Proof of Basic Properties of the Delta-Envelopes on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. Throughout, $\mathcal{D}$ carries the metric $W_{2}$ and the envelopes are those of The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair.

Claim 1. Suppose $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. By The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair (i) the function $u-\delta\mathcal{E}$ is bounded above near each point of $\mathcal{D}$ and $u^{-}_{\delta}$ is its upper semicontinuous envelope on $\mathcal{D}$. By Properties of the Upper Semicontinuous Envelope (i), $u(\nu)-\delta\,\mathcal{E}(\nu)\le u^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$, and by Properties of the Upper Semicontinuous Envelope (ii), $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$. If instead $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, then by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair (ii) the function $u+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$ and $u^{+}_{\delta}$ is its lower semicontinuous envelope; Properties of the Lower Semicontinuous Envelope, by Duality (ii) gives $u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$ and Properties of the Lower Semicontinuous Envelope, by Duality (iii) gives the lower semicontinuity.

Claim 2. Suppose first that $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The set $A_{u}(\mu)$ of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function for $u$ is nonempty by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i), so there are $c\in\mathbb{R}$ and a positive $r\in\mathbb{R}$ with $u(\nu)\le c$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfying $W_{2}(\nu,\mu)\le r$; for every such $\nu$, claim 4 of Elementary Order Arithmetic in an Ordered Field turns $u(\nu)\le c$ into $-c\le-u(\nu)$. Hence $-c$ belongs, with the same $r$, to the set $B_{-u}(\mu)$ of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function for the function $-u$, so that set is nonempty; as $\mu$ was arbitrary, $-u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i). Hence, by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair (ii) applied to $-u$, the function $(-u)+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$ and $(-u)^{+}_{\delta}$ is its lower semicontinuous envelope on $\mathcal{D}$.

Write $g=u-\delta\mathcal{E}$, a function on $\mathcal{D}$. For $\nu\in\mathcal{D}$,

$$\bigl((-u)+\delta\mathcal{E}\bigr)(\nu)=-u(\nu)+\delta\,\mathcal{E}(\nu)=-\bigl(u(\nu)-\delta\,\mathcal{E}(\nu)\bigr)=(-g)(\nu),$$

by the distributivity axiom of Ordered Field and claim 2 of Zero Products and Elementary Identities in a Field, so $(-u)+\delta\mathcal{E}=-g$. Applying Properties of the Lower Semicontinuous Envelope, by Duality (i) to the function $-g$ gives $\bigl(-(-g)\bigr)^{*}=-\bigl((-g)_{*}\bigr)$, that is $g^{*}=-\bigl((-g)_{*}\bigr)$, since $-(-g)=g$ by claim 2 of Zero Products and Elementary Identities in a Field; taking additive inverses, $(-g)_{*}=-\bigl(g^{*}\bigr)$. Therefore

$$(-u)^{+}_{\delta}=\bigl((-u)+\delta\mathcal{E}\bigr)_{*}=(-g)_{*}=-\bigl(g^{*}\bigr)=-\,u^{-}_{\delta}\qquad\text{on }\mathcal{D}.$$

Suppose now that $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. The same argument, run with the nonempty set $B_{u}(\mu)$ in place of $A_{u}(\mu)$ and with claim 4 of Elementary Order Arithmetic in an Ordered Field turning $c\le u(\nu)$ into $-u(\nu)\le-c$, shows that $-u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$; and with $g'=u+\delta\mathcal{E}$, which is bounded below near each point of $\mathcal{D}$ by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair (ii) applied to $u$, one has $(-u)-\delta\mathcal{E}=-g'$ on $\mathcal{D}$ and, by Properties of the Lower Semicontinuous Envelope, by Duality (i) applied to $g'$, $(-g')^{*}=-\bigl(g'_{*}\bigr)$, whence $(-u)^{-}_{\delta}=-\,u^{+}_{\delta}$ on $\mathcal{D}$.

Claim 3. Suppose $u$ is continuous and $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

Local bounds. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Continuous Map Between Metric Spaces, applied with the positive number $1$, there is a positive $\rho\in\mathbb{R}$ such that $|u(\nu)-u(\mu)|<1$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)<\rho$; by claim 9 of Properties of the Absolute Value in an Ordered Field such a $\nu$ satisfies $u(\mu)-1<u(\nu)<u(\mu)+1$, hence also $u(\mu)-1\le u(\nu)\le u(\mu)+1$, a strict inequality implying the corresponding weak one in the ordered field $\mathbb{R}$. Let $r=\rho\,2^{-1}$, positive and smaller than $\rho$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)\le r$ satisfies $W_{2}(\nu,\mu)<\rho$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, hence the two bounds above. So $u(\mu)+1\in A_{u}(\mu)$ and $u(\mu)-1\in B_{u}(\mu)$ in the notation of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, and $u$ is bounded above near each point and bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i). Both $\delta$-envelopes are therefore defined.

The envelopes. The restriction of $u$ to $\mathcal{D}$ is continuous on $\mathcal{D}$ relative to $\mathcal{D}$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map, hence both upper and lower semicontinuous on $\mathcal{D}$ by claim 2 of Semicontinuity Under Negation and Characterization of Continuity. Since $0\le\delta$, the function $\delta\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, and $-(\delta\mathcal{E})$ is upper semicontinuous on $\mathcal{D}$ by claim 1 of Semicontinuity Under Negation and Characterization of Continuity. By claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions the sum of the restriction of $u$ and $-(\delta\mathcal{E})$, which is $u-\delta\mathcal{E}$ by claim 2 of Zero Products and Elementary Identities in a Field, is upper semicontinuous on $\mathcal{D}$; so $u^{-}_{\delta}=(u-\delta\mathcal{E})^{*}=u-\delta\mathcal{E}$ on $\mathcal{D}$ by Properties of the Upper Semicontinuous Envelope (iv). Likewise the sum of the restriction of $u$ and $\delta\mathcal{E}$, namely $u+\delta\mathcal{E}$, is lower semicontinuous on $\mathcal{D}$ by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, so $u^{+}_{\delta}=(u+\delta\mathcal{E})_{*}=u+\delta\mathcal{E}$ on $\mathcal{D}$ by Properties of the Lower Semicontinuous Envelope, by Duality (v).