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arXiv · 1807.05624

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

Soon-Mo Jung
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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.

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Soon-Mo Jung. 2018-07-15. The conjecture of Ulam on the invariance of measure on Hilbert cube. https://arxiv.org/abs/1807.05624

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