Proof of The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty

Each result cited is universally quantified over the data in its own statement. Throughout, $\mathcal{D}$ is a nonempty subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$: it contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, and $\mathcal{D}_{\Sigma}$ is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty.

Proof of clause 1. The set $S=\{\mathcal{E}(\sigma):\sigma\in\mathcal{D}\}$ is a nonempty set of real numbers bounded below by $e_{0}$, so it has a greatest lower bound $s$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, unique by Uniqueness of the Supremum and of the Infimum. Put $c=s+1$ and $K=\{\sigma\in\mathcal{D}:\mathcal{E}(\sigma)\le c\}$, which is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ by Wasserstein-Coercive Penalty Pairs §coercive.

For each $n\in\mathbb{N}$ the number $\tfrac{1}{n}$ is positive (claim 7 of Elementary Order Arithmetic in an Ordered Field), so by Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ there is $\sigma_{n}\in\mathcal{D}$ with $\mathcal{E}(\sigma_{n})<s+\tfrac{1}{n}$. Since $1\le n$ for every $n\in\mathbb{N}$ by claim 4 of Properties of the Order on the Natural Numbers, we have $\tfrac{1}{n}\le1$ by claims 4 and 5 of Elementary Arithmetic in an Ordered Field, so $\mathcal{E}(\sigma_{n})<c$ and $\sigma_{n}\in K$.

By Sequentially Compact Subset of a Metric Space there are a strictly increasing $\varphi:\mathbb{N}\to\mathbb{N}$ and $\mu_{\min}\in K$ such that the sequence whose $k$-th term is $\sigma_{\varphi(k)}$ converges to $\mu_{\min}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. In particular $\mu_{\min}\in\mathcal{D}$, so $s\le\mathcal{E}(\mu_{\min})$.

Let $\epsilon\in\mathbb{R}$ be positive. Since $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$, Lower Semicontinuous Function on a Subset of a Metric Space provides a positive $\rho\in\mathbb{R}$ such that every $\sigma\in\mathcal{D}$ with $W_{2}(\sigma,\mu_{\min})<\rho$ satisfies $\mathcal{E}(\mu_{\min})-\epsilon<\mathcal{E}(\sigma)$. By The Archimedean Property of the Real Numbers choose $m\in\mathbb{N}$ with $\tfrac{1}{\epsilon}\le m$, so that $\tfrac{1}{m}\le\epsilon$ (claims 4 and 5 of Elementary Arithmetic in an Ordered Field); by convergence and by $k\le\varphi(k)$ (Strictly Increasing Sequences of Natural Numbers Dominate Their Index) there is $k\in\mathbb{N}$ with $m\le\varphi(k)$ and $W_{2}(\sigma_{\varphi(k)},\mu_{\min})<\rho$. Then, using $\tfrac{1}{\varphi(k)}\le\tfrac{1}{m}\le\epsilon$,

so $\mathcal{E}(\mu_{\min})<s+2\epsilon$ by claim 1 of Elementary Order Arithmetic in an Ordered Field. As $\epsilon$ was an arbitrary positive real number, Comparison of Real Numbers with Arbitrary Positive Slack gives $\mathcal{E}(\mu_{\min})\le s$, whence $\mathcal{E}(\mu_{\min})=s$. Since $s$ is a lower bound of $S$, $\mathcal{E}(\mu_{\min})\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$.

Proof of clause 2. Since $(\Omega,\mathcal{F},P)$ is rich and $\mathcal{D}$ is nonempty, The Wasserstein Distance and the Mean-Square Distance of Random Vectors §onto provides, for a chosen $\sigma\in\mathcal{D}$, an $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\sigma$; then $X\in\mathcal{D}^{\Lambda}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §preimages, so $\mathcal{D}^{\Lambda}$ is nonempty.

Let $X\in\mathcal{D}^{\Lambda}$ and put $\sigma=\mathcal{L}(X)\in\mathcal{D}$. Then $\hat{u}(X)=u(\sigma)-\delta\mathcal{E}(\sigma)$ and $\hat{v}(X)=v(\sigma)+\delta\mathcal{E}(\sigma)$. From $e_{0}\le\mathcal{E}(\sigma)$ and $0<\delta$ we get $\delta e_{0}\le\delta\mathcal{E}(\sigma)$ by claim 5 of Elementary Arithmetic in an Ordered Field. Two weak inequalities may be added: if $x\le y$ and $x'\le y'$ then $0\le y-x$ and $0\le y'-x'$ by claim 3 of Elementary Arithmetic in an Ordered Field, so $0\le(y-x)+(y'-x')$ by claim 2 of that lemma, and this number is $(y+y')-(x+x')$, so that $x+x'\le y+y'$ by claim 3 again. Adding $u(\sigma)\le b$ and $\delta e_{0}\le\delta\mathcal{E}(\sigma)$ in this way gives $u(\sigma)+\delta e_{0}\le b+\delta\mathcal{E}(\sigma)$, and claim 3 of Elementary Arithmetic in an Ordered Field turns this into $\hat{u}(X)=u(\sigma)-\delta\mathcal{E}(\sigma)\le b-\delta e_{0}$, the two differences $\bigl(b+\delta\mathcal{E}(\sigma)\bigr)-\bigl(u(\sigma)+\delta e_{0}\bigr)$ and $\bigl(b-\delta e_{0}\bigr)-\bigl(u(\sigma)-\delta\mathcal{E}(\sigma)\bigr)$ being equal. Adding $b'\le v(\sigma)$ and $\delta e_{0}\le\delta\mathcal{E}(\sigma)$ in the same way gives $b'+\delta e_{0}\le v(\sigma)+\delta\mathcal{E}(\sigma)=\hat{v}(X)$, whence $-\hat{v}(X)\le-b'-\delta e_{0}$ by claim 4 of Elementary Order Arithmetic in an Ordered Field.

Proof of clause 3. We verify the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §sequential for $\hat{u}$, the metric space being $(L^{2}(\Omega;\mathbb{R}^{d}),d_{L^{2}})$ and the subset $\mathcal{D}^{\Lambda}$. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}^{\Lambda}$ converging to $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and let $t\in\mathbb{R}$ satisfy $t\le\hat{u}(X_{n})$ for every $n$. Write $\sigma_{n}=\mathcal{L}(X_{n})\in\mathcal{D}$ and $\sigma=\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

The laws lie in a sequentially compact set. For every $n$, $t\le u(\sigma_{n})-\delta\mathcal{E}(\sigma_{n})\le b-\delta\mathcal{E}(\sigma_{n})$, so $\delta\mathcal{E}(\sigma_{n})\le b-t$ and hence $\mathcal{E}(\sigma_{n})\le c$, where $c=\delta^{-1}(b-t)$, by claims 4, 5 and 7 of Elementary Arithmetic in an Ordered Field and claim 7 of Elementary Order Arithmetic in an Ordered Field. Thus every $\sigma_{n}$ lies in $K=\{\tau\in\mathcal{D}:\mathcal{E}(\tau)\le c\}$, which is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ by Wasserstein-Coercive Penalty Pairs §coercive.

The limit lies in $\mathcal{D}^{\Lambda}$. By The Wasserstein Distance and the Mean-Square Distance of Random Vectors §inequality, $W_{2}(\sigma_{n},\sigma)\le\lVert X_{n}-X\rVert_{L^{2}}$ for every $n$; since the right-hand side converges to $0$ and $W_{2}(\sigma_{n},\sigma)$ is nonnegative, the sequence whose $n$-th term is $\sigma_{n}$ converges to $\sigma$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, by Convergent Sequence in a Metric Space and Order Properties of Limits of Real Sequences. By Sequentially Compact Subset of a Metric Space some subsequence of $(\sigma_{n})_{n\in\mathbb{N}}$ converges to a point $\tau\in K$; that subsequence also converges to $\sigma$ by A Subsequence of a Convergent Sequence Has the Same Limit, and limits in a metric space are unique by Uniqueness of Limits in a Metric Space, so $\sigma=\tau\in K\subseteq\mathcal{D}$. Hence $X\in\mathcal{D}^{\Lambda}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §preimages, and $\hat{u}(X)$ is defined.

The value passes to the limit. Let $\epsilon\in\mathbb{R}$ be positive. Since $u$ is upper semicontinuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, Upper Semicontinuous Function on a Subset of a Metric Space provides a positive $\rho_{1}$ with $u(\tau)<u(\sigma)+\epsilon$ for every $\tau\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\tau,\sigma)<\rho_{1}$; since $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by Basic Properties of a Wasserstein-Coercive Penalty Pair §lsc, Lower Semicontinuous Function on a Subset of a Metric Space provides a positive $\rho_{2}$ with $\mathcal{E}(\sigma)-\epsilon<\mathcal{E}(\tau)$ for every $\tau\in\mathcal{D}$ with $W_{2}(\tau,\sigma)<\rho_{2}$. Let $\rho$ be the least of $\rho_{1}$ and $\rho_{2}$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), which is positive, and choose $n$ with $W_{2}(\sigma_{n},\sigma)<\rho$. Then

using claim 5 of Elementary Arithmetic in an Ordered Field to multiply $\mathcal{E}(\sigma)-\epsilon<\mathcal{E}(\sigma_{n})$ by the positive $\delta$, claim 4 of Elementary Order Arithmetic in an Ordered Field to reverse the sign, and claim 3 of Elementary Order Arithmetic in an Ordered Field to add. As $\epsilon$ was an arbitrary positive real number and $1+\delta$ is positive, Comparison of Real Numbers with Arbitrary Positive Slack gives $t\le\hat{u}(X)$. By Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §sequential, $\hat{u}$ has closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})$.

For $-\hat{v}$ the argument is the same, with two changes: from $t\le-\hat{v}(X_{n})=-v(\sigma_{n})-\delta\mathcal{E}(\sigma_{n})\le-b'-\delta\mathcal{E}(\sigma_{n})$ one gets $\mathcal{E}(\sigma_{n})\le\delta^{-1}(-b'-t)$, which again places the laws in a sequentially compact sublevel set; and in the last step the lower semicontinuity of $v$ gives a positive $\rho_{1}$ with $v(\sigma)-\epsilon<v(\tau)$, hence $-v(\tau)<-v(\sigma)+\epsilon$, for $\tau$ within $\rho_{1}$ of $\sigma$.

Proof of clause 4. By clause 2 the set $\mathcal{D}^{\Lambda}$ is nonempty, and $\hat{u}$ and $-\hat{v}$ are bounded above there; by clause 3 they have closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})$. Let $\Phi_{0}:\mathcal{D}^{\Lambda}\times\mathcal{D}^{\Lambda}\to\mathbb{R}$ have value $\hat{u}(X)+(-\hat{v})(Y)-\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}$ at $(X,Y)$. By Closed Superlevel Sets of a Sum on a Product Space and of a Doubled Function §doubled, applied with $L^{2}(\Omega;\mathbb{R}^{d})$ in the role of the space written $H$ there, with $S_{1}=S_{2}=\mathcal{D}^{\Lambda}$, with $u_{1}=\hat{u}$ and $u_{2}=-\hat{v}$, the function $\Phi_{0}$ has closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$.

The function $g$ on $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$ with value $\mu\,|\zeta-q|^{2}$ at $\zeta$ is continuous: the map $\zeta\mapsto|\zeta-q|$ is continuous by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §lipschitz, and claims 1, 3 and 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space give the continuity of its square and of the constant multiple. Since $\Psi(X,Y)=\Phi_{0}(X,Y)-g(X,Y)$, Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §perturbation shows that $\Psi$ has closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$.

Finally, for $(X,Y)\in\mathcal{D}^{\Lambda}\times\mathcal{D}^{\Lambda}$ the numbers $\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}$ and $\mu\,|(X,Y)-q|^{2}$ are nonnegative, $\alpha$ and $\mu$ being nonnegative and squares of norms being nonnegative (claim 5 of Elementary Arithmetic in an Ordered Field), so by clause 2 and claims 2 and 3 of Elementary Arithmetic in an Ordered Field,