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Turing degrees and randomness for continuous measures

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every Δ20 -degree contains an NCR element.

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Work Title Turing degrees and randomness for continuous measures
Access
Open Access
Creators
  1. Mingyang Li
  2. Jan Reimann
Keyword
  1. Continuous Measures
  2. Turing Degrees
  3. Continuous Measure
  4. Turing Degree
  5. Probability Measure
  6. Non Random Sampling
  7. Continuous Probability
  8. Null Set
  9. Hausdorff Measure
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Archive for Mathematical Logic
Publication Date February 1, 2024
Publisher Identifier (DOI)
  1. https://doi.org/10.1007/s00153-023-00873-7
Deposited May 26, 2025

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Work History

Version 1
published

  • Created

    May 26, 2025 18:38 by jsr25

  • Updated

    May 26, 2025 18:38 by [unknown user]

  • Added Creator Mingyang Li

    May 26, 2025 18:40 by jsr25

  • Added Creator Jan Reimann

    May 26, 2025 18:40 by jsr25

  • Updated Keyword, Publisher, Publisher Identifier (DOI), and 2 more Show Changes

    May 26, 2025 18:40 by jsr25

    Keyword
    • Continuous Measures, Turing Degrees, Continuous Measure, Turing Degree, Probability Measure, Non Random Sampling, Continuous Probability, Null Set, Hausdorff Measure
    Publisher
    • Archive for Mathematical Logic
    Publisher Identifier (DOI)
    • https://doi.org/10.1007/s00153-023-00873-7
    Description
    • We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every Δ20 -degree contains an NCR element.
    Publication Date
    • 2024-02-01
  • Updated

    May 26, 2025 18:40 by jsr25

  • Updated

    May 26, 2025 18:40 by jsr25

  • Updated Creator Mingyang Li

    May 26, 2025 18:40 by jsr25

  • Updated Creator Jan Reimann

    May 26, 2025 18:40 by jsr25

  • Added NCR and T-degrees sn-article.pdf

    May 26, 2025 18:40 by jsr25

  • Updated

    May 26, 2025 18:42 by jsr25

  • Updated License Show Changes

    May 26, 2025 18:45 by jsr25

    License
    • https://rightsstatements.org/page/InC/1.0/
  • Published

    May 26, 2025 18:45 by jsr25

  • Updated

    May 26, 2025 22:05 by [unknown user]