Independence, relative randomness, and PA degrees

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 δ⁰ ₂-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 ≱T A, then A ⊕ P ≥_T ; ∅.