Extensions of the optimal partition problem for a measure space
type: Item
instance of: scholarly article
title: Extensions of the optimal partition problem for a measure space
zbMATH Open document ID: 0876.28003
DOI: 10.1007/BF01210438
author: É. M. Nazarov
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 525733
zbMATH Keywords: optimal partition problem
zbMATH Keywords: extremal problems
MaRDI profile type: Publication
cites work: Optimal constrained selection of a measurable subset
cites work: Q3663565
cites work: Q3786676
full work available at URL: https://doi.org/10.1007/bf01210438
OpenAlex ID: W2096419800
description: scientific article; zbMATH DE number 525733