# Do there exist measurable transformations which are recurrent over spaces of infinite measure?

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)?

• What about real line with Lebesgue measure and the map $T(c)=-c$? Nov 5, 2020 at 19:11
• @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. Nov 5, 2020 at 19:20