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A Subtle Symmetry of Lebesgue’s Measure

Author

Listed:
  • Muhammed Uludağ

    (Galatasaray University)

  • Hakan Ayral

    (Galatasaray University)

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.

Suggested Citation

  • Muhammed Uludağ & Hakan Ayral, 2019. "A Subtle Symmetry of Lebesgue’s Measure," Journal of Theoretical Probability, Springer, vol. 32(1), pages 527-540, March.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:1:d:10.1007_s10959-017-0804-y
    DOI: 10.1007/s10959-017-0804-y
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