A portion of Lemma 3.7.2 of Silva's "Invitation to Ergodic Theory" states the following:
Lemma. Let $(X, \mathcal{S}, \mu)$ be a $\sigma$-finite measure space and let $T: X \to X$ be a measure-preserving transformation. Then, the following two statements are equivalent
- If $A$ and $B$ are sets of positive measure, then there exists an integer $n > 0$ such that $$T^{-n}(A) \cap B \neq \emptyset.$$
- If $A$ and $B$ are sets of positive measure, then there exists an integer $n > 0$ such that $$\mu(T^{-n}(A) \cap B) > 0.$$
Clearly, only (1) $\implies$ (2) needs to be shown, and the proof is short:
Let $A, B$ be sets of positive measure. For the sake of contradiction, suppose $\mu(T^{-n}(A) \cap B) =0 $ for all $n$. Let $A_0 = A \backslash \bigcup_{n = 1}^\infty T^{-n}(A) \cap B$. Then $\mu(A_0) > 0$ but $T^{-n}(A_0) \cap B = \emptyset$, a contradiction.
I'm having trouble seeing the argument for $$A_0 := A \backslash \bigcup_{n = 1}^\infty T^{-n}(A) \cap B \implies T^{-n}(A_0) \cap B = \emptyset.$$
What exactly is the reasoning for this claim? Any hints, discussion, and solutions are appreciated.
I'm not sure about the proof given by the book but I figured another one.
Consider $B_0=B \cap \cup_{n=1}^{\infty} T^{-n}(A)$, the point in $B$ that eventually reach $A$, and $B_1=B \backslash \cup_{n=1}^{\infty} T^{-n}(A)$, point in $B$ that never reach $A$.
$B_0$ and $B_1$ are both measurable and their disjoint union is $B$ so $\mu(B)=\mu(B_0)+\mu(B_1)$.
We have $\mu(B_1)=0$ other by $(1)$ we would find $k$ such that $B_1 \cap T^{-k}(A) \ne \emptyset$ which is not.
Now we have $\mu(B_0)=\mu(B)$ and it gives that for at least a $n$ $\mu(B \cap T^{-n}(A))>0$