Hausdorff theorems on measurable functions and a new class of uniform functions

Abstract

Borel, Lebesgue, and Hausdorff described all families A(T) of functions f: T → ℝ on a set T possessing the same properties as the set C(T, G) of all continuous functions with respect to an open topology G. Those are just the families M(T, S) of all functions S-measurable with respect to some σ-additive and multiplicative ensembles S on T. The problem of description of all families A(T) of bounded functions possessing the same properties as the set Cb(T, G) of all continuous bounded functions remained open. In this paper a solution to this problem is given with the help of a new class of functions U(T, C) uniform with respect to some multiplicative families of finite coverings on T.