A Subtle Symmetry of Lebesgue’s Measure

Muhammed Uludağ () and Hakan Ayral
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Muhammed Uludağ: Galatasaray University
Hakan Ayral: Galatasaray University

Journal of Theoretical Probability, 2019, vol. 32, issue 1, 527-540

Abstract: 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 $$\mathrm {PGL}(2,\mathbb {Z})$$ PGL ( 2 , Z ) . Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.

Keywords: Farey tree; Modular group; Boundary; Measure; Lebesgue measure; Stern–Brocot tree; Calkin–Wilf tree; 28C10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-017-0804-y

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