Elementary numerosity and measures

TL;DR

This paper introduces elementary numerosity, a non-Archimedean set function resembling finite cardinality, and demonstrates its connection to measures, showing that non-atomic finitely additive measures can be derived from it.

Contribution

It defines elementary numerosity and establishes a fundamental link between this concept and non-atomic finitely additive measures.

Findings

Every non-atomic finitely additive measure can be obtained from an elementary numerosity.

Elementary numerosity takes values in a non-Archimedean field and generalizes finite cardinality.

Applications of the main result are discussed in the paper.

Abstract

In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite cardinality. We investigate the relationships between this notion and the notion of measure. The main result is that every non-atomic finitely additive measure is obtained from a suitable elementary numerosity by simply taking its ratio to a unit. In the last section we give applications to this result.