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 $\lambda >0$.