### 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 μ so that μ{A,B} = 0 and {A,B} is {μ × μ}-random. We prove that if A is computably enumerable (c.e.), then no δ^{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 c.e. and P is of PA degree so that P ≱T A, then A ⊕ P ≥_{T} ; ∅.

Original language | English (US) |
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Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Notre Dame Journal of Formal Logic |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - Feb 5 2014 |

### All Science Journal Classification (ASJC) codes

- Logic

## Fingerprint Dive into the research topics of 'Independence, relative randomness, and PA degrees'. Together they form a unique fingerprint.

## Cite this

*Notre Dame Journal of Formal Logic*,

*55*(1), 1-10. https://doi.org/10.1215/00294527-2377842