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