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1997 | 74 | 1 | 109-121
Tytuł artykułu

The uniqueness of Haar measure and set theory

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits of all points of X are uncountable. In particular, this is true if either G is a locally compact, σ-compact topological group acting continuously on X, or the space X is uniform and nonseparable, and G consists of uniformly equicontinuous unimorphisms of X.
Rocznik
Tom
74
Numer
1
Strony
109-121
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-01-15
poprawiono
1996-12-06
Twórcy
  • Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [1] D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. 260 (1987).
  • [2] D. H. Fremlin, Measure algebras, in: Handbook of Boolean Algebras, Vol. 3, J. D. Monk (ed.), Elsevier, 1989, 877-980.
  • [3] D. H. Fremlin, Real-valued-measurable cardinals, in: Set Theory of the Reals, H. Judah (ed.), Israel Math. Conf. Proc. 6, 1993, 151-304.
  • [4] P. R. Halmos, Measure Theory, Springer, New York, 1974.
  • [5] A. B. Harazišvili, Groups of motions and the uniqueness of the Lebesgue measure, Soobshch. Akad. Nauk Gruzin. SSR 130 (1988), 29-32 (in Russian).
  • [6] L. Nachbin, The Haar Integral, D. Van Nostrand, Princeton, N.J., 1965.
  • [7] J. von Neumann, On rings of operators. III, Ann. of Math. 41 (1940), 94-161.
  • [8] K. R. Parthasarathy, Introduction to Probability and Measure, Macmillan India Press, Madras, 1977.
  • [9] M. Penconek and P. Zakrzewski, The existence of non-measurable sets for invariant measures, Proc. Amer. Math. Soc. 121 (1994), 579-584.
  • [10] I. E. Segal and R. A. Kunze, Integrals and Operators, Springer, Berlin, 1978.
  • [11] P. Zakrzewski, The existence of universal invariant measures on large sets, Fund. Math. 133 (1989), 113-124.
  • [12] P. Zakrzewski, Paradoxical decompositions and invariant measures, Proc. Amer. Math. Soc. 111 (1991), 533-539.
  • [13] P. Zakrzewski, The existence of invariant probability measures for a group of transformations, Israel J. Math. 83 (1993), 343-352.
  • [14] P. Zakrzewski, Strong Fubini axioms from measure extension axioms, Comment. Math. Univ. Carolin. 33 (1992), 291-297.
  • [15] P. Zakrzewski, When do sets admit congruent partitions?, Quart. J. Math. Oxford (2) 45 (1994), 255-265.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv74i1p109bwm
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