## Abstract

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin-Löf (ML) randomness. We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent. (1) all effectively closed classes containing z have density 1 at z. (2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z. (3) z is a Lebesgue point of each lower semicomputable integrable function. We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff's pointwise ergodic theorem. Lastly, we study randomness notions related to density of and classes at a real.

Original language | English |
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Pages (from-to) | 305-331 |

Number of pages | 27 |

Journal | Bulletin of Symbolic Logic |

Volume | 22 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2016 The Association for Symbolic Logic.

### Funding

Miyabe was supported by JSPS. Nies was supported by the Marsden fund of New Zealand. Zhang was supported by a University of Auckland summer scholarship.

Funders | Funder number |
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Marsden fund of New Zealand | |

University of Auckland | |

Japan Society for the Promotion of Science |

## Keywords

- Lebesgue density
- almost-everywhere theorem
- ergodic theory
- randomness