Effectively closed sets of measures and randomness

Abstract

We show that if a real x ∈ 2 ω is strongly Hausdorff H h -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 Π 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.

Cone avoidance and randomness preservation

“…The length-invariant case of Theorem 5.3 is due to Reimann and Kjos-Hanssen (see [25,Theorem 14,Corollary 23]). Our proof of Theorem 5.3 is similar to Reimann's proof [25] of the length-invariant case.…”

“…Remark 1.12. Recall from [6,7,25,26] the notion of µ-randomness where µ is a Borel probability measure. Namely, X is said to be µ-random if X / ∈ i U i whenever U i is uniformly Σ 0 1 relative to µ and µ(U i ) ≤ 2 −i for all i.…”

“…See Theorem 5.8 below. We first prove a generalization of the Effective Capacitability Theorem from [25].…”

Computability theory, algorithmic randomness and Turing's anticipation

“…In this area, Jan Reimann [114] gave a new and simpler proof of a classical theorem called Frostman's Lemma. An even more notable example is due to Simpson [126].…”

Randomness and Semimeasures

“…We say that an enumeration of X is a representation of Θ(X). There are a number of equivalent ways to carry this out (see [Rei08] or [DM13]), the easiest one being to define Θ(X) to be the measure (if it exists and is unique) contained in B i for each i ∈ X.…”