Integration with respect to general random measures in the Riemann sense (Q2737031)

Let \(X\) be an arbitrary precompact metric space, let \(\mathcal B\) be a Borel \(\sigma\)-algebra of subsets of \(X,\) and let \((\Omega,{\mathcal F},P)\) be a probability space. By definition, a random measure \(\mu\) is a family of random variables \(\{\mu(A); A\in \mathcal B\}\) such that \(\mu(A_{n})@> P >> 0\) as \(A_{n}\to\emptyset\), and \(\mu(A\cup B)=\mu(A)+\mu(B)\) a.s. for \(A\cap B=\emptyset.\) There are no restrictions of nonnegativity and existence of the moments for the measure \(\mu.\) In this sense it means that \(\mu\) is a general random measure. The integrals of the form \(\int_{A}gd\mu\), where \(g\) is a real measurable function on \(X,\) \(\mu\) is a random measure, \(A\in\mathcal B\) were introduced by \textit{V. N. Radchenko} [``Integrals of general stochastic measures'' (1999; Zbl 0953.60035)]. The aim of this paper is to study the integrals of the form \(\int_{A}gd\mu\) with random function \(g(x,\omega)=\sum_{k=1}^{\infty}\xi_{k}(\omega)g_{k}(x),\) where \(\xi_{k}\) are random variables on \((\Omega,{\mathcal F},P),\) \(g_{k}:X\to R\) are measurable functions with \(|g_{k}(x)|\leq 1.\) The conditions of integrability of the random integrands over the random measure in the Riemann sense are proposed.