The conjecture of Ulam on the invariance of measure on Hilbert cube

Soon-Mo Jung

Published 2018-07-15Version 1

A conjecture of Ulam states that the standard probability measure $\pi$ on the Hilbert cube $I^\omega$ is invariant under the induced metric $d_a$ provided the sequence $a = \{ a_i \}$ of positive numbers satisfies the condition $\sum\limits_{i=1}^\infty a_i^2 < \infty$. In this paper, we prove this conjecture in the affirmative. More precisely, we prove that if there exists a surjective $d_a$-isometry $f : E_1 \to E_2$, where $E_1$ and $E_2$ are Borel subsets of $I^\omega$, then $\pi(E_1) = \pi(E_2)$.