Infinite measures on Cantor spaces

Abstract

We study the set $M_\infty(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $μ$ from $M_\infty(X)$ we define a defective set $M_μ= \{x \in X : for any clopen set U which contains x we have μ(U) = \infty \}$. We call a measure $μ$ from $M_\infty(X)$ non-defective ($μ\in M_\infty^0(X)$) if $μ(M_μ) = 0$. The paper is devoted to the classification of measures $μ$ from $M_\infty^0(X)$ with respect to a homeomorphism. The notions of goodness and clopen values set $S(μ)$ are defined for a non-defective measure $μ$. 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 $μ$ on a Cantor space X with $S(μ) = D$ and an aperiodic homeomorphism of X which preserves $μ$. The set $S$ of infinite ergodic R-invariant measures on non-simple stationary Bratteli diagrams consists of non-defective measures. For $μ\in S$ the set $S(μ)$ 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 $S$.