The uniqueness of Haar measure and set theory

Piotr Zakrzewski

Colloquium Mathematicae (1997)

  • Volume: 74, Issue: 1, page 109-121
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Zakrzewski, Piotr. "The uniqueness of Haar measure and set theory." Colloquium Mathematicae 74.1 (1997): 109-121. <http://eudml.org/doc/210494>.

@article{Zakrzewski1997,
abstract = {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.},
author = {Zakrzewski, Piotr},
journal = {Colloquium Mathematicae},
keywords = {real-valued measurable cardinal; invariant measure; Haar measure; locally compact space; ergodic measure; uniqueness; invariant measures; measure completion},
language = {eng},
number = {1},
pages = {109-121},
title = {The uniqueness of Haar measure and set theory},
url = {http://eudml.org/doc/210494},
volume = {74},
year = {1997},
}

TY - JOUR
AU - Zakrzewski, Piotr
TI - The uniqueness of Haar measure and set theory
JO - Colloquium Mathematicae
PY - 1997
VL - 74
IS - 1
SP - 109
EP - 121
AB - 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.
LA - eng
KW - real-valued measurable cardinal; invariant measure; Haar measure; locally compact space; ergodic measure; uniqueness; invariant measures; measure completion
UR - http://eudml.org/doc/210494
ER -

References

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  1. [1] D. H. Fremlin, Measure-additive coverings and measurable selectors, Dissertationes Math. 260 (1987). Zbl0703.28003
  2. [2] D. H. Fremlin, Measure algebras, in: Handbook of Boolean Algebras, Vol. 3, J. D. Monk (ed.), Elsevier, 1989, 877-980. 
  3. [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. [4] P. R. Halmos, Measure Theory, Springer, New York, 1974. 
  5. [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. [6] L. Nachbin, The Haar Integral, D. Van Nostrand, Princeton, N.J., 1965. 
  7. [7] J. von Neumann, On rings of operators. III, Ann. of Math. 41 (1940), 94-161. Zbl0023.13303
  8. [8] K. R. Parthasarathy, Introduction to Probability and Measure, Macmillan India Press, Madras, 1977. Zbl0395.28001
  9. [9] M. Penconek and P. Zakrzewski, The existence of non-measurable sets for invariant measures, Proc. Amer. Math. Soc. 121 (1994), 579-584. Zbl0822.03027
  10. [10] I. E. Segal and R. A. Kunze, Integrals and Operators, Springer, Berlin, 1978. Zbl0373.28001
  11. [11] P. Zakrzewski, The existence of universal invariant measures on large sets, Fund. Math. 133 (1989), 113-124. Zbl0715.28010
  12. [12] P. Zakrzewski, Paradoxical decompositions and invariant measures, Proc. Amer. Math. Soc. 111 (1991), 533-539. Zbl0716.04002
  13. [13] P. Zakrzewski, The existence of invariant probability measures for a group of transformations, Israel J. Math. 83 (1993), 343-352. Zbl0786.28012
  14. [14] P. Zakrzewski, Strong Fubini axioms from measure extension axioms, Comment. Math. Univ. Carolin. 33 (1992), 291-297. Zbl0765.03026
  15. [15] P. Zakrzewski, When do sets admit congruent partitions?, Quart. J. Math. Oxford (2) 45 (1994), 255-265. Zbl0885.28003

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