##### Work number - M 11 FILED

__Authors: __*Karpel **O**.**M**.,**Myronyuk **M**.**V**.*

A series consists of 12 works which are published during 10 years. The purpose of a series of works is the solution of problems arising in the modern mathematical analysis, in particular, properties of measures on algebraic and topological structures are studied. Authors study problems of classification of Borel measures on Cantor sets relatively homeomorphisms of these sets and problems of characterization of measures on locally compact Abelian groups.

ThecriterionisprovedforprobabilityinvariantergodicmeasuresonthepathspacesofstationaryBrattelidiagramstobehomeomorphic. The criterion of topological equivalence for infinite non-defective Borel measures on a compact Cantor space is obtained, together with the criterion for measures on a non-compact locally compact Cantor set to be homeomorphic.

We describe probability measures on the cylinder and a-adic solenoids which are characterized by the independence of the sum and the difference of two independent random variables. We describe probability measures on the two dimensional torus which are characterized by the independence of linear forms from two independent random variables. We describe probability measures on discrete groups, a-adic solenoids and Banach spaces which are characterized by the symmetry of one linear form given another.

All results are new, were presented at scientific conferences and seminars. Obtained results can be used for further research and in conducting special courses in higher education.

*The results of the series of the works are published in 12 articles (including 9 in foreign journals). All articles are published in journals with positive impact factor.*

**The total number of the authors’ publications**: 24 articlesand19conference theses.

According to SCOPUS the total citation index of works of O.M.Karpel and M.V.Myronyuk is one and three respectively.