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 $\lambda >0$.

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