Milan Journal of Mathematics

, Volume 81, Issue 1, pp 121–151

Non-Archimedean Probability


DOI: 10.1007/s00032-012-0191-x

Cite this article as:
Benci, V., Horsten, L. & Wenmackers, S. Milan J. Math. (2013) 81: 121. doi:10.1007/s00032-012-0191-x


We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity.

Mathematics Subject Classification (2010)

60A05 03H05 


Probability axioms of Kolmogorov nonstandard models fair lottery non-Archimedean fields 

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica ApplicataUniversitá degli Studi di PisaPisaItaly
  2. 2.Department of Mathematics, College of ScienceKing Saud UniversityRiyadhSaudi Arabia
  3. 3.Department of PhilosophyUniversity of BristolBristolUnited Kingdom
  4. 4.Faculty of PhilosophyUniversity of GroningenGroningenThe Netherlands

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