Multifunctions determined by integrable functions

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i

Stieltjes-Type Integrals for Metric Semigroup-Valued Functions Defined on Unbounded Intervals

Abstract We introduce the GH k integral for functions defined on (possibly) unbounded subintervals of the extended real line and with values in metric semigroups. Basic properties and convergence theorems for this integral are deduced

Lie for me: How empathy, alexithymia and emotional intelligence influence the ability to conform facial expression to a prosocial untrue verbal message

The aim of this work was to investigate the association between emotional variables (empathy, emotional intelligence, alexithymia) and lying skills. The hypothesis was that a higher emotional competence was associated to a better ability to lie. In an experimental setting, thirty-four participants were videotaped in two separate session: The first in which they were telling the truth and the second in which, motivated by a social and empathetical intention, they were lying about some emotionally arousing images they viewed. Moreover, all participant filled three self-report questionnaires: The 20-Item Toronto Alexithymia Scale, The Interpersonal Reactivity Index and The Trait Emotional Intelligence Questionnaire Short Form. The coding of facial responses was carried out using the Facial Action Coding System. The findings suggested that emotional competence is associated with the ability to lie and to conform facial expression to a prosocial untrue statement. Conversely, individuals with higher levels of alexithymia and lower emotional ability manifested more marker of discomfort as facial manipulators in lying, other than inconsistencies in verbal/nonverbal messages even telling the truth, probably due to their difficulties to identify and express emotions

Some new results on integration for multifunction

It has been proven in Di Piazza and Musiał (Set Valued Anal 13:167–179, 2005, Vector measures, integration and related topics, Birkhauser Verlag, Basel, vol 201, pp 171–182, 2010) 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). Moreover, in case of strongly measurable (multi)functions, a characterization of the Birkhoff integrability is given using a kind of Birkhoff strong property

A Fubini Theorem in Riesz spaces for the Kurzweil-Henstock Integral

A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the “extended” real plane

Convergence for varying measures

Some limit theorems of the type ∫Ωfndmn→∫Ωfdm are presented for scalar, (vector), (multi)-valued sequences of mn-integrable functions fn. The convergences obtained, in the vector and multivalued settings, are in the weak or in the strong sense

Gauge integrals and selections of weakly compact valued multifunctions

In the paper Henstock, McShane, Birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general Banach space X. In particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered