Ergodicity -- A measure preserving transformation $T$ on the space $(X, \mathcal{B} , \mu)$ is called ergodic iff $\forall B \in\mathcal{B}$ satisfying $T^{-1} B = B$ we have $\mu(B) = 0$ or 1.
Let $T$ be a measure preserving transformation of a space $(X, \mathcal{B}, \mu )$. Then the following are equivalent:
$T$ is ergodic;
For all f $\in$ L$^{1}$ (X, $\mathcal{B}$, $\mu$) satisfying $f \circ T = f$ a.e. then $f$ is constant a.e.
The book I am reading says that "we can replace $L^{1}$ in above proposition by measurable or $L^{2}$". Why is that true? Can anyone help me on this?
I suppose your measure $\mu $ is a finite measure. Suppose $T$ is erodic. $f$ is measurable and $f\circ T=f$ a.e.. Then $g\circ f \circ T=g\circ f$ a.e. for any $g$ measurable. Take a sequence of bounded measurable functions $g$ on $\mathbb R$ converging pointwise to the identity function and apply the $L^{1}$ case to each $g_n\circ f$ to see that $g_n\circ f$ is a constant for each $n$. Hence $f =\lim g_n\circ f$ is also almost surely constant.