This project proposal is focuses to study some applications of descriptive set-theoretical and logical
concepts and methods to measure theory. It seems natural, rather than considering the measurability of
given sets and functions with respect to a concrete measure
m on a base (ground) set
E, to turn attention to
the more general question of the measurability of those sets and functions with respect to a given class
M
of measures on
E. Thus, one of the central themes of this project is the study of measurability properties of
sets and real-valued functions with respect to various classes
M of measures on the base set
E.
These three results are well known:
(1) if a Mazurkiewicz set is analytic in????????, then it is also Borel in ????????;
(2) there exists a Mazurkiewicz set which is of Lebesgue measure zero and of first Baire category;
(3) there exists a Mazurkiewicz set which is Lebesguenonmeasurable and does not possess the Baire
property.
In particular, we investigate the Mazurkiewicz type sets, which are non-measurable with respect to the
classical Lebesgue measure. Let us consider the class M of the sigma-finite diffused invariant extensions of
the Lebesgue measure. With applications of Mazurkiewicz type 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)