Multi-integrals of finite variation
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Multi-integrals of finite variation Domenico Candeloro1 Anna Rita Sambucini1
· Luisa Di Piazza2
· Kazimierz Musiał3
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This work concludes a research cycle, but not the friendship that has tied us. You left us, dear Mimmo, too soon. The disease has won, but your memories will always be with us. Received: 2 December 2019 / Accepted: 14 January 2020 © Unione Matematica Italiana 2020
Abstract The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results. Keywords Finite interval variation · Multivalued integral · Decomposition of multifunctions Mathematics Subject Classification 28B20 · 26E25 · 26A39 · 28B05 · 46G10 · 54C60 · 54C65
1 Introduction In [23] was proved that a Banach space valued function is McShane integrable if and only if it is Pettis and Henstock integrable. That result has been then generalized to compact valued multifunctions Γ (see [20]), weakly compact valued multifunctions (see [6]) and bounded valued multifunctions (see [8]). Di Piazza and Marraffa [16] presented an example of a Pettis and variationally Henstock integrable function that is not variationally McShane integrable (= Bochner integrable in virtue of [18, Lemma 2]). It turns out that Fremlin’s theorem can be formulated for variational integrals if and only if the variation of the integral is finite in the following sense: sup Γ : {I1 , . . . , In } is a finite partition of [0, 1] < +∞. i
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This research was partially supported by Grant “Metodi di analisi reale per l’approssimazione attraverso operatori discreti e applicazioni” (2019) of GNAMPA – INDAM (Italy), by University of Perugia – Fondo Ricerca di Base 2019 “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, by University of Palermo – Fondo Ricerca di Base 2019 and by Progetto Fondazione Cassa di Risparmio cod. nr. 2018.0419.021 (title: Metodi e Processi di Intelligenza artificiale per lo sviluppo di una banca di immagini mediche per fini diagnostici (B.I.M.)).
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Luisa Di Piazza [email protected]
Extended author information available on the last page of the article
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Finally, in the last section, using DL or Db conditions we are able to prove that the scalar integrability of a multifunction can be obtained as a translation of the Pettis integrability (Theorem 4.1), while its Henstock integrability under DL condition is obtained using Birkhoff integrability (Theorem 4.3), both results with integrals of finite variation. This article is the last in which Domenico Candeloro was able to cooperate and to give his personal contribution, always precious, and we want to dedicate it to him, in his memory.
2 Preliminaria Throughout X is a Banach space with norm · and its dual X ∗ . The closed unit ball of X is denoted by B X . The symbol c(X ) denotes the collection of all nonempty closed convex subsets of X and cb(X ), cwk(X ) and ck(X ) denote respectively the family of all bounded, weakly compact and compact members of c(X ). Fo
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