Infinite measures on Cantor spaces

TL;DR

This paper classifies infinite non-atomic measures on Cantor spaces, introducing notions of goodness and defective sets, and explores their homeomorphic properties, including the construction of measures with prescribed value sets and their invariance under homeomorphisms.

Contribution

It provides a classification criterion for good non-defective measures on Cantor spaces and constructs measures with specific properties, extending understanding of measure homeomorphisms.

Findings

Existence of continuum classes of weakly homeomorphic measures.

Construction of measures with prescribed clopen value sets.

Characterization of measures invariant under aperiodic homeomorphisms.

Abstract

We study the set $M_\infty(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $\mu$ from $M_\infty(X)$ we define a defective set $M_\mu = \{x \in X : for any clopen set U which contains x we have \mu(U) = \infty \}$. We call a measure $\mu$ from $M_\infty(X)$ non-defective ($\mu \in M_\infty^0(X)$) if $\mu(M_\mu) = 0$. The paper is devoted to the classification of measures $\mu$ from $M_\infty^0(X)$ with respect to a homeomorphism. The notions of goodness and clopen values set $S(\mu)$ are defined for a non-defective measure $\mu$. We give a criterion when two good non-defective measures are homeomorphic and prove that there exist continuum classes of weakly homeomorphic good non-defective measures on a Cantor space. For any group-like subset $D \subset [0,\infty)$ we find a good non-defective measure $\mu$ on a Cantor space X with $S(\mu) = D$ and an aperiodic homeomorphism of X which preserves $\mu$. The set $S$ of infinite ergodic R-invariant measures on non-simple stationary Bratteli diagrams consists of non-defective measures. For $\mu \in S$ the set $S(\mu)$ is group-like, a criterion of goodness is proved for such measures. We show that a homeomorphism class of a good measure from $S$ contains countably many distinct good measures from $S$.