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On the conjecture of Ulam on the invariance of measure in the Hilbert cube

Volume 152 / 2018

Soon-Mo Jung, Eui Chul Kim Colloquium Mathematicum 152 (2018), 79-95 MSC: Primary 28C10; Secondary 28C20. DOI: 10.4064/cm7145-3-2017 Published online: 5 February 2018


A conjecture of Ulam states that, for any sequence $a = \{ a_i \}$ of positive numbers with $\sum _{i=1}^\infty a_i^2 \lt \infty $, the standard probability measure on the Hilbert cube is invariant under the induced metric $d_a$. We prove this conjecture for all decreasing sequences $a = \{ a_i \}$ of positive numbers with $a_{i+1} = o ( {a_i/\sqrt {i}})$.


  • Soon-Mo JungMathematics Section
    College of Science and Technology
    Hongik University
    30016 Sejong, Republic of Korea
  • Eui Chul KimDepartment of Mathematics Education
    College of Education
    Andong National University
    36729 Andong, Republic of Korea

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