  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.
