Convergence of integrals with respect to \(L_ 0\)-valued measures (Q1326898)

Let \(X\) be a set, \({\mathfrak B}\) a \(\sigma\)-algebra of subsets of \(X\), and let \((\Omega, {\mathcal F},P)\) be a probability space. We shall consider space \(L_ 0 (\Omega, {\mathcal F} ,P)\) (further denoted by \(L_ 0)\), consisting of real \({\mathcal F}\)-measurable functions with the topology of convergence with respect to measure \(P\). An \(L_ 0\)-valued measure is a function \(\mu:{\mathfrak B} \to L_ 0\), such that \(\mu (A_ n) @>P>>0\) for \(A_ n \downarrow \emptyset\) and for \(A \cap B=\emptyset\), \(\mu (A \cup B)=\mu (A)+\mu (B)\) \(P\)-a.e. The aim of this paper is to establish the connection between the convergence of values of \(L_ 0\)-valued measures and the convergence of integrals with respect to these measures. We have the following result: Let \(\mu_ n\), \(n\geq 1\), and \(\mu\) be \(L_ 0\)- valued measures. Then \(\forall f \in B(X)\) \(\lim_{n \to \infty} \int fd \mu_ n=\int fd \mu\), if and only if the following two conditions hold: 1) \(\forall A \in {\mathfrak B}\) \(\lim_{n \to \infty} \mu_ n (A)=\mu (A)\); 2) there exists a sequence \(\{c_ n, n \geq 1\} \subset \mathbb{R}\) such that \(\forall n\) \(0<| c_ n | \leq | c_{n-1} |/2\) and \(\forall \{k_ n, n \geq 1\} \subset N\) \(\forall A \in {\mathfrak B}\) the series \(\sum^ \infty_{n=1} c_ n\mu_{k_ n} (A)\) converges in \(L_ 0\).