Measurable selectors and set-valued Pettis integral in non-separable Banach spaces |
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Authors: | B Cascales |
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Institution: | a Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain b Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine c Departamento de Análisis Matemático, Universidad de Valencia, Avda. Doctor Moliner 50, 46100 Burjassot (Valencia), Spain d Departamento de Matematica Aplicada, Facultad de Informatica, Universidad de Murcia, 30100 Espinardo (Murcia), Spain |
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Abstract: | Kuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F:Ω→cwk(X) defined in a complete finite measure space (Ω,Σ,μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A∈Σ the Pettis integral coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F. As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F:Ω→cwk(X) admits scalarly measurable selectors; the latter is also proved when (X∗,w∗) is angelic and has density character at most ω1. In each of these two situations the Pettis integrability of a multi-function F:Ω→cwk(X) is equivalent to the uniform integrability of the family . Results about norm-Borel measurable selectors for multi-functions satisfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained. |
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Keywords: | Pettis integral Multi-function Measurable selector Multi-measure |
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