Independence, Relative Randomness, and PA Degrees
Adam R. Day, Jan Reimann · Notre Dame Journal of Formal Logic · 2014
We study pairs of reals that are mutually Martin-Löf random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen’s theorem holds for noncomputable probability measures, too. We study, for a given real A, the independence spectrum of A, the set of all B such that there exists a probability measure μ so that μ{A,B}=0 and (A,B) is (μ×μ)-random. We prove that if A is computably enumerable (c.e.), then no Δ20-set is in the independence spectrum of A. We obtain applications of this fact to PA degrees. In particular, we show that if A is c.e. and P is of PA degree so that P≱TA, then A⊕P≥T∅'.