Theory of Computing Systems

, Volume 48, Issue 3, pp 465–485 | Cite as

Randomness on Computable Probability Spaces—A Dynamical Point of View

  • Peter Gács
  • Mathieu HoyrupEmail author
  • Cristóbal Rojas


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.


Schnorr randomness Birkhoff’s ergodic theorem Computable measures 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Computer Science DepartmentBoston UniversityBostonUSA
  2. 2.INRIA NancyVandœuvre-lès-NancyFrance
  3. 3.The Fields Institute for Research in Mathematical SciencesTorontoCanada

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