USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS

Abstract

AbstractWe 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 realz, the additional randomness strengths needed for the following are equivalent.(1)all effectively closed classes containingzhave density 1 atz.(2)all nondecreasing functions with uniformly left-c.e. increments are differentiable atz.(3)zis 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.