  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 Frostman's Lemma, to
derive a collapse of randomness notions for Hausdorff measures, and to provide
a characterization of effective Hausdorff dimension similar to Frostman's
Theorem.
