Gogi Rauli Pantsulaia, Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures, Volume 5, International Journal of Advanced Research in Mathematics (Volume 5)
https://www.scipress.com/IJARM.5.8
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;ρ<sub></sub>;B<sub>ρ</sub>(G)) without isolated points there exist a metric ρ<sub>1</sub> and a group operation ⊙ in G such that B<sub>ρ</sub>(G) = B<sub>ρ1</sub>(G) and (G;ρ<sub>1</sub>;B<sub>ρ1</sub>(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where B<sub>ρ</sub>(G) and B<sub>ρ1</sub>(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ<sub>1</sub>, 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.
Keywords:
    Haar Measure, Lie Group, Polish Group, Polish Space, Two-Sided Invariant Measure