Naturalness of the Class of Lebesgue–Borel–Hausdorff Measurable Functions

It is well known that the family of all continuous functions on a topological space contains all constant functions and is closed with respect to the usual pointwise operations (addition, multiplication, finite supremum and infimum, division) and uniform convergence. The complete description of such function families (normal families) was given by Borel, Lebesgue, and Hausdorff. The normal families turned out to be exactly the families of all functions measurable with respect to multiplicative $\sigma$-additive families of sets. In 1914, Hausdorff also described the normal envelope of an arbitrary family of functions. If uniform convergence is replaced by pointwise convergence, then the notions of completely normal family and completely normal envelope arise. In 1977, Regoli described all completely normal families. They turned out to be precisely the families of all functions measurable with respect to $\sigma$-algebras of sets. Moreover, Regoli described the completely normal envelope of a specific family of functions. The present paper gives descriptive and some constructive characterizations of the completely normal envelope of an arbitrary family of functions.