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We show that if a real $x$ is strongly Hausdorff $h$-random, where $h$ is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure $mu$ such that the $mu$-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 $Pi^0_1$-classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostmans Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostmans Theorem.
We study the question, ``For which reals $x$ does there exist a measure $mu$ such that $x$ is random relative to $mu$? We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several co
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n ind
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 co
The paper considers quantitati
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only if it has a