Cardinal functions of purely atomic measures

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 $f_{\mu }(t)=|\{A\subset \mathbb{N}:\mu (A)=t\}|$. 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.