A subtle symmetry of Lebesgue's measure

TL;DR

This paper reveals a symmetry in Lebesgue measure via Farey tree boundary representation, linked to automorphisms of PGL(2,Z), leading to new measures potentially significant in arithmetic contexts.

Contribution

It introduces a novel symmetry of Lebesgue measure connected to Farey tree automorphisms and constructs three new measures with possible arithmetic importance.

Findings

Lebesgue measure represented as a boundary measure of the Farey tree

Identification of a symmetry related to Dyer's outer automorphism of PGL(2,Z)

Introduction of three new measures on the unit interval

Abstract

We represent the Lebesgue measure on the unit interval as a boundary measure of the Farey tree and show that this representation has a certain symmetry related to the tree automorphism induced by Dyer's outer automorphism of the group PGL(2,Z). Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.