Passage to the limit in R f n d μ n

Let (X,X) be a measurable space, μ1, μ2 . . . ; μ be signed measures on X and f1, f2 . . . ; f be X-measurable functions on X. Several sets of sufficient conditions for R fndμn → R fdμ and R fndμn−R fdμn → 0 are found. Two statements do not contain topological assumptions and are generalizations of the dominated convergence theorem; others concern topological spaces. Furthermore, a theorem about passage to the limit in R dn(s) R fn(s, x) n(s, dx) is proved and applied to evolution equations for measures.