New set-valued integral in a Banach space (Q492509)

In this paper \((\Omega, \mathcal{A}, \mu)\) is a complete finite measure space, \(( \mathcal{X}, \| . \|)\) a separable Banach space with \( \mathcal{X}^{*}\) its dual, \(\mathcal{P}_{wkc}(\mathcal{X})\) all convex weakly compact subsets of \(\mathcal{X}\), and \(\mathcal{M}: \mathcal{A} \to \mathcal{P}_{wkc}(\mathcal{X})\) a set-valued measure (\textit{N. S. Papageorgiou} calls them ``strong multi measures'' in his paper [Trans. Am. Math. Soc. 304, 245--265 (1987; Zbl 0634.28004)]); \(\mathcal{M}\) is a set-valued measure iff \(\sigma(x^{*}, \mathcal{M}(.))\) is a scalarly valued measure for all \(x^{*}\). A vector-valued measure \(m: \mathcal{A} \to \mathcal{X}\) is said to be a selection of the set-valued measure \(\mathcal{M}\) if \(m(A) \in \mathcal{M}(A)\) for all \(A \in \mathcal{A}\). For \(C \in \mathcal{P}_{wkc}(\mathcal{X})\), set \(|C| = \sup( \{\|x \|: x \in C \})\). If the total variation \(|\mathcal{M}|\) of \(\mathcal{M}\) is finite, then \(\mathcal{M}\) is said to be of bounded variation. For a set-valued measure \(\mathcal{M}\), a function \(f: \Omega \to \mathbb R\), \(f \; \mathcal{A}\)-measurable and an element of \(L^{1}(\Omega, \mathbb R, |\mathcal{M}|)\) is said to be \((KL) \; \mathcal{M}\)-integrable if (i) \(f\) is \(\sigma(x^{*}, \mathcal{M}(.))\)-integrable for each \(x^{*} \in \mathcal{X}^{*}\) and (ii) for each \(A \in \mathcal{A}\), there is a \(W_{A} \in \mathcal{P}_{wkc}(\mathcal{X})\) satisfying \( \sigma(x^{*}, W_{A})= \int_{A} f(\omega) d \sigma(x^{*}, \mathcal{M}(\omega))\). This paper deals with proving several basic properties and convergence theorems of \((KL)\) integration.