I want to sort of extend the Poincare Recurrence Theorem in the following way.
Poincare Recurrence: Let $f$ be a measure preserving map on a set of finite measure. Almost every point in $A$ returns to $A$. That is, for each $a\in A-C$ there exists $n$ such that $f^n(a)\in A$, where $C$ is a set of measure zero.
I've been tasked with showing there exists an $N$ such that $\{a\in A:f^N(a)\}$ has positive measure. I'm trying to find this "global" $N$, but I am having difficulty.
Assuming the set you care about is $\{a\in A:\ f^N(a)\in A\}$, what the theorem gives you is that $$\bigcup_n\{a\in A:\ f^n(a)\in A\}=A\setminus C.$$ If all the sets in the (countable!) union were nullsets, we would get that $A\setminus C$ is a nullset and thus so is $A$. So if $A$ is not a nullset, at least one set in the union is not a nullset; that is, there exists $N$ such that $\{a\in A:\ f^N(a)\in A\}$ has positive measure.