Let $(X, \mathcal{A},\mu)$ be a probability space and $T:X\rightarrow X$ an ergodic transformation. The Birkhoff averages of a function $\phi:X \rightarrow \mathbb{R}$ are defined by $$ \phi_n(x)=\frac{1}{n} \sum_{j=0}^{n-1} \phi\circ T^j(x) $$
Birkhoff ergodic theorem says that $\phi_n(x) \rightarrow c\in \mathbb{R}$ for a.e $x\in X$, where $c$ is the spacial average, under the hypothesis $\phi \in L^1(\mu)$.
Is it true, under same hypothesis, that $\varphi(x)= \sup_n \phi_n(x)$ ( which is finite a.e., by what was said), is integrable? Under some more assumptions ( like $\phi_n \geq 0$, for example)?