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 and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).

Abstract. 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 characterizations of effective notions of measurability and integrability in terms of layerwise computability. On the one hand it gives a simple way of handling effective measure theory, on the other hand it provides powerful tools to study Martin-Löf randomness, as illustrated in a sequel paper.

Abstract. 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 only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

We consider the question of computing invariant measures from an abstract point of view. We work in a general framework (computable metric spaces, computable measures and functions) where this problem can be posed precisely. We consider invariant measures as fixed points of the transfer operator and give general conditions under which the transfer operator is (sufficiently) computable. In this case, a general result ensures the computability of isolated fixed points and hence invariant measures (in given classes of "regular" measures). This implies the computability of many SRB measures.On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two interesting examples of computable dynamics, one having an SRB measure which is not computable and another having no computable invariant measure at all, showing some subtlety in this kind of problems.