Let $(X,\mathcal{A}, \mu, f)$ be an invariant system, $\mu$ be a probability, $\mathcal{B} \subset \mathcal{A}$ be a countable algebra generating $\mathcal{A}$ and $(A_k)_{k \ge 0}$ be a pairwise disjoint sequence of elements in $\mathcal{B}$. Suppose $x \in X$ is such that $\lim_n \frac{1}{n} \sum_{j=0}^{n-1} \mathcal{X}_{A_k} \circ f^j (x) = l_k \in [0,1], \forall k \ge 0$.
How do we show (if true) that $$\lim_n \frac{1}{n} \sum_{j=0}^{n-1} \mathcal{X}_{A} \circ f^j (x) =\lim_n \sum_{k=0}^\infty \frac{1}{n} \sum_{j=0}^{n-1} \mathcal{X}_{A_k} \circ f^j (x)$$ equals $$\sum_{k=0}^\infty \lim_n \frac{1}{n} \sum_{j=0}^{n-1} \mathcal{X}_{A_k} \circ f^j (x) = \sum_{k=0}^\infty l_k \in \mathbb{R} \text{ ?}$$ (i.e., that the visit frequency of a disjoint union is the series of the visit frequencies)
In order to switch limit and series, I've tried to write the series on $k$ as an integral against the counting measure to use monotone convergence theorem, dominated convergence theorem and uniform convergence with no success (I've also tried to take monotone subsequences working for all $k$'s). Fatou´s lemma also gives us a partial inequality with liminf, but I was not able to come up with the remaining inequality for the limsup. I was also not able to find arguments using a negation and 'an epsilon of room'.
This question sounds so natural that I got confused. Maybe I´m not considering enough information from the dynamical systems context. Is there a better indirect approach to get the result?
Thanks! Lucas Amorim.