T.V. Rodionov

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 σ-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.

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 σ-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 σ-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.

Postclassical families of functions proper to descriptive and prescriptive spaces

The classical works in function theory (by E. Borel, R. Baire, H. Lebesgue, F. Hausdorff, et al.) laid down the foundation of the classical descriptive theory of functions. Its initial notions are those of a descriptive space and of a measurable function on it. Measurable functions are defined in the (classical) pre-image language. However, a specific range of tasks in the theory of functions, measure theory, and integration theory that emerge on this basis necessitates using an entirely different (postclassical) cover language, which is equivalent to the preimage one in the classical case. By means of the cover language, the general concepts of a prescriptive space and of distributable and uniform functions on it are introduced and their basic properties are studied.