# A comment on ergodic theorem for amenable groups

## TL;DR

This paper extends the ergodic theorem for amenable groups by relaxing the temperedness condition on F{46}lner sequences, replacing it with a mixing condition on the function.

## Contribution

It introduces a new ergodic theorem for amenable groups that does not require tempered F{46}lner sequences, relying instead on a mixing condition.

## Key findings

- Ergodic theorem holds without tempered F{46}lner sequences.
- A mixing condition on functions replaces the need for temperedness.
- The result broadens applicability of ergodic theorems to more general group actions.

## Abstract

We prove a version of ergodic theorem for an action of an amenable group, where a F{\o} lner sequence needs not to be tempered. Instead, it is assumed that a function satisfies certain mixing condition.