Topologically equivalent measures in the Cantor space

Abstract:

The Cantor space is realized as a countable product X of two-element sets. The measures $\mu$ and $\nu$ in X are topologically equivalent if there is a homeomorphism h of X onto itself such that $\mu = \nu h$. Let $\mathcal {F}$ be the family of product measures in X which are shift invariant. The members $\mu (r)$ of $\mathcal {F}$ are in one-to-one correspondence with the real numbers r in the unit interval. The relation of topological equivalence partitions the family $\mathcal {F}$ into classes with at most countably many measures each. A class contains only the measures $\mu (r)$ and $\mu (1 - r)$ when r is a rational or a transcendental number. Equivalently, if r is rational or transcendental and $\mu (s)$ is topologically equivalent to $\mu (r)$ then $s = r$ or $s = 1 - r$.