TY - JOUR
T1 - Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures
AU - Pantsulaia, Gogi Rauli
JF - International Journal of Advanced Research in Mathematics
VL - 5
SP - 8
EP - 22
SN - 2297-6213
PY - 2016
PB - SciPress Ltd
DO - 10.18052/www.scipress.com/IJARM.5.8
UR - https://www.scipress.com/IJARM.5.8
KW - Haar Measure
KW - Lie Group
KW - Polish Group
KW - Polish Space
KW - Two-Sided Invariant Measure
AB - 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;ρ;Bρ(G)) without isolated points there exist a metric ρ1 and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, 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.
ER -