Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes }}_{\tau }Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes }}_{\tau }Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with ${\tau }_p$ the associated $L^p$-tensor product...

With an additive function φ from a Boolean ring A into a normed space two positive functions on A, called semivariations of φ, are associated. We characterize those functions as submeasures with some additional properties in the general case as well as in the cases where φ is bounded or exhaustive.

Let $X$ be a locally convex space, $m:\Sigma \to X$ be a vector measure defined on a $\sigma$-algebra $\Sigma$, and $L^1\left(m\right)$ be the associated (locally convex) space of $m$-integrable functions. Let $\Sigma \left(m\right)$ denote $\{{\chi }_{{}_E};E\in \Sigma \}$, equipped with the relative topology from $L^1\left(m\right)$. For a subalgebra $𝒜\subseteq \Sigma$, let ${𝒜}_{\sigma }$ denote the generated $\sigma$-algebra and ${\overline{𝒜}}_s$ denote the sequential closure of $\chi \left(𝒜\right)=\{{\chi }_{{}_E};E\in 𝒜\}$ in $L^1\left(m\right)$. Sets of the form ${\overline{𝒜}}_s$ arise in criteria determining separability of $L^1\left(m\right)$; see [6]. We consider some natural questions concerning ${\overline{𝒜}}_s$ and, in particular, its relation to $\chi \left({𝒜}_{\sigma }\right)$. It is shown that...

The extension theorem of bounded, weakly compact, convex set valued and weakly countably additive measures is established through a discussion of convexity, compactness and existence of selection of the set valued measures; meanwhile, a characterization is obtained for continuous, weakly compact and convex set valued measures which can be represented by Pettis-Aumann-type integral.

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...

We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_H{X}_t=F(t,X_t)P.1$, t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space $K_c\left(E\right)$, of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

We make some comments on the problem of how the Henstock-Kurzweil integral extends the McShane integral for vector-valued functions from the descriptive point of view.

A necessary condition is given for the existence of the tensor product of certain measures valued in locally convex spaces.