Luzin's (N) and randomness reflection

@article{Pauly2020LuzinsA,
  title={Luzin's (N) and randomness reflection},
  author={Arno Pauly and L. Westrick and Liang Yu},
  journal={arXiv: Logic},
  year={2020}
}
  • Arno Pauly, L. Westrick, Liang Yu
  • Published 2020
  • Mathematics
  • arXiv: Logic
  • We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $\Delta^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-Lof randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random as well. If additionally $f$ is known to have bounded variation, then we show $f… CONTINUE READING

    Figures from this paper

    References

    SHOWING 1-10 OF 29 REFERENCES
    Randomness, relativization and Turing degrees
    • 144
    • PDF
    On the close interaction between algorithmic randomness and constructive/computable measure theory
    • 1
    • PDF
    Constructive equivalence relations on computable probability measures
    • 12
    • PDF
    Algorithmic Randomness and Complexity
    • 763
    • PDF
    Proof of a conjecture of Friedman
    • 11
    • Highly Influential
    • PDF
    Computability and randomness
    • 556
    • PDF
    Some remarks on set theory
    • 87
    • PDF
    Computability of probability measures and Martin-Löf randomness over metric spaces
    • 112
    • PDF
    Borel Sets and Hyperdegrees
    • H. Friedman
    • Mathematics, Computer Science
    • J. Symb. Log.
    • 1973
    • 7
    • Highly Influential