The moment problem in \(K_ \sigma\)-space
type: Item
instance of: scholarly article
title: The moment problem in \(K_ \sigma\)-space
zbMATH Open document ID: 0834.44007
DOI: 10.1007/BF00970955
published in: Siberian Mathematical Journal
publication date: 1996-01-15T00:00:00Z
review text: We solve Hamburger's moment problem for a sequence \(\{s_k \}^\infty_{k =0}\) of vectors in a \(K_\sigma\)-space \(F\). The analogous problem was considered by \textit{S. A. Malyugin} [(*) Optimizatsiya 48(65), 124-141 (1990)]\ in the case when \(F\) is a \(K\)-space and, in its solution, the Kantorovich theorem on extension of a positive operator was used which requires order completeness for the arrival space of the operator. The intention to eliminate the order completeness is prompted by the following reasons. First, the analogous Hausdorff moment problem (the case of a bounded interval), as is easily seen, has a solution in an arbitrary \(K_\sigma\)-space \(F\). Second, it is desirable to have a solution to Hamburger's moment problem in the following statement: Suppose that a positive sequence of measurable functions \(s_n: \Omega\to \mathbb{R}\) \((n\geq 0)\) is given in a measure space \(\langle \Omega,{\mathcal B}, \nu\rangle\). It is requested to find a mapping \(\mu: {\mathcal B} (\mathbb{R})\times \Omega\to \mathbb{R}^+\) possessing the following properties: 1) for every \(\omega\in \Omega\), the function \(\mu (\cdot, \omega)\) presents a Borel measure and \[ s_n (\omega)= \int_\mathbb{R} u^n \mu(du, \omega) \qquad (n= 0, 1, 2, \dots), \] 2) for every Borel set \(A\subseteq \mathbb{R}\), the function \(\mu (A, \cdot)\) is \(\mathcal B\)-measurable. If both properties 1 are valid for \(\nu\)-almost all \(\omega\in \Omega\), the mapping \(\mu\) is usually called a random measure. Since the scalar Hamburger problem can have a nonunique solution, the question in essence reads how should one choose a solution to the scalar problem for each \(\omega\in \Omega\) so that the solutions obtained form a solution to the vector problem measurable in \(\omega\)? Under the requirement that conditions 1 and 2 hold only to within sets of \(\nu\)-measure zero the problem thus specified was partially solved in (*). Otherwise, since the space of \({\mathcal B}\)-measurable …
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 825560
zbMATH Keywords: \(K\)-sigma-space
zbMATH Keywords: Hamburger's moment problem
zbMATH Keywords: Hausdorff's moment problem
zbMATH Keywords: random measure
author: Sergey Artem'evich Malyugin
MaRDI profile type: Publication
cites work: Q5795278
cites work: Q4033276
cites work: Q3361379
cites work: On a generalization of the classical moment problem
cites work: Stone-Algebra-Valued Measures and Integrals
cites work: Q4000054
cites work: Q5518036
cites work: Q5737276
cites work: Quasi-Radon measures
full work available at URL: https://doi.org/10.1007/bf00970955
OpenAlex ID: W2067679683
description: scientific article; zbMATH DE number 825560