Extensions of the optimal partition problem for a measure space
Let \((S,\Sigma,\mu)\) be a space with a measure, where \(\Sigma\) is a \(\sigma\)-algebra over the set \(S\) \((S\neq\emptyset)\), and \(\mu\) is a finite positive countably additive measure on \((S,\Sigma)\). We use \(D(S,\Sigma)\) to denote the family of all possible finite disjoint partitions of \(S\) by elements of \(\Sigma\). For a given set-theoretic function \(f_0:\Sigma\to\mathbb{R}\) and a nonempty subfamily of partitions \(D_0,D_0\subset D(S,\Sigma)\), we consider the extremal problem \[ \sum_{E\in\Delta} f_0(E)\mu(E)\to\inf,\quad\Delta\in D_0,\tag{1} \] whose value we denote by \(c_0\). The fundamental difficulty presented by the indicated problem is that there are no linear or topological structures on the set \(D(S,\Sigma)\). In the present paper, we present a procedure for imbedding the set of admissible solutions to problem (1) in certain linear normed spaces, and subsequently compactifying the set of generalized solutions in the weak topology. Various examples of extremal problems in which the quality criterion is determined by a set-theoretic function on a space with a measure were also considered by \textit{R. J. T. Morris} [J. Math. Anal. Appl. 70, 546-562 (1979; Zbl 0417.49032)] and \textit{V. A. Kaminskij} [Mat. Zametki 47, No. 1, 81-91 (1990; Zbl 0706.90097)].
scientific article; zbMATH DE number 525733