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

TL;DR

This paper characterizes the conditions under which the Dunford-Schwartz ergodic theorem holds in infinite measure spaces, establishing a precise criterion related to the finiteness of level sets of functions.

Contribution

It provides a necessary and sufficient condition for the pointwise validity of the Dunford-Schwartz ergodic theorem in infinite measure spaces.

Findings

The theorem holds if and only if the measure of the set where f exceeds any positive lambda is finite.

The result applies to functions in the sum of L^1 and L^∞ spaces.

It clarifies the limitations of ergodic theorems in infinite measure contexts.

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 $\lambda > 0$.