Abstract
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.
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Communicated by P. De Lucia.
Dedicated to Prof. Hans Weber on the occasion of his 70th birthday with deep esteem.
This research was supported by the Grant Prot. No. U2015/001379 of GNAMPA—INDAM (Italy); by University of Perugia—Dept. of Mathematics and Computer Sciences—Grant No. 2010.011.0403 and by University of Palermo (Italy).
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Candeloro, D., Di Piazza, L., Musiał, K. et al. Some new results on integration for multifunction. Ricerche mat 67, 361–372 (2018). https://doi.org/10.1007/s11587-018-0376-x
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DOI: https://doi.org/10.1007/s11587-018-0376-x
Keywords
- Multifunction
- Set-valued Pettis integral
- Set-valued variationally Henstock and Birkhoff integrals
- Selection