I have the following exercise:
"Show that if a measure-preserving system $(X, \mathcal B, \mu, T)$ has the property that for any $A,B \in \mathcal B$ there exists $N$ such that $\mu(A \cap T^{-n} B) = \mu(A)\mu(B)$ for all $n \geq N$, then $\mu(A) = 0$ or $1$ for all $A \in \mathcal B$"
Now the back of the book states that I should fix $B$ with $0 < \mu(B) < 1$ and then find $A$ using the Baire Category Theorem. Edit: I'm now pretty sure that this "$B$" is what "$A$" is in the required result.
Edit: This stopped being homework so I removed the tag. Any approach would be nice. I have some idea where I approximate $A$ with $T^{-n} B^C$ where the $n$ will be an increasing sequence and then taking the $\limsup$ of the sequence. I'm not sure if it is correct. I will add it later on.
My attempt after @Did's comment: "proof": First pick $B$ with $0 < \mu(B) < 1$. Then set $A_0 = B^C$ and determine the smallest $N_0$ such that
$\mu(A_0 \cap T^{-N_0} B) = \mu(A_0) \mu(B)$
Continue like this and set
$A_k = T^{-N_{k - 1}} B^C$
Now we note that the $N_k$ are a strictly increasing sequence, since suppose not, say $N_{k} \leq N_{k - 1}$ then $\mu \left ( T^{-N_{k - 1}} B^C \cap T^{-N_{k - 1}} B \right ) = 0 \neq \mu(B^C) \mu(B) > 0$
Set $A = \limsup_n A_n$, then note that \begin{align} \sum_n \mu(A_n) = \sum_n \mu(B^C) = \infty \end{align}
So $\mu(A) = 1$, by the Borel-Cantelli lemma. Well, not yet, because we are also required to show that the events are independent, so it is sufficient to show that $\mu(A_{k + 1} \cap A_k) = \mu(A_{k + 1 })\mu(A_k)$
We know that $\mu(T^{N_k} B^C \cap T^{N_{k + 1}} B) = \mu(B^C)\mu(B)$. So does a similar result now hold if we replace $B$ with $B^C$ in the second part?
Note: \begin{align} \mu(A \cap T^{-M} B^C) &= \mu(A \setminus (T^{-M} B \cap A))\\\ &= \mu(A) - \mu(A)\mu(B) \\\ &= \mu(A) - \mu(A \cap T^{-M} B)\\\ &= \mu(A)\mu(B^C) \end{align} which is what was required.
For this $A$ and $B$ we can find an $M$ and a $k$ such that $N_k \leq M < N_{k + 1}$. Now note that $\limsup_n A \cap T^{-M} B = \limsup_n (A \cap T^{-M} B)$.
Further, $\sum_n \mu(A_n \cap T^{-N_{k +1}}) = \mu(A_0 \cap T^{-N_{k + 1}}) + \ldots + \mu(A_{k + 1} \cap T^{N_{k + 1}}) < \infty$ So again by the Borel-Cantelli Lemma we have $\mu(\limsup_n A_n \cap T^{-M} B) = 0$. Thus we get
$\mu(A) \mu(B) = \mu(B) = \mu(A \cap T^{-M} B) = 0$
which is a contradiction since $\mu(B) > 0$. So, such $B$'s violate the condition.
Added: Actually the metric on the space of events $d(A,B) = \mu(A \Delta B)$ can work together with Baire's Category Theorem.