Measure-preserving semiflows and one-parameter Koopman semigroups

  • Nikolai Edeko
  • Moritz Gerlach
  • Viktoria KühnerEmail author
Research Article


For a finite measure space \(\mathrm {X}\), we characterize strongly continuous Markov lattice semigroups on \(\mathrm {L}^p(\mathrm {X})\) by showing that their generator A acts as a derivation on the dense subspace \(D(A)\cap \mathrm {L}^\infty (\mathrm {X})\). We then use this to characterize Koopman semigroups on \(\mathrm {L}^p(\mathrm {X})\) if \(\mathrm {X}\) is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.


Measure-preserving semiflow Koopman semigroup Derivation Topological model 



We express our sincere gratitude towards the anonymous referee for their insightful observations and detailed comments.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Nikolai Edeko
    • 1
  • Moritz Gerlach
    • 2
  • Viktoria Kühner
    • 1
    Email author
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Institut für MathematikUniversität PotsdamPotsdamGermany

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