Upper and Lower Semicontinuous Envelopes of a Real-Valued Function

Let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, let $\mathbb{R}$ be the ordered field of real numbers, and let $u:S\to\mathbb{R}$. For $x\in S$ put

$$A_{u}(x)=\bigl\{\,c\in\mathbb{R}\ :\ \text{there is a positive }r\in\mathbb{R}\text{ such that }u(y)\le c\text{ for every }y\in S\text{ with }d(y,x)\le r\,\bigr\},$$ $$B_{u}(x)=\bigl\{\,c\in\mathbb{R}\ :\ \text{there is a positive }r\in\mathbb{R}\text{ such that }c\le u(y)\text{ for every }y\in S\text{ with }d(y,x)\le r\,\bigr\}.$$

1. (Boundedness near each point)(i) We say that $u$ is bounded above near each point of $S$ if $A_{u}(x)$ is nonempty for every $x\in S$, and that $u$ is bounded below near each point of $S$ if $B_{u}(x)$ is nonempty for every $x\in S$.

2. (Upper semicontinuous envelope)(ii) Suppose $u$ is bounded above near each point of $S$, and let $x\in S$. By the metric axioms $d(x,x)=0$, and $0\le r$ for every positive $r$, so $x$ is itself one of the points $y$ quantified over in the description of $A_{u}(x)$; consequently $u(x)\le c$ for every $c\in A_{u}(x)$, that is, $A_{u}(x)$ is bounded below by $u(x)$. Being nonempty as well, $A_{u}(x)$ has a greatest lower bound, unique, by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. The upper semicontinuous envelope of $u$ is the function $u^{*}:S\to\mathbb{R}$ given by

$$u^{*}(x)=\inf A_{u}(x)\qquad\text{for }x\in S .$$

3. (Lower semicontinuous envelope)(iii) Suppose $u$ is bounded below near each point of $S$, and let $x\in S$. Exactly as in clause 2, every $c\in B_{u}(x)$ satisfies $c\le u(x)$, so $B_{u}(x)$ is bounded above by $u(x)$; being nonempty it has a least upper bound by Least Upper Bound Property of the Real Numbers, unique by Uniqueness of the Supremum and of the Infimum. The lower semicontinuous envelope of $u$ is the function $u_{*}:S\to\mathbb{R}$ given by

$$u_{*}(x)=\sup B_{u}(x)\qquad\text{for }x\in S .$$