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:

June 2016

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Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

[1] R. Baker, Lebesgue measure, on R∞. Proc. Amer. Math. Soc., 113(4) (1991), 1023-1029.

[2] R. Baker, Lebesgue measure, on R∞. II. Proc. Amer. Math. Soc., 132(9) (2004), 2577-2591.

[3] J. Cichon , A. Kharazishvili, Weglorz B., Subsets of the real line, 177 p. Wydawnictwo Uniwersytetu Lodzkiego, Lodz, (1995).

[4] A. Maleki , An applications of ultrafilters to the Haar measure, Afr. Diaspora J. Math., 14(1) (2012) , 54-64.

[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).

[7] J. von Neumann, Invariant measures, xvi+134 pp. Amer. Math. Soc., Providence, RI, (1999).

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