Let $(S, \mathcal{A}, \mu)$ be a probability space, and let $\theta$ be a µ-measure preserving transformation on it. Let $g \ge 0$ with $g\in L^1$. I need proof that $\lim\limits_{n\to\infty} \dfrac{\sum_{k=1}^n g(\theta^k x)}{n} <\infty$.
Use the pointwise ergodic theorem ($\theta$ isn't ergodic), I have $ \lim\limits_{n\to\infty} \dfrac{\sum_{k=1}^n g(\theta^k x)}{n} = \overline{g} $ with $ \overline{g} \in L^1$. Since $ \overline{g} \in L^1$ and $\overline{g} \ge 0$ so $\overline{g}<\infty$ a.e. Hence we obtain $ \lim\limits_{n\to\infty} \dfrac{\sum_{k=1}^n g(\theta^k x)}{n} = \overline{g} <\infty$.
Is my proof right?