  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 $A_{i}$ with effectively summable measures,
there are computable points which are not contained in infinitely many $A_{i}$.
  As a consequence of this we obtain the existence of computable points which
follow the \emph{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 Lipshitz 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.
