Proof of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity

Throughout, $V\cap U$ is nonempty by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (v), and envelopes, local bounds and semicontinuity of functions on $U$ or on $V\cap U$ refer to these sets as subsets of $(H,d_{H})$, as fixed in Hilbert Triples: Standing Notation and Background (v); the sets $A_{v}(x)$ and $B_{v}(x)$ of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function are formed accordingly. Real arithmetic and order facts are those of The Real Numbers: Standing Notation and Background. Two elementary observations are used repeatedly.

(i) Restriction preserves semicontinuity. If $v$ is a function on $V$ that is lower semicontinuous at $x\in V\cap U$ relative to $V$, then its restriction to $V\cap U$ is lower semicontinuous at $x$ relative to $V\cap U$: the defining condition requires, for each $\varepsilon>0$, some $\eta>0$ such that $v(x)-\varepsilon<v(y)$ for every point $y$ of the set with $d_{H}(x,y)<\eta$, and an $\eta$ that works for all such $y\in V$ works for all such $y\in V\cap U\subseteq V$. The same argument applies to upper semicontinuity (with $v(y)<v(x)+\varepsilon$), and to a function on $U$ restricted to $V\cap U\subseteq U$.

(ii) The functions $\pm\delta h$ on $V\cap U$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (iv), $h$ is lower semicontinuous on $V$; by (i) its restriction to $V\cap U$ is lower semicontinuous on $V\cap U$, hence so is $\delta h$ (claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, as $0\le\delta$), while $-\delta h=\delta(-h)$ is upper semicontinuous on $V\cap U$ by claim 1 of Semicontinuity Under Negation and Characterization of Continuity and claim 2 of Sums and Nonnegative Multiples of Semicontinuous Functions.

Claim 1. Suppose $u$ is bounded above near each point of $U$. By The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i), $u^{-}_{\delta}$ is the upper semicontinuous envelope of the function $u-\delta h$ on the nonempty subset $V\cap U$ of $(H,d_{H})$, which is bounded above near each point of $V\cap U$. Hence Properties of the Upper Semicontinuous Envelope (ii) shows that $u^{-}_{\delta}$ is upper semicontinuous on $V\cap U$ and bounded above near each point of $V\cap U$, and Properties of the Upper Semicontinuous Envelope (i) gives $u(x)-\delta h(x)\le u^{-}_{\delta}(x)$ for $x\in V\cap U$. The second half follows in the same way from The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (ii), Properties of the Lower Semicontinuous Envelope, by Duality (iii) and Properties of the Lower Semicontinuous Envelope, by Duality (ii).

Claim 2. For $x\in U$, $c\in\mathbb{R}$ and a real $r>0$, the statement that $u(y)\le c$ for every $y\in U$ with $d_{H}(y,x)\le r$ is equivalent, by claim 4 of Elementary Order Arithmetic in an Ordered Field, to the statement that $-c\le(-u)(y)$ for every such $y$. Thus $c\in A_{u}(x)$ if and only if $-c\in B_{-u}(x)$, so $A_{u}(x)$ is nonempty if and only if $B_{-u}(x)$ is; that is, $u$ is bounded above near each point of $U$ if and only if $-u$ is bounded below near each point of $U$. In that case put $v=(-u)+\delta h$ on $V\cap U$, so that $v=-(u-\delta h)$. The function $v$ is bounded below near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi) applied to $-u$, so Properties of the Lower Semicontinuous Envelope, by Duality (i) applied to $v$ gives $(-v)^{*}=-v_{*}$ on $V\cap U$, that is, $(u-\delta h)^{*}=-\bigl((-u)+\delta h\bigr)_{*}$, which by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i) and The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (ii) reads $u^{-}_{\delta}=-(-u)^{+}_{\delta}$; multiplying by $-1$ gives $(-u)^{+}_{\delta}=-u^{-}_{\delta}$. The second assertion is the first applied to $-u$ in place of $u$, since $-(-u)=u$: $-u$ is bounded below near each point of $U$ if and only if $u$ is bounded above, and then $u^{+}_{\delta}=(-(-u))^{+}_{\delta}=-(-u)^{-}_{\delta}$, that is, $(-u)^{-}_{\delta}=-u^{+}_{\delta}$.

Claim 3. Suppose $u$ is bounded above near each point of $U$. The metric space $(U,d_{H})$ carries the restriction of $d_{H}$ (Real Hilbert Spaces: Standing Notation and Background (iii)), so a sequence in $U$ converges in $(U,d_{H})$ to a point of $U$ if and only if it converges to that point in $(H,d_{H})$, the defining condition involving only the distances $d_{H}(x_{m},x)$. We verify the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i) for the function $u^{-}_{\delta}$ on the subset $V\cap U$ of the metric space $(U,d_{H})$. Let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $V\cap U$ converging to $x\in U$, and let $t\in\mathbb{R}$ satisfy $t\le u^{-}_{\delta}(x_{m})$ for every $m$. We must show $x\in V$ and $t\le u^{-}_{\delta}(x)$.

Since $u$ is bounded above near $x\in U$, there are $c\in\mathbb{R}$ and a real $r>0$ with $u(y)\le c$ for every $y\in U$ with $d_{H}(y,x)\le r$. Choose $M_{0}\in\mathbb{N}$ with $d_{H}(x_{m},x)<\tfrac{r}{2}$ for every $m\ge M_{0}$. For each $m\in\mathbb{N}$ let $\varepsilon_{m}$ be the lesser of $\tfrac{1}{m}$ and $\tfrac{r}{2}$, and choose, by Properties of the Upper Semicontinuous Envelope (v) (countable choice), a point $z_{m}\in V\cap U$ with

$$d_{H}(z_{m},x_{m})\le\varepsilon_{m}\qquad\text{and}\qquad\bigl|u(z_{m})-\delta h(z_{m})-u^{-}_{\delta}(x_{m})\bigr|<\varepsilon_{m}.$$

For $m\ge M_{0}$ the triangle inequality gives $d_{H}(z_{m},x)\le d_{H}(z_{m},x_{m})+d_{H}(x_{m},x)<\tfrac{r}{2}+\tfrac{r}{2}=r$, so $u(z_{m})\le c$; and claim 9 of Properties of the Absolute Value in an Ordered Field gives $u(z_{m})-\delta h(z_{m})>u^{-}_{\delta}(x_{m})-\varepsilon_{m}\ge t-1$, using $t\le u^{-}_{\delta}(x_{m})$ and $\varepsilon_{m}\le\tfrac1m\le1$, the last because $1\le m$ (claim 4 of Properties of the Order on the Natural Numbers, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background (i)) gives $\tfrac1m\le1$ on multiplying by $\tfrac1m\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field). Hence $\delta h(z_{m})<u(z_{m})-t+1\le c-t+1$, and multiplying by $\delta^{-1}\ge0$ (claims 4 and 5 of Elementary Arithmetic in an Ordered Field), $h(z_{m})\le c'$ with $c'=(c-t+1)/\delta$, for every $m\ge M_{0}$. Moreover $(z_{m})$ converges to $x$ in $(H,d_{H})$: given $\varepsilon>0$, choose $N_{1}\in\mathbb{N}$ with $\tfrac{1}{N_{1}}<\tfrac{\varepsilon}{2}$ (The Archimedean Property of the Real Numbers) and $N_{2}\in\mathbb{N}$ with $d_{H}(x_{m},x)<\tfrac{\varepsilon}{2}$ for $m\ge N_{2}$; for $m$ at least the larger of $N_{1}$ and $N_{2}$, the triangle inequality of Metric Space and $\tfrac1m\le\tfrac{1}{N_{1}}$ (from $N_{1}\le m$, as above) give $d_{H}(z_{m},x)\le d_{H}(z_{m},x_{m})+d_{H}(x_{m},x)<\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon$.

The sequence $(w_{j})_{j\in\mathbb{N}}$ with $w_{j}=z_{M_{0}+j}$ is a subsequence of $(z_{m})$, the indices $n_{j}=M_{0}+j$ being strictly increasing by claim 6 of Properties of the Order on the Natural Numbers; so it converges to $x$ in $(H,d_{H})$ by A Subsequence of a Convergent Sequence Has the Same Limit, it lies in $V$, and it satisfies $h(w_{j})\le c'$ for every $j$, since $M_{0}+j\ge M_{0}$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (iv), $x\in V$. Thus $x\in V\cap U$.

Finally, let $\varepsilon>0$. By claim 1, $u^{-}_{\delta}$ is upper semicontinuous at $x$ relative to $V\cap U$, so there is a real $\eta>0$ with $u^{-}_{\delta}(y)<u^{-}_{\delta}(x)+\varepsilon$ for every $y\in V\cap U$ with $d_{H}(x,y)<\eta$. Choosing $m$ with $d_{H}(x_{m},x)<\eta$ gives $t\le u^{-}_{\delta}(x_{m})<u^{-}_{\delta}(x)+\varepsilon$. As $\varepsilon>0$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack (i) gives $t\le u^{-}_{\delta}(x)$. This verifies the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i), so $u^{-}_{\delta}$ has closed superlevel sets in $(U,d_{H})$; by A Real Function with Closed Superlevel Sets on a Subset of a Metric Space (ii) this means that $\{x\in V\cap U:t\le u^{-}_{\delta}(x)\}$ is closed in $(U,d_{H})$ for every $t\in\mathbb{R}$.

If instead $u$ is bounded below near each point of $U$, then by claim 2, $-u$ is bounded above near each point of $U$ and $(-u)^{-}_{\delta}=-u^{+}_{\delta}$ on $V\cap U$. Applying what was just proved to $-u$, the function $-u^{+}_{\delta}$ has closed superlevel sets in $(U,d_{H})$, and its superlevel set at height $t$ is $\{x\in V\cap U:t\le-u^{+}_{\delta}(x)\}=\{x\in V\cap U:u^{+}_{\delta}(x)\le-t\}$ (claim 4 of Elementary Order Arithmetic in an Ordered Field); as $t$ ranges over $\mathbb{R}$ so does $-t$, which gives the stated form.

Claim 4. Suppose $u(x)\le C$ for every $x\in U$. For each $x\in U$, $C\in A_{u}(x)$ (with the positive radius $1$, claim 6 of Elementary Order Arithmetic in an Ordered Field), so $u$ is bounded above near each point of $U$. Let $g:V\cap U\to\mathbb{R}$ be given by $g(x)=C-\delta h(x)$. By (ii), $-\delta h$ is upper semicontinuous on $V\cap U$; the constant function $C$ is upper semicontinuous on $V\cap U$ (the defining inequality $C<C+\varepsilon$ holds at every point); hence $g$ is upper semicontinuous on $V\cap U$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. Since $u(y)-\delta h(y)\le C-\delta h(y)=g(y)$ for every $y\in V\cap U$ (claim 3 of Elementary Arithmetic in an Ordered Field), Properties of the Upper Semicontinuous Envelope (iii) gives $u^{-}_{\delta}(x)\le g(x)=C-\delta h(x)$ for $x\in V\cap U$. Multiplying $\tfrac12|x|_{H}^{2}\le h(x)$ (The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (i)) by $\delta>0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\tfrac{\delta}{2}|x|_{H}^{2}\le\delta h(x)$, hence $C-\delta h(x)\le C-\tfrac{\delta}{2}|x|_{H}^{2}$.

If instead $-C\le u(x)$ for every $x\in U$, then $(-u)(x)\le C$ for every $x\in U$, so by the first part $-u$ is bounded above near each point of $U$ and $(-u)^{-}_{\delta}(x)\le C-\delta h(x)\le C-\tfrac{\delta}{2}|x|_{H}^{2}$ for $x\in V\cap U$. By claim 2, $u$ is bounded below near each point of $U$ and $(-u)^{-}_{\delta}=-u^{+}_{\delta}$; multiplying by $-1$ gives $-C+\tfrac{\delta}{2}|x|_{H}^{2}\le-C+\delta h(x)\le u^{+}_{\delta}(x)$.

Claim 5. Suppose $u$ and $v$ are bounded above near each point of $U$, so that $u-\delta h$, $v-\delta h$ and $u-\delta' h$ are bounded above near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi) (note $0<\delta\le\delta'$). For $y\in V\cap U$, $u(y)-\delta h(y)\le v(y)-\delta h(y)$ by claim 3 of Elementary Arithmetic in an Ordered Field, so Properties of the Upper Semicontinuous Envelope (vi) gives $u^{-}_{\delta}\le v^{-}_{\delta}$ on $V\cap U$. Also $0\le(\delta'-\delta)h(y)$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (i) and claim 5 of Elementary Arithmetic in an Ordered Field, so $u(y)-\delta' h(y)\le u(y)-\delta h(y)$, and Properties of the Upper Semicontinuous Envelope (vi) gives $u^{-}_{\delta'}\le u^{-}_{\delta}$ on $V\cap U$. If $u$ and $v$ are bounded below near each point of $U$, the same two comparisons, $u+\delta h\le v+\delta h$ and $u+\delta h\le u+\delta' h$ on $V\cap U$, together with Properties of the Lower Semicontinuous Envelope, by Duality (vii), give $u^{+}_{\delta}\le v^{+}_{\delta}$ and $u^{+}_{\delta}\le u^{+}_{\delta'}$.

Claim 6. Suppose $u$ is upper semicontinuous on $U$. For $x\in U$, upper semicontinuity at $x$ with $\varepsilon=1$ gives a real $\eta>0$ with $u(y)<u(x)+1$ for every $y\in U$ with $d_{H}(x,y)<\eta$; with $r=\tfrac{\eta}{2}$, every $y\in U$ with $d_{H}(y,x)\le r$ satisfies $d_{H}(x,y)<\eta$, hence $u(y)\le u(x)+1$. Thus $u(x)+1\in A_{u}(x)$, and $u$ is bounded above near each point of $U$. By (i) the restriction of $u$ to $V\cap U$ is upper semicontinuous on $V\cap U$, by (ii) so is $-\delta h$, and so is their sum $u-\delta h$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. Hence Properties of the Upper Semicontinuous Envelope (iv) gives $(u-\delta h)^{*}=u-\delta h$ on $V\cap U$, that is, $u^{-}_{\delta}(x)=u(x)-\delta h(x)$. If $u$ is lower semicontinuous on $U$, then for $x\in U$ lower semicontinuity at $x$ with $\varepsilon=1$ gives a real $\eta>0$ with $u(x)-1<u(y)$ for every $y\in U$ with $d_{H}(x,y)<\eta$, so with $r=\tfrac{\eta}{2}$ one has $u(x)-1\in B_{u}(x)$; thus $u$ is bounded below near each point of $U$; the restriction of $u$ to $V\cap U$ and $\delta h$ are lower semicontinuous on $V\cap U$ by (i) and (ii), hence so is $u+\delta h$ by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, and Properties of the Lower Semicontinuous Envelope, by Duality (v) gives $u^{+}_{\delta}(x)=u(x)+\delta h(x)$. Finally, if $u$ is continuous on $U$ then it is both upper and lower semicontinuous on $U$ by claim 2 of Semicontinuity Under Negation and Characterization of Continuity, so both conclusions hold.

Claim 7. Put $\gamma=\delta'-\delta$, so that $0\le\gamma$ by claim 3 of Elementary Arithmetic in an Ordered Field, and $0<\delta'$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Suppose $u$ is bounded above near each point of $U$, so that $u-\delta h$ and $u-\delta' h$ are bounded above near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi) and $u^{-}_{\delta}$ and $u^{-}_{\delta'}$ are defined on $V\cap U$ by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i). Let $g:V\cap U\to\mathbb{R}$ be given by $g(x)=u^{-}_{\delta}(x)-\gamma h(x)$. By claim 1, $u^{-}_{\delta}$ is upper semicontinuous on $V\cap U$; by (ii) applied with $\gamma$ in place of $\delta$, which used only $0\le\delta$, the function $-\gamma h$ is upper semicontinuous on $V\cap U$; hence $g$ is upper semicontinuous on $V\cap U$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. For $y\in V\cap U$, claim 1 gives $u(y)-\delta h(y)\le u^{-}_{\delta}(y)$, and adding $-\gamma h(y)$ to both sides (claim 3 of Elementary Arithmetic in an Ordered Field) gives

$$u(y)-\delta' h(y)=\bigl(u(y)-\delta h(y)\bigr)-\gamma h(y)\le u^{-}_{\delta}(y)-\gamma h(y)=g(y),$$

where $\delta' h(y)=\delta h(y)+\gamma h(y)$ by distributivity in $\mathbb{R}$. Since $u^{-}_{\delta'}$ is the upper semicontinuous envelope of $u-\delta' h$ on $V\cap U$ (The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i)), Properties of the Upper Semicontinuous Envelope (iii) gives $u^{-}_{\delta'}(x)\le g(x)=u^{-}_{\delta}(x)-\gamma h(x)$ for every $x\in V\cap U$, which is the first assertion after adding $\gamma h(x)$ to both sides.

If instead $u$ is bounded below near each point of $U$, then by claim 2, $-u$ is bounded above near each point of $U$, $(-u)^{-}_{\delta}=-u^{+}_{\delta}$ and $(-u)^{-}_{\delta'}=-u^{+}_{\delta'}$ on $V\cap U$. The first assertion applied to $-u$ gives $-u^{+}_{\delta'}(x)+\gamma h(x)\le-u^{+}_{\delta}(x)$ for $x\in V\cap U$; multiplying by $-1$ (claim 4 of Elementary Order Arithmetic in an Ordered Field) gives $u^{+}_{\delta}(x)\le u^{+}_{\delta'}(x)-\gamma h(x)$, that is, $u^{+}_{\delta}(x)+\gamma h(x)\le u^{+}_{\delta'}(x)$.