Applications of Effective Probability Theory to Martin-Löf Randomness

We pursue the study of the framework of layerwise computability introduced in [HR09a] and give three applications. (i) We prove a general version of Birkhoff’s ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable. This result significantly improves [V’y97, Nan08]. (ii) We provide a general framework for deriving sharper theorems for random points, sensitive to the speed of convergence. This offers a systematic approach to obtain results in the spirit of Davie [Dav01]. (iii) Proving an effective version of a theorem of Ulam, we positively answer a question raised in [Fou08]: can random Brownian paths reach any random number? All this shows that layerwise computability is a powerful framework to study Martin-Löf randomness, with a wide range of applications.