# The uniqueness of Haar measure and set theory

Colloquium Mathematicae (1997)

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

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topZakrzewski, 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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