Locally essentially bounded activations with null discontinuity closure

Assume Countable Choice for the Lebesgue-measure convention. A real-valued function $\sigma :\mathbb{R}\to \mathbb{R}$ belongs to the activation class $\mathcal{M}$ when it is Lebesgue measurable, is locally essentially bounded, and the closure of its discontinuity set has one-dimensional Lebesgue measure zero. Here locally essentially bounded means that for every compact interval $J\subset \mathbb{R}$ there is a finite $M_J$ such that $\mathrm{\mid }\sigma (t)\mathrm{\mid }\le M_J$ for Lebesgue-almost every $t\in J$. The discontinuity set is $\{t:\sigma \text{ is not continuous at }t\}$. The function is a polynomial almost everywhere when there is a real algebraic polynomial $p$ such that $\sigma =p$ outside a Lebesgue-null set.

The subclass ${\mathcal{M}}_{\mathrm{p}\mathrm{t}}$ consists of those $\sigma \in \mathcal{M}$ that are also pointwise locally bounded: ${\sup }_{t\in J}\mathrm{\mid }\sigma (t)\mathrm{\mid }<\mathrm{\infty }$ for every compact interval $J$. This extra condition distinguishes pointwise uniform approximation from approximation in the essential-supremum norm.

Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$

Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Leshno et al., Definitions 3–5 and §4: the class $\mathcal{M}$

Moreau, Chapter 2 §2.3: the narrower piecewise-continuous class used for pointwise uniform approximation