Measurable selectors and set-valued Pettis integral in non-separable Banach spaces

Abstract

Kuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multifunctions 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 A F dμ 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 ω 1 . 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 {sup x * (F (·)): x * ∈ B X * } ⊂ R Ω . Results about norm-Borel measurable selectors for multi-functions sat-✩ B. Cascales and J. Rodríguez were supported by MEC and FEDER (project MTM2005-08379) and Fundación Séneca (project 00690/PI/04). J. Rodríguez was also supported by the "Juan de la Cierva" Programme (MEC and FSE).isfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained.

Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

“…If Γ is a Pettis integrable fuzzy mapping on Ω, for each r ∈ (0, 1] the multifunction Γ r is Pettis integrable. Therefore S( Γ 1 ) = ∅ (see [8] or [22]).…”

Gauge integrals and selections of weakly compact valued multifunctions

“…[11,Theorem 4.1]) that is λ-continuous. As also observed in [11, section 3], this means that the embedded measure i(J G ) is a countably additive measure with values in l ∞ (B(X * )).…”

A Note on Comparison Between Birkhoff and McShane-Type Integrals for Multifunctions

“…So j • F is Pettis integrable, and then, by [9,Proposition 4.4], F is Pettis integrable and for all A ∈ Σ and x * ∈ B X * we get:…”

“…In [5] Boccuto and Sambucini and in [10,11,8,9] Cascales, Kadets and Rodríguez introduced the McShane and the Birkhoff multivalued integrals respectively, while in [13,14] Di Piazza and Musia l introduced the Kurzweill-Henstock one. Since these kinds of integration lie strictly between Bochner and Pettis integrability (both in the single-valued and in multivalued cases) it is natural to study the possible relationships between the Birkhoff and McShane integrals and with the other multivalued integrals, in particular with the Pettis and Aumann Pettis studied also in [3,4,15].…”