Cardinal functions of purely atomic measures

TL;DR

This paper investigates the possible ranges of cardinal functions of purely atomic measures and explores the set-theoretic and topological properties of uniquely obtained measure values.

Contribution

It characterizes which subsets of the codomain can be realized as the range of the cardinal function of purely atomic measures.

Findings

Identifies conditions for sets to be ranges of the cardinal function.

Analyzes the set-theoretic properties of uniquely obtained measure values.

Provides examples of measures with prescribed cardinal function ranges.

Abstract

Let $\mu$ be a purely atomic measure. By $f_\mu:[0,\infty)\to\{0,1,2,\dots,\omega,\mathfrak{c}\}$ we denote its cardinal function $f_{\mu}(t)=\vert\{A\subset\mathbb N:\mu(A)=t\}\vert$. We study the problem for which sets $R\subset\{0,1,2,\dots,\omega,\mathfrak{c}\}$ there is a measure $\mu$ such that $R$ is the range of $f_\mu$. We are also interested in the set-theoretic and topological properties of the set of $\mu$-values which are obtained uniquely.