Radon–Nikodým Theorems for Finitely Additive Multimeasures

Piazza, L. Di; Porcello, Giovanni

doi:10.4171/zaa/1545

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…Proof. In the statement of [17,Proposition 4.4] the hypothesis V σ(x * ,Φ) ≪ λ was substituted by the stronger condition V Φ ≪ λ, but in the proof the V σ(x * ,Φ) ≪ λ is the condition used.…”

Section: Henstock and Mcshane Integrability Of Cwk(x)-valued Multifun...mentioning

confidence: 99%

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

Candeloro

Piazza

Musiał

et al. 2017

Annali di Matematica

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The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems (Theorem 3.2, Theorem 4.2 and Theorem 5.3). As applications of such decompositions, we deduce characterizations of Henstock (Theorem 3.3) and H (Theorem 4.3) integrable multifunctions, together with an extension of a well-known theorem of Fremlin [22, Theorem 8].

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Section: Henstock and Mcshane Integrability Of Cwk(x)-valued Multifun...mentioning

confidence: 99%

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

Candeloro

Piazza

Musiał

et al. 2017

Annali di Matematica

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show abstract

“…Similar problems were studied afterwards, e.g. in [30,35] as an extension of [21,31], later in [4,5], and also recently deeply examined in [11,15] both in the countably and the finitely additive case using different notions of integrals. Here we will undertake a similar investigation and we will consider fuzzy multisubmeasures defined on an algebra and taking convex compact values in an arbitrary Banach space X .…”

Section: Introductionmentioning

confidence: 93%

A multivalued version of the Radon-Nikodym theorem, via the single-valued Gould integral

Candeloro¹,

Croitoru²,

Gavriluţ³

et al. 2015

Preprint

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In this paper we consider a Gould-type integral of real functions with respect to a compact and convex valued non necessarily additive measure. In particular we will introduce the concept of integrable multimeasure and, thanks to this notion, we will establish an exact Radon-Nikodým theorem relative to a fuzzy multisubmeasure which is new also in the finite dimensional case. Some results concerning the Gould integral are also obtained.

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“…For the definition of the variational measure V Φ associated to a finitely additive interval measure Φ : I → R we refer the reader to [5,22]. In particular, we recall that the variational measure V φ of the primitive φ of a variationally Henstock integrable mapping is a (possibly unbounded) λcontinuous measure on L: see [37, Proposition 3.3.1].…”

Section: Preliminary Factsmentioning

confidence: 99%

Some new results on integration for multifunction

Candeloro

Piazza

Musiał³

et al. 2018

Ricerche mat

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It has been proven in [20] and [18] that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable (Theorem 3.4).keywords Multifunction, set-valued Pettis integral, set-valued variationally Henstock and Birkhoff integrals, selection M.S.

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Radon–Nikodým Theorems for Finitely Additive Multimeasures

Abstract: Abstract. In this paper we deal with interval multimeasures. We show some Radon-Nikodým theorems for such multimeasures using multivalued Henstock or Henstock-Kurzweil-Pettis derivatives. We do not use the separability assumption in the results.

Cited by 7 publications

References 9 publications

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

A multivalued version of the Radon-Nikodym theorem, via the single-valued Gould integral

Some new results on integration for multifunction

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