The validity space of Dunford-Schwartz ergodic theorem for infinite measure

Abstract: We show that if $(\Omega,\mu)$ is an infinite measure space, the pointwise Dunford-Shwartz ergodic theorem holds for $f \in \mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$ if and only if $\mu{f>\lambda}<\infty$ for all $\lambda > 0$.