Effectively closed sets of measures and randomness

Annals of Pure and Applied Logic 156 (1):170-182 (2008)
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Abstract

We show that if a real x2ω is strongly Hausdorff -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 continuous transformations and a basis theorem for -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

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References found in this work

Degrees of Unsolvability of Continuous Functions.Joseph S. Miller - 2004 - Journal of Symbolic Logic 69 (2):555 - 584.
On partial randomness.Cristian S. Calude, Ludwig Staiger & Sebastiaan A. Terwijn - 2006 - Annals of Pure and Applied Logic 138 (1):20-30.

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