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Some new results on integration for multifunction

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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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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, 06123, Perugia, Italy

    Domenico Candeloro & Anna Rita Sambucini

  2. Department of Mathematics, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Luisa Di Piazza

  3. Institut of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Kazimierz Musiał

Authors
  1. Domenico Candeloro
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  2. Luisa Di Piazza
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  3. Kazimierz Musiał
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  4. Anna Rita Sambucini
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Corresponding author

Correspondence to Luisa Di Piazza.

Additional information

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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