We prove an analogue of Topsøe's criterion for relative compactness of a family of probability measures which are regular with respect to a family sets. We consider measures whose values are compact convex sets in a locally convex linear topological space.

We introduced the notion of $(𝐗,dist,𝒱)$-boundedness of a filtered family of operators in the Musielak-Orlicz sequence space $X_{\varphi }$ of multifunctions. This notion is used to get the convergence theorems for the families of $𝐗$-linear operators, $𝐗$-dist-sublinear operators and $𝐗$-dist-convex operators. Also, we prove that $X_{\varphi }$ is complete.

We introduce the spaces $M_{Y,\varphi }^1$, $M_{Y,\varphi }^{o,n}$, ${\tilde{M}}_{Y,\varphi }^o$ and $M_{Y,𝐝,\varphi }^o$ of multifunctions. We prove that the spaces $M_{Y,\varphi }^1$ and $M_{Y,𝐝,\varphi }^o$ are complete. Also, we get some convergence theorems.