### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Logic

### Cite this

*Notre Dame Journal of Formal Logic*,

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

}

*Notre Dame Journal of Formal Logic*, vol. 55, no. 1, pp. 1-10. https://doi.org/10.1215/00294527-2377842

**Independence, relative randomness, and PA degrees.** / Day, Adam R.; Reimann, Jan Severin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Independence, relative randomness, and PA degrees

AU - Day, Adam R.

AU - Reimann, Jan Severin

PY - 2014/2/5

Y1 - 2014/2/5

N2 - 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 δ0 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 ; ∅.

AB - 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 δ0 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 ; ∅.

UR - http://www.scopus.com/inward/record.url?scp=84893367418&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893367418&partnerID=8YFLogxK

U2 - 10.1215/00294527-2377842

DO - 10.1215/00294527-2377842

M3 - Article

AN - SCOPUS:84893367418

VL - 55

SP - 1

EP - 10

JO - Notre Dame Journal of Formal Logic

JF - Notre Dame Journal of Formal Logic

SN - 0029-4527

IS - 1

ER -