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.

For multifunctions $F:T\times X\to 2^Y$, measurable in the first variable and semicontinuous in the second one, a relation is established between being product measurable and being superpositionally measurable.

The purpose of this paper is the investigation of the necessary and sufficient conditions under which a given multifunctions admits a cliquish and measurable selection. Our investigation also covers the search for quasicontinuous selections for multifunctions which are continuous with respect to the generalized notion of the semi-quasicontinuity.

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $\Gamma :X\to 2^{\Phi }$ be a cyclic ${\Phi }^{\gamma (·,·)}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the ${\Phi }^{\gamma (·,·)}$-subdifferential of f, $\Gamma \left(x\right)\subset {\partial }_{\Phi }^{\gamma (·,·)}{f|}_x$.