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.
@article{bwmeta1.element.bwnjournal-article-cmv74i1p109bwm,
author = {Piotr Zakrzewski},
title = {The uniqueness of Haar measure and set theory},
journal = {Colloquium Mathematicae},
volume = {72},
year = {1997},
pages = {109-121},
zbl = {0893.28009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv74i1p109bwm}
}
Zakrzewski, Piotr. The uniqueness of Haar measure and set theory. Colloquium Mathematicae, Tome 72 (1997) pp. 109-121. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv74i1p109bwm/
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