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 $u$ has penalty-subordinate growth from above. By The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus 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 §bounds, $u(\nu)-\delta\,\mathcal{E}(\nu)\le u^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$, and by Properties of the Upper Semicontinuous Envelope §usc, $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$. If instead $u$ has penalty-subordinate growth from below, then by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus 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 §bounds gives $u^{+}_{\delta}(\nu)\le u(\nu)+\delta\,\mathcal{E}(\nu)$ and Properties of the Lower Semicontinuous Envelope, by Duality §lsc gives the lower semicontinuity.

Claim 2. Growth. Let $\delta'\in\mathbb{R}$ be positive and $C\in\mathbb{R}$. For $\nu\in\mathcal{D}$, claim 4 of Elementary Order Arithmetic in an Ordered Field (used in both directions, together with $-(-s)=s$ from claim 2 of Zero Products and Elementary Identities in a Field) shows that $u(\nu)\le C+\delta'\,\mathcal{E}(\nu)$ holds if and only if $-\bigl(C+\delta'\,\mathcal{E}(\nu)\bigr)\le-u(\nu)$, and $-\bigl(C+\delta'\,\mathcal{E}(\nu)\bigr)=-C-\delta'\,\mathcal{E}(\nu)$ by the distributivity axiom of Ordered Field and claim 2 of Zero Products and Elementary Identities in a Field. Comparing Penalty-Subordinate Growth of a Function on the Penalty Domain §above for $u$ with Penalty-Subordinate Growth of a Function on the Penalty Domain §below for $-u$, $u$ has penalty-subordinate growth from above if and only if $-u$ has penalty-subordinate growth from below. The same computation with $u$ replaced by $-u$, and $-(-u)=u$, shows that $u$ has penalty-subordinate growth from below if and only if $-u$ has penalty-subordinate growth from above.

Envelopes. Suppose first that $u$ has penalty-subordinate growth from above. Then $-u$ has penalty-subordinate growth from below, and by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus 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 $\nu\in\mathcal{D}$,

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 §duality 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

Suppose now that $u$ has penalty-subordinate growth from below, so that $-u$ has penalty-subordinate growth from above, as shown above. 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 §plus applied to $u$, one has $(-u)-\delta\mathcal{E}=-g'$ on $\mathcal{D}$ and, by Properties of the Lower Semicontinuous Envelope, by Duality §duality applied to $g'$, $(-g')^{*}=-\bigl(g'_{*}\bigr)$, whence $(-u)^{-}_{\delta}=-\,u^{+}_{\delta}$ on $\mathcal{D}$.

Claim 3. Suppose $u$ is continuous on $\mathcal{D}$ and $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$. Then $u$ is 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 $u$ 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, when $u$ has penalty-subordinate growth from above, $u^{-}_{\delta}=(u-\delta\mathcal{E})^{*}=u-\delta\mathcal{E}$ on $\mathcal{D}$ by Properties of the Upper Semicontinuous Envelope §fixed. Likewise the sum 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 applied to the lower semicontinuous $u$ and $\delta\mathcal{E}$, so, when $u$ has penalty-subordinate growth from below, $u^{+}_{\delta}=(u+\delta\mathcal{E})_{*}=u+\delta\mathcal{E}$ on $\mathcal{D}$ by Properties of the Lower Semicontinuous Envelope, by Duality §fixed.

Claim 4. Suppose $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ and let $C,\eta\in\mathbb{R}$ satisfy $0\le\eta\le\delta$. Then $0\le\delta-\eta$ by claim 3 of Elementary Arithmetic in an Ordered Field, so $(\delta-\eta)\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, and its negative is upper semicontinuous on $\mathcal{D}$ by claim 1 of Semicontinuity Under Negation and Characterization of Continuity. The constant function with value $C$ on $\mathcal{D}$ is continuous by claim 1 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, hence upper semicontinuous by claim 2 of Semicontinuity Under Negation and Characterization of Continuity; so the function $g:\mathcal{D}\to\mathbb{R}$, $g(\nu)=C-(\delta-\eta)\,\mathcal{E}(\nu)$, is upper semicontinuous on $\mathcal{D}$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions and claim 2 of Zero Products and Elementary Identities in a Field.

Suppose $u$ has penalty-subordinate growth from above and $u(\nu)\le C+\eta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$. Adding $-\delta\,\mathcal{E}(\nu)$ to both sides, by the compatibility of the order with addition in the ordered field $\mathbb{R}$, and using $\eta\,\mathcal{E}(\nu)-\delta\,\mathcal{E}(\nu)=-(\delta-\eta)\,\mathcal{E}(\nu)$ (distributivity and claim 2 of Zero Products and Elementary Identities in a Field), gives $u(\nu)-\delta\,\mathcal{E}(\nu)\le g(\nu)$ for every $\nu\in\mathcal{D}$. By Properties of the Upper Semicontinuous Envelope §least, applied to $u-\delta\mathcal{E}$ and the upper semicontinuous majorant $g$, $u^{-}_{\delta}(\nu)=(u-\delta\mathcal{E})^{*}(\nu)\le g(\nu)$ for every $\nu\in\mathcal{D}$, which is the first assertion.

Suppose $u$ has penalty-subordinate growth from below and $-C-\eta\,\mathcal{E}(\nu)\le u(\nu)$ for every $\nu\in\mathcal{D}$. By claim 2, $-u$ has penalty-subordinate growth from above, and $-u(\nu)\le C+\eta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$ by the computation in the first paragraph of the proof of claim 2. The first assertion, applied to $-u$, gives $(-u)^{-}_{\delta}(\nu)\le g(\nu)$, and $(-u)^{-}_{\delta}(\nu)=-u^{+}_{\delta}(\nu)$ by claim 2; so $-g(\nu)\le u^{+}_{\delta}(\nu)$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, and $-g(\nu)=-C+(\delta-\eta)\,\mathcal{E}(\nu)$ by claim 2 of Zero Products and Elementary Identities in a Field, for every $\nu\in\mathcal{D}$.