Effectively closed sets of measures and randomness

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Abstract

We show that if a real x ∈ 2ω is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π10-classes applied to closed sets of probability measures. We use the main result to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman's Theorem.

Original languageEnglish (US)
Pages (from-to)170-182
Number of pages13
JournalAnnals of Pure and Applied Logic
Volume156
Issue number1
DOIs
StatePublished - Nov 2008

All Science Journal Classification (ASJC) codes

  • Logic

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