A constructive Borel–Cantelli lemma. Constructing orbits with required statistical properties

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel–Cantelli lemma: given a sequence (constructive in some way) of sets Ai with effectively summable measures, there are computable points which are not contained in infinitely many Ai.As a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and a certain “logarithmic” speed of convergence of Birkhoff averages over Lipschitz observables. This is applied to uniformly hyperbolic systems, piecewise expanding maps, systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base.