Let $(X,\mu)$ be a probability space and $(\Omega,\beta)$ be the corresponding Bernoulli space, that is, $\Omega = X^\mathbb N$ and $\beta = \mu^\mathbb N$. Let $T:\Omega \to \Omega$ be the (one-sided) shift map.
I'm looking for a reference for the following result, which I believe is true:
If a measurable function $\varphi$ is constant (almost everywhere) on the fibers of $T^{n}$ for every $n\geq 1$ then $\varphi$ must be constant almost everywhere. This sounds like a well-known result, but I couldn't find it in any ergodic theory book I know.
Remark: I don't assume $X$ to be finite. If it helps, one might suppose that $\varphi$ is in $L^1$ or $L^\infty$.