# Homogeneous probability measures on the Cantor set

## TL;DR

This paper proves that any homeomorphism between measure-zero closed subsets of the Cantor set can be extended to a measure-preserving automorphism when the set is equipped with certain probability measures, including the standard and universal rational measures.

## Contribution

It establishes the extension property for homeomorphisms on measure-zero subsets of the Cantor set under specific probability measures, expanding understanding of measure-preserving transformations.

## Key findings

- Extension of homeomorphisms to measure-preserving automorphisms
- Applicable to standard and universal rational measures
- Enhances the theory of measure-preserving dynamics on the Cantor set

## Abstract

We show that every homeomorphism between closed measure zero subsets extends to a measure preserving auto-homeomorphism, whenever the Cantor set is endowed with a suitable probability measure. This is valid both for the standard product measure, as well as for the universal homogeneous rational measure.