From Haar to Lebesgue via Domain Theory, Revised version

TL;DR

This paper demonstrates that various topological group structures on the Cantor set induce Haar measures that, through domain theory, correspond to Lebesgue measure on the interval, revealing a deep measure-theoretic connection.

Contribution

It shows that all Haar measures from different group structures on the Cantor set are equivalent to those from abelian structures and establishes measure-preserving maps to the interval.

Findings

Haar measures from different group structures are equivalent.

Maps induced by total orders on the Cantor set send Haar measure to Lebesgue measure.

Existence of Borel isomorphisms between different group structures.

Abstract

If ${\mathcal C}\simeq 2^{\mathbb N}$ denotes the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from ${\mathcal C}$ into $[0,1]$ sends Haar measure to Lebesgue measure on the interval. In fact, ${\mathcal C}$ admits many distinct topological group structures. In this note, we show that the Haar measures induced by these distinct group structures are share this property. We prove this by showing that Haar measure for any group structure is the same as Haar measure induced by a related abelian group structure. Moreover, each abelian group structure on ${\mathcal C}$ supports a natural total order that determines a map onto the unit interval that is monotone, and hence sends intervals in ${\mathcal C}$ to subintervals of the unit interval. Using techniques from domain theory, we show this implies this map sends Haar measure on ${\mathcal C}$ to Lebesgue measure on the interval, and we then use this to contract a Borel isomorphism between any two group structures on ${\mathcal C}$.