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 Π0 1-classes 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

The main goal of my thesis is the application of logical and computability-theoretic 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