Infinite measures on Cantor spaces

We study the set $M_{\mathrm{\infty }}(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $\mu$ from $M_{\mathrm{\infty }}(X)$ we define a defective set $M_\mu = {x \in X : for any clopen set U which contains x we have \mu(U) = \infty }$\mathrm{.}Wecallameasure$\mu$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 $M_{\mathrm{\infty }}^0(X)$ 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 $D\subset [0,\mathrm{\infty })$ 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 $S$.