Investigation of various problem s connected with the fundam ental concepts of m easurability of sets and functions is necessary
for further developm ent of m any branches of m odern m athem atics, such as real and com plex analysis, abstract harm onic
analysis, analysis in infinite-dim ensional topological vector spaces, ergodic theory, probability theory, the theory of stochastic
processes, gam e theory, general topology, etc. In this respect, the notion of m easurability m ay be treated or construed in
different ways, accordingly to the specific features of problem s under consideration.
The m ain goal of this project is to exam ine various approaches to the notion of m easurability and to the closely related notion of
sm allness of sets in infinite-dim ensional topological vector spaces. A dom inant part of the project is devoted to the study of shy
sets in such spaces. The corresponding m ethods will be developed for studying the properties of shy sets and for obtaining their
characterization. Special attention will be paid to describing relationships between different types of sm all sets in concrete
classes of topological vector spaces.
Also, an extensive part of the project is devoted to a m odified version of the concept of m easurability, according to which instead
of speaking of the m easurability with respect to a given single m easure on a ground (base) space E, we consider the
m easurability with respect to a given class M of m easures on E. According to this m odified version, any subset of E (respectively,
any real-valued function defined on E) becom es either absolutely m easurable with respect to M, or relatively m easurable with
respect to M, or absolutely nonm easurable with respect to M. In particular, the role of M can be played by the class M(E) of all
nonzero sigm a-finite continuous (i.e., diffused) m easures on E. In m any im portant situations, E is endowed with additional
m athem atical structures, e.g., with som e topology T or with som e transform ation group G. In the first case, it is natural to consider
the m easurability with respect to the class M of the com pletions of all nonzero sigm a-finite diffused Borel m easures on E. In the
second case, it is natural to consider the m easurability with respect to the class of all nonzero sigm a-finite com plete m easures on
E which are invariant (or quasi-invariant) with respect to all transform ations from G. The project illustrates the usefulness of such
an approach and presents a num ber of its applications in concrete problem s of m athem atical analysis.

Project members:

• Gogi Pantsulaia (Principal Investigator)

• Mariam Beriashvili (Key Personnel)

• Aleks Petre Kirtadze (Key Personnel)

• Alexander Kharazishvili (Key Personnel)