Journal of Theoretical Probability

, Volume 32, Issue 1, pp 527–540 | Cite as

A Subtle Symmetry of Lebesgue’s Measure

  • Muhammed UludağEmail author
  • Hakan Ayral


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


Farey tree Modular group Boundary Measure Lebesgue measure Stern–Brocot tree Calkin–Wilf tree 

Mathematics Subject Classification (2010)




This research is funded by a Galatasaray University Research Grant and TÜBİTAK grant 115F412. The second named author was also funded by the TÜBİTAK grant 113R017. We are grateful to an anonymous referee for shortening the proof of the expression (18).


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Authors and Affiliations

  1. 1.Department of MathematicsGalatasaray UniversityBeşiktaşTurkey

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