Upper and lower semicontinuous envelopes by local limsup and liminf

Let $A\subseteq {\mathbb{R}}^m$ be nonempty (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and let $u:A\to \mathbb{R}$. For $x\in A$ and $r>0$ put $M_r(x):=\sup \{u(y):y\in A,\mathrm{\mid }y-x\mathrm{\mid }\le r\},m_r(x):=\inf \{u(y):y\in A,\mathrm{\mid }y-x\mathrm{\mid }\le r\},$ the supremum and infimum being taken in $\bar{\mathbb{R}}=\mathbb{R}\cup \{\pm \mathrm{\infty }\}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, Greatest lower bound (infimum), Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$).

The upper semicontinuous envelope of $u$ is $u^{\ast }(x):={\inf }_{r>0}{M}_r(x)={\lim }_{r\downarrow 0}{\sup }_{\begin{array}{c}y\in A\\ \mathrm{\mid }y-x\mathrm{\mid }\le r\end{array}}u(y),$ and the lower semicontinuous envelope of $u$ is $u_{\ast }(x):={\sup }_{r>0}{m}_r(x)={\lim }_{r\downarrow 0}{\inf }_{\begin{array}{c}y\in A\\ \mathrm{\mid }y-x\mathrm{\mid }\le r\end{array}}u(y),$ both with values in $\bar{\mathbb{R}}$.

• The limits exist. For fixed $x\in A$ the map $r\mapsto M_r(x)$ is nondecreasing in $r$ (the set it is taken over grows with $r$) and the map $r\mapsto m_r(x)$ is nonincreasing in $r$, so the one-sided limits as $r\downarrow 0$ exist in $\bar{\mathbb{R}}$, with ${\inf }_{r>0}{M}_r(x)={\lim }_{r\downarrow 0}{M}_r(x)$ and ${\sup }_{r>0}{m}_r(x)={\lim }_{r\downarrow 0}{m}_r(x)$. Since $A$ is nonempty, the sets over which the suprema and infima are taken are nonempty; infinite values are kept rather than discarded.
• Comparison with $u$. For every $x\in A$ one has $m_r(x)\le u(x)\le M_r(x)$ for all $r>0$, hence $u_{\ast }(x)\le u(x)\le u^{\ast }(x)$ pointwise. If $u$ is bounded on $A$, then all values $M_r(x),m_r(x)$ lie between ${\inf }_{A}u$ and ${\sup }_{A}u$, so $u^{\ast }$ and $u_{\ast }$ are real-valued and bounded on $A$; this is a sufficient hypothesis for real-valued envelopes. Local boundedness near each point also suffices, because only arbitrarily small radii affect the defining infimum and supremum.
• Scope. These are the envelopes of $u$ on the Euclidean set $A$. The space--time cylinder of The Hamilton--Jacobi Cauchy problem and its classical solutions is used with $m=n+1$ and $A=Z$, and the envelope notation is the one appearing in the stability, Perron and comparison statements of this page (Upper and lower semicontinuity on subsets of ${\mathbb{R}}^n$ is the underlying mode of semicontinuity). No choice is used: each envelope is the value of a monotone limit indexed by $r$, hence by $r=1\mathrm{/}k$, and no sequence or point is selected.