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The existence of invariant probability measures for a group of transformations

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Abstract

SupposeG is a group of measurable transformations of aσ-finite measure space (X,A, m). A setA ∈A is weakly wandering underG if there are elementsg n ∈G such that the setsg nA, n=0, 1,…, are pairwise disjoint. We prove that the non-existence of any set of positive measure which is weakly wandering underG is a necessary and sufficient condition for the existence of aG-invariant, probability measure defined onA and dominating the measurem in the sense of absolute continuity.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 00-913, Warsaw 59, Poland

    Piotr Zakrzewski

  2. Fachbereich Mathematik, TU Berlin, Straße des 17 Juni 136, D-1000, Berlin 12, Germany

    Piotr Zakrzewski

Authors
  1. Piotr Zakrzewski
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Additional information

This paper was written while the author was visiting the Technische Universitat Berlin as a research fellow of the Alexander von Humboldt Foundation.

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Zakrzewski, P. The existence of invariant probability measures for a group of transformations. Israel J. Math. 83, 343–352 (1993). https://doi.org/10.1007/BF02784061

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  • DOI: https://doi.org/10.1007/BF02784061

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