Infinite measures on Cantor spaces

Abstract

We study the set of all infinite full non-atomic Borel measures on a Cantor space X. For a measure from , we define a defective set . We call a measure from non-defective () if . The paper is devoted to the classification of measures from with respect to a homeomorphism. The notions of goodness and clopen values set are defined for a non-defective measure . 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 , we find a good non-defective measure on a Cantor space X with and an aperiodic homeomorphism of X which preserves . The set S of infinite ergodic ℛ-invariant measures on non-simple stationary Bratteli diagrams consists of non-defective measures. For the set 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 .