a question about the definition:

given measurable dynamic system $ ( X, \mathcal{B}, T, \mu)$, $ \mu \circ T^{-1}=\mu$ ergodic.

$\phi \in L^{\infty}$ is coboundary with $\int \phi d\mu =0 $, means the decomposition: $\exists \xi $, s.t $ \phi=\xi \circ T-\xi$

$\phi$ is not coboundary, means the decomposition: $\exists \xi $, non zero function $ \beta$, s.t $ \phi=\beta+ \xi \circ T-\xi$

My question is what space this decomposition lies in? $ L^1$ or $ L^2$? that means $\xi, \beta$ are in $ L^1$ or $ L^2$?

I know some authors use $ L^2$, is this the original definition?

Thanks!