We investigate the problem of approximation of measurable multifunctions by monotone sequences of measurable simple ones. Our main tool is the Marczewski function, i.e., the characteristic function of a sequence of sets.

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

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.

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega \times X\to 2^Y$. It is shown that if $N_F$ maps a modular space $(N\left(L(\Omega ,\Sigma ,\mu ;X)\right),{\varrho }_{N,\mu })$ into subsets of a modular space $(M\left(L(\Omega ,\Sigma ,\mu ;Y)\right),{\varrho }_{M,\mu })$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in N_F\left(K\right)<\infty$.

In this paper we prove a representation result for essentially bounded multivalued martingales with nonempty closed convex and bounded values in a real separable Banach space. Then we turn our attention to the interplay between multimeasures and multivalued Riesz representations. Finally, we give the multivalued Radon-Nikodym property.

The purpose of this paper is to introduce a definition of cliquishness for multifunctions and to study the search for cliquish, quasi-continuous and Baire measurable selections of compact valued multifunctions. A correction as well as a generalization of the results of [5] are given.

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $${ℒ}^1\left[\sum ,cbf\left(X\right)\right]$$ of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $${ℒ}^1\left[\sum ,cbf\left(X\right)\right]$$ , and to derive necessary...

It is shown that product weakly measurable lower weak semi-Carathéodory multifunction is superpositionally measurable.