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.

中文翻译：

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纯原子度量的基函数

让 $\mu$ 是一个纯粹的原子度量。通过 $f_\mu:[0,\infty)\to\{0,1,2,\dots,\omega,\mathfrak{c}\}$ 我们表示它的基函数 $f_{\mu}(t) =\vert\{A\subset\mathbb N:\mu(A)=t\}\vert$. 我们研究了对于集合 $R\subset\{0,1,2,\dots,\omega,\mathfrak{c}\}$ 的问题，有一个测度 $\mu$ 使得 $R$ 是$f_\mu$。我们还对唯一获得的 $\mu$ 值集的集合论和拓扑性质感兴趣。