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Can anyone give me some hint on the following problem? Many thanks!

Given a probability space $(X, \Sigma, \mathbb{P})$ and a $\mathbb{P}$-preserving map $\tau: X\to X$, show that the following three are equivalent:

(i) $\tau$ is $\mathbb{P}$-ergodic;

(ii) For any $S\in \Sigma$ with $\mathbb{P}(S)>0$, we have $\mathbb{P}\big(\bigcup_{i=1}^\infty\tau^{-i}(S)\big)=1$;

(iii) For any $A, B\in \Sigma$ with $\mathbb{P}(A),\mathbb{P}(B)>0$, there exists $m\in \mathbb{N}$ such that $\mathbb{P}(\tau^{-m}(A)\cap B)>0$.

Actually, (i)$\Rightarrow$(ii) is trivial, and only the $\mathbb{P}$-preserving property will be employed in the proof. But how to show the others?

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I checked the original problem, and (ii) is what it is above, not a typo... – user65018 Apr 22 '13 at 1:06
up vote 1 down vote accepted

(ii) => (i)

Start with a measurable set $E$ with $\tau^{-1} E = E$. Either $P(E) = 0$, or $P(E) > 0$. What does (ii) tell you about the latter case?

(ii) => (iii) What can you say about $P(\cup_{i=1}^{\infty} \tau^{-i} (A) \cap B)$?

(iii) => (i) Start with a measurable set $E$ with $\tau^{-1} E = E$. Either $P(E) = 0$, or $P(E) > 0$. If $P(E) < 1$, in (iii) take $A = E$. Can you find a good candidate for B?

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Thanks for your hint very much! But this does not complete the proof exactly, since you did not show that (i) implies (ii). I think to employ Poincare's recurrence theorem can work. But anyway, thank you very much!! – user65018 Apr 25 '13 at 3:37
@user65018, you said in your post that (i) to (ii) is trivial, so I didn't write anything about it. – user27126 Apr 25 '13 at 3:42
Actually (i)=>(ii) is not trivial. But in the first version there was a typo in this problem: originally "$P(S)>0$" in (ii) was "$P(S)=1$", and in that case (i)=>(ii) is trivial: we even do not need to use the ergodicity condition... After modifying the problem, I did not modify my comments... Sorry for that. But thanks a lot for your comments. – user65018 Apr 25 '13 at 7:13

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