# Validity space of Dunford-Schwartz pointwise ergodic theorem

## TL;DR

This paper characterizes the conditions under which the Dunford-Schwartz pointwise ergodic theorem holds in quasi-non-atomic infinite measure spaces, linking it to the finiteness of measure of level sets of functions.

## Contribution

It provides a necessary and sufficient condition for the validity of the ergodic theorem in certain infinite measure spaces.

## Key findings

- Ergodic theorem holds iff measure of {f ≥ λ} is finite for all λ>0
- Characterizes the validity of the theorem in quasi-non-atomic spaces
- Links measure properties of functions to ergodic convergence

## 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 $\lambda>0$.