Uniform sets for infinite measure-preserving systems

TL;DR

This paper introduces the concept of uniform sets in infinite measure-preserving systems and shows their existence is tied to generating the sigma-algebra, leading to a classification result for ergodic transformations.

Contribution

It defines uniform sets for infinite measure-preserving systems and proves their existence correlates with generating the sigma-algebra, enabling a new isomorphism classification.

Findings

Uniform sets exist when they generate the sigma-algebra.

Any ergodic, measure-preserving transformation is isomorphic to a minimal homeomorphism.

The resulting homeomorphism admits a unique, up to scaling, invariant Radon measure.

Abstract

The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist as much as they generate the underlying $\sigma$-algebra. This leads to the result that any ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space is isomorphic to a minimal homeomorphism on a locally compact metric space which admits a unique, up to scaling, invariant Radon measure.