Proof of Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. Throughout, $\mathcal{D}$ carries the metric $W_{a}$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space, and the envelopes are those of The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair. Elementary order and arithmetic of real numbers (the field axioms, $-(-s)=s$, $-(s+t)=-s-t$, and the compatibility of the order with addition, with multiplication by nonnegative numbers and with negation, which reverses it) is carried by The Real Numbers: Standing Notation and Background §background, in force through Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §background; it is used below without further mention. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, which is nonempty by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, so $\mathcal{D}$ is a nonempty subset of the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$, as Properties of the Upper Semicontinuous Envelope and Properties of the Lower Semicontinuous Envelope, by Duality require.

Claim 1. Suppose $u$ has penalty-subordinate growth from above. By The Delta-Envelopes of a Function on the Domain of a Noise 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 Domain of a Noise 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 $\nu\in\mathcal{D}$, since negation reverses the order and $-(-s)=s$, the inequality $u(\nu)\le C+\delta'\,\mathcal{E}(\nu)$ holds if and only if $-\bigl(C+\delta'\,\mathcal{E}(\nu)\bigr)\le-u(\nu)$, and $-\bigl(C+\delta'\,\mathcal{E}(\nu)\bigr)=-C-\delta'\,\mathcal{E}(\nu)$. Comparing Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §above for $u$ with Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §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 Domain of a Noise 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 $\nu\in\mathcal{D}$,

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

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 $-(-g)=g$; 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 $u$ 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 Domain of a Noise 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 $u-\delta\mathcal{E}$, 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 $0\le\delta-\eta$, 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 at every point of $\mathcal{D}$ relative to $\mathcal{D}$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants, hence upper semicontinuous on $\mathcal{D}$ 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.

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 and using $\eta\,\mathcal{E}(\nu)-\delta\,\mathcal{E}(\nu)=-(\delta-\eta)\,\mathcal{E}(\nu)$ 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, for every $\nu\in\mathcal{D}$, the sign reversal Elementary Order Arithmetic in an Ordered Field §sign-reversal applied to $-C-\eta\,\mathcal{E}(\nu)\le u(\nu)$ gives $-u(\nu)\le-\bigl(-C-\eta\,\mathcal{E}(\nu)\bigr)=C+\eta\,\mathcal{E}(\nu)$. The first assertion, applied to $-u$, gives $(-u)^{-}_{\delta}(\nu)\le g(\nu)$, and $(-u)^{-}_{\delta}(\nu)=-u^{+}_{\delta}(\nu)$ by Claim 2; so $-g(\nu)\le u^{+}_{\delta}(\nu)$, and $-g(\nu)=-C+(\delta-\eta)\,\mathcal{E}(\nu)$, for every $\nu\in\mathcal{D}$.

Preliminaries for claims 5 to 7. Suppose the pair is noise-closed. By Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §bounded-below fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$; by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §lsc, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}^{a}_{\rho},W_{a})$. For positive $\delta$ and $\mu\in\mathcal{D}$, multiplying $e_{0}\le\mathcal{E}(\mu)$ by the nonnegative $\delta$ gives $\delta e_{0}\le\delta\,\mathcal{E}(\mu)$.

Claim 5. Suppose $u(\mu)\le b$ for every $\mu\in\mathcal{D}$, and let $\delta$ be positive. Put $C=b-\delta e_{0}$. For $\mu\in\mathcal{D}$, adding $b-\delta e_{0}$ to both sides of $\delta e_{0}\le\delta\,\mathcal{E}(\mu)$ gives $b\le C+\delta\,\mathcal{E}(\mu)$, so $u(\mu)\le C+\delta\,\mathcal{E}(\mu)$. As $\delta$ was arbitrary, $u$ has penalty-subordinate growth from above (Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §above). Suppose instead $b\le u(\mu)$ for every $\mu\in\mathcal{D}$, and put $C=-b-\delta e_{0}$. For $\mu\in\mathcal{D}$, adding $b-\delta\,\mathcal{E}(\mu)$ to both sides of $\delta e_{0}\le\delta\,\mathcal{E}(\mu)$ gives $-C-\delta\,\mathcal{E}(\mu)=b+\delta e_{0}-\delta\,\mathcal{E}(\mu)\le b\le u(\mu)$, so $u$ has penalty-subordinate growth from below (Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §below).

Claim 6. Suppose $u(\mu)\le b$ for every $\mu\in\mathcal{D}$. By Claim 5, $u$ has penalty-subordinate growth from above; since $0\,\mathcal{E}(\mu)=0$, $u(\mu)\le b+0\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. Apply Claim 4 with $C=b$ and $\eta=0$, for which $0\le\eta\le\delta$ holds, $\mathcal{E}$ being lower semicontinuous on $\mathcal{D}$: it gives $u^{-}_{\delta}(\mu)\le b-(\delta-0)\,\mathcal{E}(\mu)=b-\delta\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. If instead $b\le u(\mu)$ for every $\mu\in\mathcal{D}$, then $u$ has penalty-subordinate growth from below by Claim 5 and $-(-b)-0\,\mathcal{E}(\mu)=b\le u(\mu)$; the second part of Claim 4, with $C=-b$ and $\eta=0$, gives $b+\delta\,\mathcal{E}(\mu)\le u^{+}_{\delta}(\mu)$ for every $\mu\in\mathcal{D}$.

Claim 7, from above. Suppose $u$ has penalty-subordinate growth from above, and put $\kappa=\delta'-\delta$, which is positive. The function $-\mathcal{E}$ on $\mathcal{D}$, with value $-\mathcal{E}(\nu)$ at $\nu$, is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by claim 1 of Semicontinuity Under Negation and Characterization of Continuity, $\mathcal{E}$ being lower semicontinuous there. By Claim 1, $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ and

$$u(\nu)-\delta\,\mathcal{E}(\nu)\le u^{-}_{\delta}(\nu)\qquad(\nu\in\mathcal{D}).$$

Let $h:\mathcal{D}\to\mathbb{R}$ have value $h(\nu)=u^{-}_{\delta}(\nu)+\kappa\bigl(-\mathcal{E}(\nu)\bigr)=u^{-}_{\delta}(\nu)-\kappa\,\mathcal{E}(\nu)$. By claims 2 and 1 of Sums and Nonnegative Multiples of Semicontinuous Functions, applied at every point of $\mathcal{D}$ with the nonnegative multiplier $\kappa$, $h$ is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$. For $\nu\in\mathcal{D}$, adding $-\kappa\,\mathcal{E}(\nu)$ to both sides of the last display gives

$$u(\nu)-\delta'\,\mathcal{E}(\nu)=\bigl(u(\nu)-\delta\,\mathcal{E}(\nu)\bigr)-\kappa\,\mathcal{E}(\nu)\le h(\nu).$$

The function $u-\delta'\mathcal{E}$ is bounded above near each point of $\mathcal{D}$ and $u^{-}_{\delta'}$ is its upper semicontinuous envelope, by The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §minus read with $\delta'$. So Properties of the Upper Semicontinuous Envelope §least, in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ with the nonempty subset $\mathcal{D}$, the function $u-\delta'\mathcal{E}$ and the upper semicontinuous majorant $h$, gives $u^{-}_{\delta'}(\mu)\le h(\mu)=u^{-}_{\delta}(\mu)-\kappa\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$; adding $\kappa\,\mathcal{E}(\mu)$ to both sides is the claim.

Claim 7, from below. Suppose $u$ has penalty-subordinate growth from below. By Claim 2, applied with $\delta$ and with $\delta'$, $-u$ has penalty-subordinate growth from above, $(-u)^{-}_{\delta}=-u^{+}_{\delta}$ and $(-u)^{-}_{\delta'}=-u^{+}_{\delta'}$ on $\mathcal{D}$. The part from above, applied to $-u$, gives $-u^{+}_{\delta'}(\mu)+\kappa\,\mathcal{E}(\mu)\le-u^{+}_{\delta}(\mu)$ for every $\mu\in\mathcal{D}$; adding $u^{+}_{\delta}(\mu)+u^{+}_{\delta'}(\mu)-\kappa\,\mathcal{E}(\mu)$ to both sides gives $u^{+}_{\delta}(\mu)\le u^{+}_{\delta'}(\mu)-\kappa\,\mathcal{E}(\mu)$. $\blacksquare$