Some combinatorial problems on the measurability of functions with respect to invariant extensions of the Lebesgue measure
A. B. Kharazishvili (2010)
Acta Universitatis Carolinae. Mathematica et Physica
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Some combinatorial problems on the measurability of functions with respect to invariant extensions of the Lebesgue measure
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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...