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Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures

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;B_ρ(G)) without isolated points there exist a metric ρ₁ and a group operation ⊙ in G such that B_ρ(G) = B_ρ1(G) and (G;ρ₁;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 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.

Info:

Periodical:

International Journal of Advanced Research in Mathematics (Volume 5)

Pages:

8-22

DOI:

https://doi.org/10.18052/www.scipress.com/IJARM.5.8

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:

June 2016

Authors:

Gogi Rauli Pantsulaia

Keywords:

Haar Measure, Lie Group, Polish Group, Polish Space, Two-Sided Invariant Measure

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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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[5] V. Yakovenko, Derivation of the Lorentz Transformation, Lecture note for course Phys171H, Introductory Physics: Mechanics and Relativity, Department of Physics, University of Maryland, College Park, 15 November, (2004).

[6] R.P. Halmos, Measure Theory, xi+304 pp. D. Van Nostrand Company, Inc., New York, (1950).

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[8] A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. of Math., (2) 34 (1) (1933), 147-169.

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