Short Communications: Convergence of Integrals of UnboundedReal Functions in Random Measures

Abstract

$\sm$-additive random measures and integrals with respect to them of real valued functions are considered in the most general setting. The statement of convergence of $\int f\,d\mu_n\tlp\int f\,d\mu$, $\ny$, is proved under conditions similar to uniform integrability. An analogue of the Valle--Poussin theorem is established. A criterion is given for the relation $\int f_ng d\mu\tlp\int g\,d\eta$, $\ny$, to hold for all bounded g.