Proof of The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum

Throughout, $A_{u}(x)$ and $B_{u}(x)$ denote the sets attached to a function and a point of its domain in Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, the ambient metric space being $(M,d)$. Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named at the point of use. We write $u^{*}\vee v^{*}$ and $u_{*}\wedge v_{*}$ for the functions $S\to\mathbb{R}$ whose values at $y$ are $\max\{u^{*}(y),v^{*}(y)\}$ and $\min\{u_{*}(y),v_{*}(y)\}$.

Claim 1. Suppose $u$ and $v$ are bounded above near each point of $S$ and let $x\in S$. By Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds there are $c_{1},c_{2}\in\mathbb{R}$ and positive $r_{1},r_{2}\in\mathbb{R}$ such that $u(y)\le c_{1}$ for every $y\in S$ with $d(y,x)\le r_{1}$, and $v(y)\le c_{2}$ for every $y\in S$ with $d(y,x)\le r_{2}$. Put $c=\max\{c_{1},c_{2}\}$ and $r=\min\{r_{1},r_{2}\}$; by claim 2 of Elementary Properties of the Minimum of Two Elements, $r$ is $r_{1}$ or $r_{2}$ and so is positive, and by claim 1 of that lemma $r\le r_{1}$ and $r\le r_{2}$. Let $y\in S$ satisfy $d(y,x)\le r$. Then $d(y,x)\le r_{1}$ and $d(y,x)\le r_{2}$ by transitivity of the order, so $u(y)\le c_{1}$ and $v(y)\le c_{2}$; and $c_{1}\le c$ and $c_{2}\le c$ by claim 1 of Elementary Properties of the Maximum of Two Elements, so $u(y)\le c$ and $v(y)\le c$ by transitivity. Hence $(u\vee v)(y)\le c$ by claim 3 of Elementary Properties of the Maximum of Two Elements, and $(u\wedge v)(y)\le(u\vee v)(y)\le c$ by claim 5 of Elementary Properties of the Minimum of Two Elements and transitivity. Thus $c$ belongs to both $A_{u\vee v}(x)$ and $A_{u\wedge v}(x)$, and as $x\in S$ was arbitrary both functions are bounded above near each point of $S$.

If instead $u$ and $v$ are bounded below near each point of $S$, the same argument with $c=\min\{c_{1},c_{2}\}$, the inequalities reversed, claim 1 and claim 3 of Elementary Properties of the Minimum of Two Elements in place of the corresponding claims for the maximum, and claim 5 of Elementary Properties of the Minimum of Two Elements again, shows that $c$ belongs to $B_{u\wedge v}(x)$ and to $B_{u\vee v}(x)$.

Claim 2. Suppose $u$ and $v$ are bounded above near each point of $S$; by claim 1 so is $u\vee v$, so all three upper semicontinuous envelopes below are defined. Fix $x\in S$.

For every $y\in S$ we have $u(y)\le(u\vee v)(y)$ and $v(y)\le(u\vee v)(y)$ by claim 1 of Elementary Properties of the Maximum of Two Elements. Hence claim 6 of Properties of the Upper Semicontinuous Envelope, applied to the pair $u$, $u\vee v$ and to the pair $v$, $u\vee v$, gives $u^{*}(x)\le(u\vee v)^{*}(x)$ and $v^{*}(x)\le(u\vee v)^{*}(x)$, and therefore

by claim 3 of Elementary Properties of the Maximum of Two Elements.

Conversely, $u^{*}$ and $v^{*}$ are upper semicontinuous on $S$ by claim 2 of Properties of the Upper Semicontinuous Envelope, so $u^{*}\vee v^{*}$ is upper semicontinuous on $S$ by claim 2 of The Maximum of Two Upper Semicontinuous Functions. For $y\in S$ we have $u(y)\le u^{*}(y)$ and $v(y)\le v^{*}(y)$ by claim 1 of Properties of the Upper Semicontinuous Envelope, while $u^{*}(y)\le(u^{*}\vee v^{*})(y)$ and $v^{*}(y)\le(u^{*}\vee v^{*})(y)$ by claim 1 of Elementary Properties of the Maximum of Two Elements; by transitivity $u(y)\le(u^{*}\vee v^{*})(y)$ and $v(y)\le(u^{*}\vee v^{*})(y)$, so $(u\vee v)(y)\le(u^{*}\vee v^{*})(y)$ by claim 3 of Elementary Properties of the Maximum of Two Elements. Claim 3 of Properties of the Upper Semicontinuous Envelope, applied to $u\vee v$ and the upper semicontinuous majorant $u^{*}\vee v^{*}$, now gives $(u\vee v)^{*}(x)\le\max\{u^{*}(x),v^{*}(x)\}$.

The two inequalities give $(u\vee v)^{*}(x)=\max\{u^{*}(x),v^{*}(x)\}$ by antisymmetry of the order.

Claim 3. Suppose $u$ and $v$ are bounded below near each point of $S$; by claim 1 so is $u\wedge v$. Fix $x\in S$.

For every $y\in S$, $(u\wedge v)(y)\le u(y)$ and $(u\wedge v)(y)\le v(y)$ by claim 1 of Elementary Properties of the Minimum of Two Elements, so claim 7 of Properties of the Lower Semicontinuous Envelope, by Duality, applied to the pair $u\wedge v$, $u$ and to the pair $u\wedge v$, $v$, gives $(u\wedge v)_{*}(x)\le u_{*}(x)$ and $(u\wedge v)_{*}(x)\le v_{*}(x)$; hence $(u\wedge v)_{*}(x)\le\min\{u_{*}(x),v_{*}(x)\}$ by claim 3 of Elementary Properties of the Minimum of Two Elements.

Conversely, $u_{*}$ and $v_{*}$ are lower semicontinuous on $S$ by claim 3 of Properties of the Lower Semicontinuous Envelope, by Duality, so $u_{*}\wedge v_{*}$ is lower semicontinuous on $S$ by claim 3 of The Maximum of Two Upper Semicontinuous Functions. For $y\in S$ we have $(u_{*}\wedge v_{*})(y)\le u_{*}(y)\le u(y)$ and $(u_{*}\wedge v_{*})(y)\le v_{*}(y)\le v(y)$ by claim 1 of Elementary Properties of the Minimum of Two Elements, claim 2 of Properties of the Lower Semicontinuous Envelope, by Duality and transitivity, so $(u_{*}\wedge v_{*})(y)\le(u\wedge v)(y)$ by claim 3 of Elementary Properties of the Minimum of Two Elements. Claim 4 of Properties of the Lower Semicontinuous Envelope, by Duality, applied to $u\wedge v$ and the lower semicontinuous minorant $u_{*}\wedge v_{*}$, gives $\min\{u_{*}(x),v_{*}(x)\}\le(u\wedge v)_{*}(x)$.

The two inequalities give the asserted equality by antisymmetry of the order.