## Effectively closed sets of measures and randomness

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Venue: | Ann. Pure Appl. Logic |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Reimann_effectivelyclosed,

author = {Jan Reimann},

title = {Effectively closed sets of measures and randomness},

journal = {Ann. Pure Appl. Logic},

year = {},

pages = {170--182}

}

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### Abstract

We show that if a real x ∈ 2ω is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µ-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 Π0 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. 1