ or of $v$ has been used. This is claim 4.