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

In the setting of The Real Numbers: Standing Notation and Background, let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, and let $u,v:S\to\mathbb{R}$. The order of $\mathbb{R}$ is a total order, so that any two real numbers $a,b$ have a maximum $\max\{a,b\}$ and a minimum $\min\{a,b\}$; as in The Maximum of Two Upper Semicontinuous Functions, $u\vee v$ and $u\wedge v$ denote the functions $S\to\mathbb{R}$ whose values at $y\in S$ are $\max\{u(y),v(y)\}$ and $\min\{u(y),v(y)\}$.

That a function on $S$ is bounded above near each point of $S$, or bounded below near each point of $S$, and its upper semicontinuous envelope $(\cdot)^{*}$ and lower semicontinuous envelope $(\cdot)_{*}$, are as defined there, with the ambient metric space $(M,d)$. Then the following hold.

1. (Local bounds)(i) If $u$ and $v$ are bounded above near each point of $S$, then so are $u\vee v$ and $u\wedge v$. If $u$ and $v$ are bounded below near each point of $S$, then so are $u\vee v$ and $u\wedge v$.

2. (Upper envelope of a maximum)(ii) Suppose $u$ and $v$ are bounded above near each point of $S$. Then

$$(u\vee v)^{*}(x)=\max\{u^{*}(x),v^{*}(x)\}\qquad\text{for every }x\in S .$$

3. (Lower envelope of a minimum)(iii) Suppose $u$ and $v$ are bounded below near each point of $S$. Then

$$(u\wedge v)_{*}(x)=\min\{u_{*}(x),v_{*}(x)\}\qquad\text{for every }x\in S .$$