Independence, Relative Randomness, and PA Degrees

Abstract

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 $\mu$ so that $\mu \{A,B\}=0$ and $(A,B)$ is $(\mu \times \mu )$-random. We prove that if $A$ is computably enumerable (c.e.), then no ${\Delta }_2^0$-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 $P{\ngeqq }_{\mathrm{T}}A$, then $A\oplus P{\ge }_{\mathrm{T}}\varnothing \text{'}$.