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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.


all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.


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 ${\rm{\Pi }}_n^0$ and ${\rm{\Sigma }}_1^1$ classes at a real.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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