Half-relaxed limits of a locally bounded family

Let $U\subseteq {\mathbb{R}}^m$ be open (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 $(u_{\epsilon }{)}_{\epsilon \in (0,1)}$ be a family of real-valued functions on $U$ that is locally bounded: for every compact $K\subseteq U$ there is $M_K<\mathrm{\infty }$ with $\mathrm{\mid }{u}_{\epsilon }(y)\mathrm{\mid }\le M_K$ for all $\epsilon \in (0,1)$ and $y\in K$. The upper half-relaxed limit and the lower half-relaxed limit of the family, computed in $\bar{\mathbb{R}}$ (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), Epsilon characterisation of the supremum), are $\bar{u}(z):={\mathrm{lim\hspace{0.17em}sup}}_{\begin{array}{c}\epsilon \downarrow 0\\ U\ni y\to z\end{array}}{u}_{\epsilon }(y):={\inf }_{\delta >0}\sup \{u_{\epsilon }(y):0<\epsilon <\delta ,y\in U,\mathrm{\mid }y-z\mathrm{\mid }<\delta \},$ $\underset{‾}{u}(z):={\mathrm{lim\hspace{0.17em}inf}}_{\begin{array}{c}\epsilon \downarrow 0\\ U\ni y\to z\end{array}}{u}_{\epsilon }(y):={\sup }_{\delta >0}\inf \{u_{\epsilon }(y):0<\epsilon <\delta ,y\in U,\mathrm{\mid }y-z\mathrm{\mid }<\delta \}\mathrm{.}$ The quantifiers range over the index $\epsilon$ and the point $y$ simultaneously, so the limit records the behaviour of the whole family near $z$, not the limit of the single family of values $(u_{\epsilon }(z){)}_{\epsilon }$; both envelopes are local and depend only on the germ of the family at $z$.

• Basic properties. If the family converges locally uniformly on $U$ to a continuous function $u$, then $\bar{u}=\underset{‾}{u}=u$. If the family is only locally bounded, then $\underset{‾}{u}\le \bar{u}$ pointwise, and $\bar{u}$ is upper semicontinuous while $\underset{‾}{u}$ is lower semicontinuous on $U$: the expressions are again monotone limits of local suprema and infima over families, and the proof of The envelopes are the least upper and greatest lower semicontinuous functions applies verbatim with the family indexed by $\delta$. Every value is kept in $\bar{\mathbb{R}}$; local boundedness makes both envelopes real-valued on each compact subset of $U$.
• Why the joint limit. Under Countable Choice, there are pairs $({\epsilon }_j,y_j)\to (0,z)$ with $u_{{\epsilon }_j}(y_j)\to \underset{‾}{u}(z)$, and likewise for $\bar{u}(z)$. For finite lower limit, take the infima over $0<\epsilon <1\mathrm{/}j$, $\mathrm{\mid }y-z\mathrm{\mid }<1\mathrm{/}j$ and choose a point within $1\mathrm{/}j$ of each infimum; these infima increase to $\underset{‾}{u}(z)$. The upper case is dual, with suprema decreasing to $\bar{u}(z)$. Infinite values use diverging finite thresholds. The definition itself is set-based and selects no subsequence or point; only this sequential characterization uses Countable Choice. This is the limit notion consumed by Half-relaxed limits of sub- and supersolutions with vanishing perturbations.