Independence, Relative Randomness, and PA Degrees

We study pairs of reals that are mutually Martin-L{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgens 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 $mutimesmu$-random. We prove that if $A$ is r.e., then no $Delta^0_2$ 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 $P otgeq_{T} A$, then $A oplus P geq_{T} 0$.