Properties of the Upper Semicontinuous Envelope

Let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, let $\mathbb{R}$ be the ordered field of real numbers with absolute value $|\cdot|$, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers, and let $\mathbb{N}$ be the natural numbers.

Let $u:S\to\mathbb{R}$ be bounded above near each point of $S$, and let $u^{*}$ be its upper semicontinuous envelope; the sets $A_{u}(x)$ are those of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function.

Then the following hold.

1. (Bounds)(i) $u(x)\le u^{*}(x)$ for every $x\in S$.

2. (Upper semicontinuity)(ii) $u^{*}$ is upper semicontinuous on $S$, and $u^{*}$ is itself bounded above near each point of $S$.

3. (Least upper semicontinuous majorant)(iii) If $v:S\to\mathbb{R}$ is upper semicontinuous on $S$ and $u(y)\le v(y)$ for every $y\in S$, then $u^{*}(x)\le v(x)$ for every $x\in S$.

4. (Fixed points)(iv) $u$ is upper semicontinuous on $S$ if and only if $u^{*}(x)=u(x)$ for every $x\in S$.

5. (Approximation)(v) For every $x\in S$ and every positive $\varepsilon\in\mathbb{R}$ there is $z\in S$ with

$$d(z,x)\le\varepsilon\qquad\text{and}\qquad|u(z)-u^{*}(x)|<\varepsilon .$$

Consequently, for every $x\in S$ there is a sequence $(x_{k})_{k\in\mathbb{N}}$ in $S$ which converges to $x$ in $(M,d)$ and for which $(u(x_{k}))_{k\in\mathbb{N}}$ converges to $u^{*}(x)$ in $(\mathbb{R},d_{\mathbb{R}})$.

6. (Monotonicity)(vi) If $v:S\to\mathbb{R}$ is bounded above near each point of $S$ and $u(y)\le v(y)$ for every $y\in S$, then $u^{*}(x)\le v^{*}(x)$ for every $x\in S$.