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Effectively closed sets of measures and randomness
 Ann. Pure Appl. Logic
"... We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, 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 µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to con ..."
Abstract

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We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, 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 µmeasure 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 Π0 1classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1
THESIS SUMMARY
"... The main goal of my thesis is the application of logical and computabilitytheoretic techniques to better understand the foundational nature of structures and theorems from different branches of mathematics. To achieve this goal, I examined the effective (i.e. computable) content of structures and t ..."
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The main goal of my thesis is the application of logical and computabilitytheoretic techniques to better understand the foundational nature of structures and theorems from different branches of mathematics. To achieve this goal, I examined the effective (i.e. computable) content of structures and theorems from several different branches of math, including computability theory and model theory (Chapter 1), fractal dimension