The moment problem in \(K_ \sigma\)-space
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 functions in just a \(K_\sigma\)-space and the space \({\mathcal P} (\mathbb{R})\) of polynomials can not be dense in \(L_1 (\mu)\), another technique is needed for solving this moment problem.
scientific article; zbMATH DE number 825560