I was asked to prove this:
Let $\mu$ be a positive measure in $[0,1]$ and $T:[0,1]\to[0,1]$ a measurable bijection , $\mu([0,1])=1$ and $A\subseteq[0,1]$ such that $\mu(A)>0$.
Find conditions on the measure such that there is a positive integer $n$ such that $\mu(A\cap T^n(A))>0$. Under such conditions prove or give a counterexample the following:
For all positive integer $k$ there exists another positive integer $n$ such that $\mu(A\cap T^k(A)\cap T^{2k}(A)\cap\dots\cap T^{nk}(A))>0$.
I have been looking for help and I have found Poincaré Recurrence Theorem and Fustenberg's one, but I don`t know very well how to link them with what I want to prove.
Any suggestion? Thanks.