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

TL;DR

This paper proves Ulam's conjecture that the standard measure on the Hilbert cube remains invariant under a specific metric, completing previous partial proofs by classifying cylinders and addressing overlooked cases.

Contribution

It provides a complete proof of Ulam's conjecture by classifying cylinders and handling degenerate cases previously neglected.

Findings

Confirmed invariance of measure under the metric for all cases

Classified cylinders into non-degenerate and degenerate types

Resolved the conjecture fully by addressing overlooked cases

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 $\sum\limits_{i=1}^\infty a_i^2 < \infty$. 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.