Proof of Properties of the Lower Semicontinuous Envelope, by Duality

Conventions. The order $\le$ and the arithmetic of $\mathbb{R}$ are those of the ordered field of real numbers, and $-(-t)=t$ for every $t\in\mathbb{R}$. The sets $A_{w}(x)$ and $B_{w}(x)$ attached to a function $w$ are those of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function.

Claim 1. Let $x\in S$, let $c\in\mathbb{R}$ and let $r\in\mathbb{R}$ be positive. For $y\in S$, claim 4 of Elementary Order Arithmetic in an Ordered Field shows that $-u(y)\le c$ holds if and only if $-c\le u(y)$ holds, the two being obtained from each other by reversing signs and using $-(-t)=t$. Hence $r$ witnesses $c\in A_{-u}(x)$ if and only if $r$ witnesses $-c\in B_{u}(x)$, and therefore $c\in A_{-u}(x)$ if and only if $-c\in B_{u}(x)$.

Consequently $A_{-u}(x)$ is nonempty if and only if $B_{u}(x)$ is: if $b\in B_{u}(x)$ then $-b\in A_{-u}(x)$, since $-(-b)=b$, and if $c\in A_{-u}(x)$ then $-c\in B_{u}(x)$. So $-u$ is bounded above near each point of $S$ if and only if $u$ is bounded below near each point of $S$.

Assume now that this holds, and put $\beta=u_{*}(x)=\sup B_{u}(x)$. If $c\in A_{-u}(x)$ then $-c\in B_{u}(x)$, so $-c\le\beta$ and hence $-\beta\le c$ by claim 4 of Elementary Order Arithmetic in an Ordered Field; thus $-\beta$ is a lower bound for $A_{-u}(x)$. If $m\in\mathbb{R}$ is any lower bound for $A_{-u}(x)$, then for every $b\in B_{u}(x)$ we have $-b\in A_{-u}(x)$, so $m\le-b$ and hence $b\le-m$ by that same claim 4; thus $-m$ is an upper bound for $B_{u}(x)$, so $\beta\le-m$ because $\beta$ is the least upper bound, and hence $m\le-\beta$. Therefore $-\beta$ is the greatest lower bound of $A_{-u}(x)$, that is, $(-u)^{*}(x)=-\beta=-u_{*}(x)$.

Claim 2. By claim 1 the function $-u$ is bounded above near each point of $S$, so Properties of the Upper Semicontinuous Envelope (i) gives $-u(x)\le(-u)^{*}(x)$, which is $-u_{*}(x)$ by claim 1. Reversing signs by claim 4 of Elementary Order Arithmetic in an Ordered Field gives $u_{*}(x)\le u(x)$.

Claim 3. By claim 1 the function $-u$ is bounded above near each point of $S$, so Properties of the Upper Semicontinuous Envelope (ii) applies to it: the function $(-u)^{*}$ is upper semicontinuous on $S$ and is bounded above near each point of $S$. By claim 1, $(-u)^{*}=-u_{*}$, the function whose value at $y$ is the additive inverse of $u_{*}(y)$. Claim 1 of Semicontinuity Under Negation and Characterization of Continuity, applied to the function $u_{*}$ at each point of $S$, therefore shows that $u_{*}$ is lower semicontinuous on $S$. Finally, applying claim 1 above with $u_{*}$ in place of $u$ — legitimate because $-u_{*}$ is bounded above near each point of $S$ — shows that $u_{*}$ is bounded below near each point of $S$.

Claim 4. Let $v:S\to\mathbb{R}$ be lower semicontinuous on $S$ with $v(y)\le u(y)$ for every $y\in S$. By claim 1 of Semicontinuity Under Negation and Characterization of Continuity, $-v$ is upper semicontinuous on $S$, and by claim 4 of Elementary Order Arithmetic in an Ordered Field, $-u(y)\le-v(y)$ for every $y\in S$. By claim 1 the function $-u$ is bounded above near each point of $S$, so Properties of the Upper Semicontinuous Envelope (iii), applied with $-u$ in the role of $u$ and $-v$ in the role of $v$, gives $(-u)^{*}(x)\le-v(x)$ for every $x\in S$. By claim 1 this reads $-u_{*}(x)\le-v(x)$, and reversing signs gives $v(x)\le u_{*}(x)$.

Claim 5. By claim 1 of Semicontinuity Under Negation and Characterization of Continuity, $u$ is lower semicontinuous on $S$ if and only if $-u$ is upper semicontinuous on $S$, which by Properties of the Upper Semicontinuous Envelope (iv) holds if and only if $(-u)^{*}(x)=-u(x)$ for every $x\in S$. By claim 1 the left-hand side is $-u_{*}(x)$, and $-u_{*}(x)=-u(x)$ holds if and only if $u_{*}(x)=u(x)$, additive inverses in a field being unique. This proves claim 5.

Claim 6. Let $x\in S$. By Properties of the Upper Semicontinuous Envelope (v), applied to $-u$, there is a sequence $(x_{k})_{k\in\mathbb{N}}$ in $S$ converging to $x$ in $(M,d)$ such that the sequence with terms $-u(x_{k})$ converges to $(-u)^{*}(x)=-u_{*}(x)$ in $(\mathbb{R},d_{\mathbb{R}})$. For $s,t\in\mathbb{R}$ we have

$$d_{\mathbb{R}}(-s,-t)=|-s-(-t)|=|-(s-t)|=|s-t|=d_{\mathbb{R}}(s,t)$$

by claim 2 of Properties of the Absolute Value in an Ordered Field. Hence for every positive $\eta$ the same index $N$ that witnesses $d_{\mathbb{R}}(-u(x_{k}),-u_{*}(x))<\eta$ for $k\ge N$ witnesses $d_{\mathbb{R}}(u(x_{k}),u_{*}(x))<\eta$, so $(u(x_{k}))$ converges to $u_{*}(x)$ in $(\mathbb{R},d_{\mathbb{R}})$.

Claim 7. Let $v:S\to\mathbb{R}$ be bounded below near each point of $S$ with $u(y)\le v(y)$ for every $y\in S$. By claim 4 of Elementary Order Arithmetic in an Ordered Field, $-v(y)\le-u(y)$ for every $y\in S$, and by claim 1 both $-v$ and $-u$ are bounded above near each point of $S$. Hence Properties of the Upper Semicontinuous Envelope (vi), applied with $-v$ in the role of $u$ and $-u$ in the role of $v$, gives $(-v)^{*}(x)\le(-u)^{*}(x)$ for every $x\in S$; by claim 1 this reads $-v_{*}(x)\le-u_{*}(x)$, and reversing signs gives $u_{*}(x)\le v_{*}(x)$.