Convergence of integrals of unbounded real functions in random measures

$\sigma$-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 $\int f d\mu_n\stackrel{\mathsf{P}}{\longrightarrow}\int f d\mu$, $n\to\infty$, is proved under conditions similar to uniform integrability. An analogue of the Valle–Poussin theorem is established. A criterion is given for the relation $\int f_ng d\mu\stackrel{\mathsf{P}}{\longrightarrow}\int g d\eta$, $n\to\infty$, to hold for all bounded $g$.