Mathematica Eterna
Open Access

Abstract

Passage to the limit in R fndµn

Andriy Yurachkivsky

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 dνn(s) R fn(s, x)ψn(s, dx) is proved and applied to evolution equations for measures.