Using almost-everywhere theorems from analysis to study randomness

Kenshi Miyabe, André Nies, Jing Zhang

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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 languageEnglish
Pages (from-to)305-331
Number of pages27
JournalBulletin of Symbolic Logic
Issue number3
StatePublished - 1 Sep 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 The Association for Symbolic Logic.


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


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