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Dec 6 at 18:53 comment added Anthony Quas Interesting question. I feel that it should be true that the random variables $N_A(n)/\mu(A)$ and $N_B(n)/\mu(B)$ approach a common distribution (here $N_A(n)=\sum_{j=1}^n \mathbf 1_A(T^jx)$ where $x$ is distributed according to $\mu_A$), but haven't yet managed to write down a proof. I am thinking about making the, hopefully mild, assumption that $T$ is invertible. And then I would use the ratio ergodic theorem (so that for a.e. $x$, the ratio of the number of visits to $A$ versus $B$ up to $n$ should be $\mu(A)/\mu(B)$). The statement I made would imply a positive answer to your question.
Dec 6 at 8:44 comment added abcdmath @AnthonyQuas Thank you for pointing that out. I apologize for this oversight.
Dec 6 at 8:21 history edited abcdmath CC BY-SA 4.0
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Dec 6 at 7:44 comment added Anthony Quas In an infinite measure space, $\lim_{n\to\infty}\frac 1n\sum_{k=1}^n \chi_A\circ T^k(x)=0$ a.e.
Dec 6 at 6:21 history edited abcdmath CC BY-SA 4.0
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Dec 6 at 6:20 comment added abcdmath @AnthonyQuas Here $X$ is an infinite measure space, and thank you for pointing out the dependence of $\mathcal{B}(A,t)$ on $n$. I have updated my question accordingly.
Dec 6 at 0:01 comment added Anthony Quas Is $X$ a finite measure space or an infinite measure space? If the latter, as your post suggests, then you are misquoting the ergodic theorem.
Dec 5 at 23:57 comment added Anthony Quas Your set $\mathcal B(A,t)$ has an $n$ in its definition. Do you want $\mathcal B(A,t)$ to be $\mathcal B_n(A,t)$? And then I'm guessing that the definition of $\theta_n$ would involve $\mathcal B_n$; not just $\mathcal B$?
Dec 5 at 12:24 history asked abcdmath CC BY-SA 4.0