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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 $\\Delta^0_2$ set is in the independence spectrum of $A$. 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