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_{};B_{ρ}(G)) without isolated points there exist a metric ρ_{1} and a group operation ⊙ in G such that B_{ρ}(G) = B_{ρ1}(G) and (G;ρ_{1};B_{ρ1}(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where B_{ρ}(G) and B_{ρ1}(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ_{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.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 5)

Pages:

8-22

Citation:

G. R. Pantsulaia "Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures", International Journal of Advanced Research in Mathematics, Vol. 5, pp. 8-22, 2016

Online since:

Jun 2016

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Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

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