On the representation of sets in finite measure spaces (Q1917664)

Let \((X,M,\mu)\) be an atomless measure space, \(I\) an index set, \(\{A_i\}_{i\in I}\subset M\) a family of sets with finite measure, and \(\{\alpha_i\}_{i\in I}\). \(\{A_i\}_{i\in I}\) is \(\alpha\)-\textit{representable} if there is a family \(\{E_i\}_{i\in I}\subset M\) such that \(E_i\subset A_i\) and \(\mu(E_i)=\alpha_i\) for each \(i\in I\), and \(\mu(E_i\cap E_j)=0\) whenever \(i\neq j\). \textit{B. Bollobás} and \textit{N. Th. Varopoulos} [Math. Proc. Camb. Philos. Soc. 78, 323-325 (1975; Zbl 0304.28001)] proved that \(\{A_i\}_{i\in I}\) is \(\alpha\)-representable iff \(\mu(\bigcup_{i\in J}A_i)\geq\sum_{i\in J}\mu(A_i)\) for every finite set \(J\subset I\). In the paper under review, the author gives a simple proof of this result in the case \(\mu(X)<\infty\).