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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