Effectively closed sets of measures and randomness

TL;DR

The paper establishes a connection between strong Hausdorff $h$-randomness and randomness for certain continuous measures, introducing new measure construction methods and applying them to classical results like Frostman's Lemma.

Contribution

It introduces a novel measure construction technique based on effective transformations and applies it to unify and extend results in randomness and Hausdorff measure theory.

Findings

Strong Hausdorff $h$-randomness implies measure-based randomness.

New proof of Frostman's Lemma using effective measure construction.

Collapse of various randomness notions for Hausdorff measures.

Abstract

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.