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Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
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A counterexample for the Mean Ergodic Theorem in $L_\infty$
The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, Ergodic …