Measurability and semi-continuity of multifunctions

Integration of multifunctions with closed convex values in arbitrary Banach spaces

“…An important key ingredient in such a decomposition is the existence of selections "integrable in the same sense" as the corresponding multifunction. The existence of scalarly measurable selections of arbitrary weakly compact valued scalarly measurable multifunctions has been showed by Cascales, Kadets and Rodriguez in [11].…”

Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

“…Let G : Ω → F (X) be any scalarly measurable fuzzy mapping that is dominated by Γ . Then, for each r ∈ (0, 1], the multifunction G r has scalarly measurable selections (see [9]). It follows by the hypothesis that each scalarly measurable selection of G r is Pettis integrable.…”

“…control theory, optimization or mathematical economics), a wide theory was developed for set-valued integrability in separable Banach spaces (see [1], [6], [18] and reference inside, [12,13], [15], [20]). Recently several authors studied set-valued integrability without separability assumptions in Banach spaces (see [2], [3], [5] [7,9], [10], [14] and [22]). It is well known the useful role played by fuzzy mappings in the theory of fuzzy sets and their applications.…”

Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures

“…The choice to deal with these types of integration is motivated by the fact that Bochner integrability of selections is a strong condition; moreover selection theorems for the Auman-Bochner integral are stated in the separable context. In order to overcome this problem contributions have been given also in [16,17,21,22].…”

“…A very important tool for the study of set-valued integration is the Kuratowski and Ryll-Nardzewski's theorem which guarantees the existence of measurable selectors, though it has the handicap of the requirement of separability for the range space. Recently this result was extended to the non separable case in [16,17,24] for other kinds of multivalued integrals or for the ck(X)-valued case, while here a selection theorem is given for cwk(X)-valued, H-integrable multifunctions.…”