Milan Journal of Mathematics

, Volume 81, Issue 1, pp 121-151

First online:

Non-Archimedean Probability

  • Vieri BenciAffiliated withDipartimento di Matematica Applicata, Universitá degli Studi di PisaDepartment of Mathematics, College of Science, King Saud University Email author 
  • , Leon HorstenAffiliated withDepartment of Philosophy, University of Bristol
  • , Sylvia WenmackersAffiliated withFaculty of Philosophy, University of Groningen

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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