I am not sure if this is the correct forum for my present question. Returning to it after a rather long period and hence, have forgotten the conventions.
I am unable to understand the proof of theorem attributed to U. Krengel as given in Aaronson's ''An Introduction to Infinite Ergodic Theory'' (Th. 1.4.4, pp. 37-39).
Let $T$ be a non-singular transformation of a probability space $( X, \mathcal{B}, m )$. Either, there exists an $m$-abs. cont. $T$-invariant probability measure on $X$, or for all $h \in L^1 (m)$, $$ \frac{1}{n} \sum_{k = 0}^{n - 1} \hat{T}^k h \xrightarrow{m} 0 \textrm{ as } n \rightarrow \infty. $$
The proof is supposed to be using the Lemma below or its $L^1$ version.
Let $( X, \mathcal{B}, m )$ be a prob. space and for all $n \geq 1,\ f_n \in L^2 (m)$ such that $\sup_n \| f_n \|_2 < \infty$. Then, $\exists f \in L^2 (m)$ and $n_k \rightarrow \infty$ s. t. whenever $\{ n'_j \}$ is a subsequence of $\{ n_k \}$, $$ \frac{1}{K^2} \sum_{j = 1}^{K^2} f_{n'_j} \rightarrow f \textrm{ a. e. as } K \rightarrow \infty. $$
The proof is rather long to be typed out here but can someone who has read it previously explain what is going on. It will be appreciated. Thanks.