Effectively closed sets of measures and randomness

We show that if a real x ∈ 2^ω is strongly Hausdorff H^h-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.