@article{pantsulaia2016,
author = {Pantsulaia, Gogi Rauli},
title = {Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures},
year = {2016},
month = {6},
volume = {5},
pages = {8--22},
journal = {International Journal of Advanced Research in Mathematics},
doi = {10.18052/www.scipress.com/IJARM.5.8},
keywords = {Polish Space, Polish Group, Lie Group, Haar Measure, Two-Sided Invariant Measure},
abstract = {It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;$\rho$;B[removed info]$\rho$(G)) without isolated points there exist a metric [removed info]$\rho$1 and a group operation ⊙ in G such that B$\rho$(G) = B$\rho$1(G) and (G;$\rho$1;B$\rho$1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure $\mu$ , where B$\rho$[removed info](G) and B[removed info]$\rho$1(G) denote Borel [removed info]$\sigma$-algebras of subsets of G generated by metrics $\rho$ [removed info]and $\rho$[removed info]1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.}
}