Postclassical Families of Functions Proper for Descriptive and Prescriptive Spaces

Abstract

The classics of function theory (E. Borel, H. Lebesgue, R. Baire, W. H. Young, F. Hausdorff, et al.) have laid down the foundation of the classical descriptive theory of functions. Its initial notions are the notions of a descriptive space and of a measurable function on it. Measurable functions were defined in the classical preimage language. However, a specific range of tasks in theory of functions, measure theory, and integration theory emergent on this base necessitates the usage of the entirely different postclassical cover language, equivalent to the preimage language in the classical case. By means of the cover language, the general notions of a prescriptive space and distributable and uniform functions on it are introduced in this paper and their basic properties are studied.