The validity space of Dunford-Schwartz pointwise ergodic theorem

We show that if a σ-finite infinite measure space (Ω,μ) is quasi-non-atomic, then the Dunford-Schwartz pointwise ergodic theorem holds for f∈L1(Ω)+L∞(Ω) if and only ifμ{|f|≥λ}<∞ for all λ>0.