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

Abstract

A conjecture of Ulam states that the standard probability measure $\pi$ on the Hilbert cube $I^{\omega }$ is invariant under the induced metric $d_a$ when the sequence $a=\{a_i\}$ of positive numbers satisfies the condition $\sum _{i=1}^{\infty }{a}_i^2<\infty$. This conjecture was proved in \cite{jung1} when $E_1$ is a non-degenerate subset of $M_a$. In this paper, we prove the conjecture of Ulam completely by classifying cylinders as non-degenerate and degenerate cylinders and by treating the degenerate case that was overlooked in the previous paper.

arXiv commentIn a paper of Jung [The conjecture of Ulam on the invariance of measure on Hilbert cube, J. Math. Anal. Appl. 481 (2020), no. 2, 123500], the conjecture of Ulam on the invariance of measure was proved in the affirmative only for non-degenerate case. In this paper we provide complete proof for Ulam's conjecture, whether the related sets are degenerate or non-degenerate. 48 pages, 1 figure