Randomness for Non-computable Measures

Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gács, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probability measure, we turn our attention to Levin’s neutral measures, for which all sequences are random. We show that every PA degree computes a neutral measure. We also show that a neutral measure has no least Turing degree representation and explain why the framework of the continuous degrees (a substructure of the enumeration degrees studied by Miller) can be used to determine the computational complexity of neutral measures. This allows us to show that the Turing ideals below neutral measures are exactly the Scott ideals. Since X ∈ 2 is an atom of a neutral measure μ if and only if it is computable from (every representation of) μ, we have a complete understanding of the possible sets of atoms of a neutral measure. One simple consequence is that every neutral measure has a Martin-Löf random atom. 1. Defining randomness Let X be an element of Cantor space and μ a Borel probability measure on Cantor space. What should it mean for X to be random with respect to μ? In the case that μ is the Lebesgue measure, then the theory of μrandomness is well developed (for recent treatises on the subject the reader is referred to Downey and Hirschfeldt, and Nies [2, 13]). In fact if μ is a computable measure, then early work of Levin showed that μ-randomness can be seen as essentially a variant on randomness for Lebesgue measure [10]. This leaves the question of how to define randomness if μ is non-computable. We will show that the two approaches that have previously been used to define μ-randomness, for non-computable μ, are equivalent. Later, in Theorem 4.12, we will provide another characterization of μ-randomness using the enumeration degrees. Last compilation: September 8, 2011 Last time the following date was changed: November 22, 2010. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 03D30. The second author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.