6533b824fe1ef96bd12809a2
RESEARCH PRODUCT
Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values
Kazimierz MusiałLuisa Di Piazzasubject
Pettis integralDiscrete mathematicsMathematics::Functional AnalysisPure mathematicsIntegrable systemGeneral MathematicsMultifunction McShane integral Henstock integral Pettis integral Henstock--Kurzweil--Pettis integral selectionMathematics::Classical Analysis and ODEsBanach spaceRegular polygonFunction (mathematics)Separable spaceSettore MAT/05 - Analisi MatematicaLocally integrable functionMathematicsdescription
Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial (Monatsh Math 148:119–126, 2006) proved that if \(X\) is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of \(X\) is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1).
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-12-24 | Monatshefte für Mathematik |