Validity space of Dunford-Schwartz pointwise ergodic theorem

Abstract: We show that if a $\sigma-$finite infinite measure space $(\Omega,\mu)$ is quasi-non-atomic, then the Dunford-Schwartz pointwise ergodic theorem holds for $f\in \mathcal L^1(\Omega)+\mathcal L^{\infty}(\Omega)$ if and only if $\mu{f\ge \lambda}<\infty$ for all $\lambda>0$.