Set-theoretical methods can be used to prove interesting and significant results in various fields of mathematics. Actually, the
axiomatic system of Zermelo-Frenkel with Axiom of Choice is the foundation of Mathematics in the sense that essentially all
Mathematics can be formalized within ZFC. Over the past 30 years there has been extensive work on the interactions and
applications of descriptive set theory to other areas of mathematics, e.g. to mathematical analysis, functional analysis,
harmonic analysis, and so on. Also, intensive studies in mathematical analysis, the theory of functions of a real variable, and
measure theory inspired and stimulated further research on various kinds of paradoxical point sets.
The presented project proposal is devoted to a special, but very important, case of the extension problem for real-valued
partial functions. Namely, we discuss therein several variants of the so-called measure extension problem and we pay our
attention to purely set-theoretical aspects of this problem. In our project we study and elaborate the classical method of
extending measures, developed originally by Marczewski, which is in fact purely set-theoretical.
In the project we will also discuss some point sets, which have an extraordinary descriptive property and play an important
role in measure theory. In particular, we will be applying such the point sets to the study of the measure extension problem.
For instance, we consider Mazurkiewicz sets in connection with the measure extension problem, uniform sets and the
measure extension problem, Bernstein type constructions and absolutely nonmeasurable functions.

Project members:

• Mariam Beriashvili (Principal Investigator)