A Class of Uniform Functions and Its Relationship with the Class of Measurable Functions

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set $T$ whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on $T$. These families turn out to be exactly the families of all functions measurable with respect to some $\sigma$-additive and multiplicative ensembles on $T$. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on $T$. It is proved that the class of uniform functions differs from the class of measurable functions.