Effective Geometric Measure Theory 

(Abstract)

It is often pointed out that Martin-Loef randomness is an adequate formulation of randomness since it is robust with respect to different approaches to randomness: measure-theoretic typicality, unpredictability, incompressibility. However, this robustness appears rather fragile when one extends the study beyond Lebesgue measure to geometric outer measures, in particular Hausdorff measures.

We study one of the central results of geometric measure theory, Frostman's Lemma, in the effective setting. We show that it yields an unexpected dichotomy of randomness notions which does not appear in the Lebesgue case. We further discuss how various characterizations of Hausdorff dimension give rise to an apparently linear hierarchy of randomness notions, and how these in turn can be used to derive results about computational properties of Martin-Loef random reals.