Abstract:
We study the set $M(X)$ of full non-atomic Borel measures $\mu$ on a non-compact locally compact Cantor set $X$. The set $\mathfrak{M}_\mu = \{x \in X\colon \text{for any compact open set}\ U \ni x \text{ we have}\ \mu(U) = \infty \}$ is called defective. $\mu$ is non-defective if $\mu(\mathfrak{M}_\mu) = 0$. The set $M^0(X) \subset M(X)$ consists of probability and infinite non-defective measures. We classify the measures from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and the compact open values set $S(\mu)$ are defined. A criterion when two good measures are homeomorphic is given. For a group-like set $D$ and a locally compact zero-dimensional metric space $A$ we find a good non-defective measure $\mu$ on $X$ such that $S(\mu) = D$ and $\mathfrak{M}_\mu$ is homeomorphic to $A$. We give a criterion when a good measure on $X$ can be extended to a good measure on the compactification of $X$.
Key words and phrases:Borel measures, locally compact Cantor set, compactification, invariant measures.
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