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On the existence of invariant probability measures

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Abstract

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA 0⊂A is a convergence class for ℳ such that, for everyA ∈A 0, the sequence ((1/n) Σ =0/n−1 i 1 A ∘T i) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstr. 39, D-8000, München 2, FRG

    Wolfgang Adamski

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  1. Wolfgang Adamski
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Adamski, W. On the existence of invariant probability measures. Israel J. Math. 65, 79–95 (1989). https://doi.org/10.1007/BF02788175

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  • DOI: https://doi.org/10.1007/BF02788175

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