Properties of the Lower Semicontinuous Envelope, by Duality

Let $(M,d)$ be a metric space, let $S\subseteq M$ be nonempty, let $\mathbb{R}$ be the ordered field of real numbers, 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 below near each point of $S$, and let $u_{*}$ be its lower semicontinuous envelope; the sets $A_{u}(x)$ and $B_{u}(x)$ are those of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function. Let $-u:S\to\mathbb{R}$ be the function whose value at $y\in S$ is the additive inverse of $u(y)$.

Then the following hold.

1. (Duality)(i) For every $x\in S$ and every $c\in\mathbb{R}$, $c$ belongs to $A_{-u}(x)$ if and only if $-c$ belongs to $B_{u}(x)$. Consequently $-u$ is bounded above near each point of $S$ if and only if $u$ is bounded below near each point of $S$, and in that case

$$(-u)^{*}(x)=-u_{*}(x)\qquad\text{for every }x\in S .$$

2. (Bounds)(ii) $u_{*}(x)\le u(x)$ for every $x\in S$.

3. (Lower semicontinuity)(iii) $u_{*}$ is lower semicontinuous on $S$, and $u_{*}$ is itself bounded below near each point of $S$.

4. (Greatest lower semicontinuous minorant)(iv) If $v:S\to\mathbb{R}$ is lower semicontinuous on $S$ and $v(y)\le u(y)$ for every $y\in S$, then $v(x)\le u_{*}(x)$ for every $x\in S$.

5. (Fixed points)(v) $u$ is lower semicontinuous on $S$ if and only if $u_{*}(x)=u(x)$ for every $x\in S$.

6. (Approximation)(vi) 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 $(\mathbb{R},d_{\mathbb{R}})$.

7. (Monotonicity)(vii) If $v:S\to\mathbb{R}$ is bounded below near each point of $S$ and $u(y)\le v(y)$ for every $y\in S$, then $u_{*}(x)\le v_{*}(x)$ for every $x\in S$.