Outer measure preserving ergodic transformations generate the Carath\'eodory definition of measurable sets

TL;DR

This paper demonstrates that outer measure preserving ergodic transformations universally generate the Carathéodory definition of measurable sets, extending previous specific examples to a broad class of transformations.

Contribution

It proves that the property of generating Carathéodory's definition holds for all outer measure preserving ergodic transformations on measure spaces.

Findings

All outer measure preserving ergodic transformations generate Carathéodory's definition.

Previously known specific examples are encompassed by this general class.

The result unifies understanding of measure generation in ergodic theory.

Abstract

It is known that there are specific examples of ergodic transformations on measure spaces for which the calculation of the outer measure of transformation invariant sets leads to a condition closely resembling Carath\'eodory's condition for sets to be measurable. It is then natural to ask what functions are capable of `generating', that is leading to, the Carath\'eodory definition in the same way. The present work answers this question by showing that the property of generating Carath\'eodory's definition holds for the general class of outer measure preserving ergodic transformations on measure spaces. We further show that the previously found specific examples of functions generating Carath\'eodory's definition fall into this family of transformations.