An Application of Martin-Löf Randomness to Effective Probability Theory

  • Mathieu Hoyrup
  • Cristóbal Rojas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5635)


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.


Algorithmic randomness universal test computable analysis effective probability theory Lebesgue integration layerwise computability 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Mathieu Hoyrup
    • 1
  • Cristóbal Rojas
    • 2
  1. 1.LORIA - 615, rue du jardin botaniqueVandœuvre-lès-NancyFrance
  2. 2.Institut de Mathématiques de LuminyMarseille Cedex 9France

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