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 non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so 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 r.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 r.e.\ and $P$ is of PA degree so that $P/{\ge }_{T}A$, then $A\oplus P{\ge }_T{0}^{\prime }$.