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In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on Martin-Löf randomness and the existence of a universal randomness test. We then prove(More)
In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable(More)
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)(More)
We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and(More)
We consider the dynamical behavior of Martin-Löf random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior(More)
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(More)
This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of ran-domness (with respect to computable measures) in terms of randomness deficiency functions. A formula that(More)
We show that computability of the Radon-Nikodym derivative of a measure µ absolutely continuous w.r.t. some other measure λ can be reduced to a single application of the non-computable operator EC, which transforms enumeration of sets (in N) to their characteristic functions. We also give a condition on the two measures (in terms of the computability of the(More)
We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of(More)
A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior of a typical point of the system. It was proved in [2] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative,(More)