2012/03/30 by Karpel, O.
#28C15 (Secondary) #28D05 #37A05 #37B05 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1204.0027
We study the set M(X) of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set X. For an infinite measure μ in M(X), the set \mathfrakMμ= \x ∈ X : for any compact open set U \ni x we have μ(U) = ∞ \ is called defective. We call μ non-defective if μ(\mathfrakMμ) = 0. The class M0(X) ⊂ M(X) consists of probability measures and infinite non-defective measures. We classify measures μ from M0(X) with respect to a homeomorphism. The notions of goodness and compact open values set S(μ) are defined. A criterion when two good measures from M0(X) are homeomorphic is given. For any group-like D ⊂ [0,1) we find a good probability measure μ on X such that S(μ) = D. For any group-like D ⊂ [0,∞) and any locally compact, zero-dimensional, metric space A we find a good non-defective measure μ on X such that S(μ) = D and \mathfrakMμ is homeomorphic to A. We consider compactifications cX of X and give a criterion when a good measure μ∈ M0(X) can be extended to a good measure on cX.