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