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On non-singular transformations of a measure space. I

  • Wolfgang Krieger
Article

Abstract

We consider a Lebesgue measure space (M, ∇, m). By an automorphism of (M, ∇, m) we mean a bi-measurable transformation of (M, ∇, m) that together with its inverse is non-singular with respeot to m. We study an equivalence relation between these automorphisms that we call the weak equivalence. Two automorphisms S and T are weakly equivalent if there is an automorphism U such that for almost all xε M U maps the S-orbit of x onto the T-orbit of U x. Ergodicity, the existence of a finite invariant measure, the existenoe of a σ-finite infinite invariant measure, and the non-existence of such measures are invariants of weak equivalenoe. In this paper and in its sequel we solve the problem of weak equivalenoe for a class of automorphisms that comprises all ergodic automorphisms that admit a σ-finite invariant measure, and also certain ergodic automorphisms that do not admit such a measure.

Keywords

Stochastic Process Probability Theory Equivalence Relation Lebesgue Measure Invariant Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • Wolfgang Krieger
    • 1
  1. 1.Mathematisches Institut der UniversitÄtMünchen

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