I have been reading some papers in Infinite Ergodic theory, and I think that the following fact is a standard argument in the subject, but I have not been able to figure out why it is true. I will state it in the form that I need it, so some hypothesis could be not necessary.
Let $(X,\mathbb{B},\mu,T)$ be a $\sigma-$finite measure space equipped with a $\mu-$invariant transformation $\sigma:X\to X$. We also suppose that the transformation $\sigma$ is conservative and ergodic. The for a measurable function $f:X\to \mathbb{R}$, if $f(Tx)\geq f(x)$ for every $x$, then $f$ is constant a.e.
Maybe if I can get that $f(Tx)\leq f(x)$ then by ergodicity it would follow that $f$ is constant a.e., but I do not know how to prove that inequality. Any idea would be appreciated. Thanks in advance.