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I am trying to find a measurable space $(X, \mathcal{B})$, with measure $\mu$ such that $\mu(X)=\infty$, and with a corresponding measure preserving transformation $T:X\rightarrow X$ such that $T$ is recurrent over $X$. That is, $T$ is such that for all $B\in \mathcal{B}$, there exists infinitely many $n>0$ such that $T^{n}(B) \cap B \not= \emptyset$.

Does such a dynamical system exist? If so, can you provide example or reference? If no such system exists, is there a particular reason (ie. it is not possible, or one has not been found)?

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    $\begingroup$ What about real line with Lebesgue measure and the map $T(c)=-c$? $\endgroup$ Nov 5, 2020 at 19:11
  • $\begingroup$ @WhoKnowsWho ok that certainly answers the question. Are there any more complex examples though, ie a map that is topologically transitive instead of recurrent then? If you would like, I could edit the question, and you can answer and I can close this. $\endgroup$
    – user918212
    Nov 5, 2020 at 19:20

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Your question is related to, what is called, infinite ergodic theory. There are many interesting examples. An excellent reference to learn about this is the book of Jon Aaronson "An introduction to infinite ergodic theory"

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