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.