The presented research is concerned with the application of descriptive set theoretical and
geometrical concepts and methods to measure theory. The main goal of the project is to study the
measurability properties of sets and real-valued functions with respect to various classes
M of
measures on the base set
E. More precisely, for a class M of measures, the measurability of given
sets and functions admits the following three aspects:
a) absolute measurability with respect to M;
b) relative measurability with respect to M;
c) absolute non-measurability with respect to M.
We investigate the uniform subsets of the Euclidian plane and their measurability properties with
respect to various classes M. It is well knowns, that Luzin was started study of uniform subsets and
it is well known the Luzini problem of the existence of a countable family of uniform sets, whose
union is identical with the Euclidean plane. The Uniform sets plays an important role in measure
theory, in particular in the study of measure extension problem. For example, in the plane we can
construct the extension of the classical two dimensional Lebesgue measure, such that all uniform
subsets in direction Oy-axis are measurable with respect of constructed extension.
In presented project we consider the uniform subsets of Euclidian plane, also uniform subsets of
n-dimensional Euclidian space and with applications of uniform sets we study the absolutely and
relatively measurability properties of sets and functions with respect to the M class

Project members:

• Mariam Beriashvili (Principal Investigator)