Can you explain for me why author write " Based on $N^*f(x)<\infty$ and theorem 2, we have $$ \lim\limits_{n\to\infty} \dfrac{1}{n} \sum_{k=1}^{n} f(T^k x)ξ_k(\omega) ?$$ Take $a_k=f(T^k x)$, in theorem 3 we only have $\sup_n \dfrac{1}{n} \# \{ k: a_k/k \ge 1/n \}<\infty$. How should you do change it to have the conditions of theorem 2?
