Turing degrees and randomness for continuous measures

Abstract

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every \(\Delta ^0_2\)-degree contains an NCR element.