Abstract
Let (X, ρ) be a complete separable metric space and M be the set of all probability Borel measures on X. We show that if the space M is equipped with the weak topology, the set of measures having the upper (resp. lower) correlation dimension zero is residual. Moreover, the upper correlation dimension of a typical (in the sense of Baire category) measure is estimated by means of the local lower entropy and local upper entropy dimensions of X.
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