Using almost-everywhere theorems from analysis to study randomness

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.