Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that $\mathbb{P}[X_n \in A $ for $\infty$ many $n\geq 0] = 1$.
The question specifies, that I shall do this in two possbile ways: with (i) and without (ii) the ergodic theorem.
Ad (i): I use
$f: \mathbb{R} \rightarrow \{0, 1\}, X_n \mapsto \mathbb{1}_{A_n}$, where $A_n := \{X_n \in A\}$.
$f$ is measurable and therefore $(\mathbb{1}_{A_n})_{n\geq 0}$ is ergodic and stationary. Therefore $\frac{1}{n}\sum_{n\geq0}\mathbb{1}_{A_n}\rightarrow \mathbb{E}[\mathbb{1}_{A_0}] = \mathbb{P}[{A_0}]>0$.
Thus $\sum_{n\geq0}\mathbb{1}_{A_n} = \infty$ a. s., $\mathbb{P}[\sum_{n\geq0}\mathbb{1}_{A_n} = \infty] = 1$ and $\{\sum_{n\geq0}\mathbb{1}_{A_n} = \infty\} = \{X_n \in A $ for $\infty$ many $n\geq 0\}$. Is this correct?
Ad (ii):
I tried to do this the same way, but arguing with the generalized Lemma of Borel-Cantelli:
$\{X_n \in A $ for $\infty$ many $n\geq 0\} = \limsup_{n\rightarrow \infty} A_n = \{\sum_{n\geq1}\mathbb{E}[{A_n}|\sigma(X_0, ... X_{n-1})] = \infty\}$.
But at this point I fail to see, how to proceed, since the two expressions in the conditional expectation are neither independent, nor is $A_n$ $\sigma(X_0, ... X_{n-1})$-measurable. Any hints?
Item i) is correct.
For ii), notice that the set $S:=\{\omega, X_n(\omega)\in A\mbox{ for infinitely many }n\}$ is invariant. Therefore, its probability is either $0$ or $1$. Since for each $n$, $\mu\left(\bigcup_{j\geqslant n}\{X_j\in A\}\right)$ is positive and greater than $\mathbb P(A)$, the measure of $\bigcap_n\bigcup_{j\geqslant n}\{X_j\in A\}$ is positive.