On the conjecture of Ulam on the invariance of measure in the Hilbert cube
Volume 152 / 2018
Colloquium Mathematicum 152 (2018), 79-95
MSC: Primary 28C10; Secondary 28C20.
DOI: 10.4064/cm7145-3-2017
Published online: 5 February 2018
Abstract
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 $a_{i+1} = o ( {a_i/\sqrt {i}})$.
