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 non-computable probability measures, too. We study, for a given real A, the independence spectrum of A, the set of all B so 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 r.e., then no ∆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 r.e. and P is of PA degree so that P 6≥T A, then A⊕ P ≥T ∅′. 1. Independence and relative randomness The property of independence is central to probability theory. Given a probability space with measure μ, we call two measurable sets A and B independent if