Randomness for non-computable measures
Adam R. Day, Joseph S. Miller · Transactions of the American Mathematical Society · 2013
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 ω X\in 2^\omega is an atom of a neutral measure μ \mu if and only if it is computable from (every representation of) μ \mu , 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.