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.